{ "CO1007-syllabus-info-0000": { "id": "CO1007-syllabus-info-0000", "text": "Course: Discrete Structures for Computing (ID: CO1007)\nCredits: 4\nSemester: 20221", "metadata": { "doc_type": "syllabus", "course_id": "CO1007", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_001.png", "page_index": 0, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:01:25+07:00" } }, "CO1007-syllabus-assessments-0000": { "id": "CO1007-syllabus-assessments-0000", "text": "Lectures: 45 hours. Evaluation: Midterm Exam (30%), Final Exam (50%),\nTutorial:\nLabs/Practices:\nProjects: 45 hours. Weight: 20%.\nSelf-study: 135 hours.\nOthers:", "metadata": { "doc_type": "syllabus", "course_id": "CO1007", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_001.png", "page_index": 0, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:01:25+07:00" } }, "CO1007-syllabus-raw-0000": { "id": "CO1007-syllabus-raw-0000", "text": "09:44,04/09/2022 DCMH.CO1007_Discrete Structures for Computing HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY BK TP.HCM TRUONG DAI HOC BACH KHOA - DHQG-HCM DCMH.C01007.5.1 Dai Hoc Qu6c Gia TP.HCM Vietnam National University - HCMC Truo'ng Dai Hoc Bach Khoa Ho Chi Minh City University of Technology Khoa Khoa hoc va Ky thuat May tinh Faculty of Computer Science and Engineering DE CUONG HOC PHAN Course Syllabus 1. Thong tin vé hoc phan (Course information) 1.1. Thng tin töng quan (General information) Ten hoc phan: Cau trüc roi rac cho khoa hoc may tinh Course title: Discrete Structures for Computing Ma hoc phan (Course ID): CO1007 - S6 tin chi (Credits): 4 (ETCS: 8 ) Hoc ky ap dung (Applied from semester): 20221 To chúc hoc phan (Course format): Hinh thúrc hoc tap Só tiét/gi S6 tin chi Ghi chü (Teaching/study type) (Hours) (Credits) (Notes) Ly thuyét (LT) 45 (Lectures) Thao luan (ThL)/Thuc hanh tai lóp (TH) 0 (Tutorial) Thi nghiém (TNg)/Thuc tap xuöng (TT) 0 (Labs/Practices) Bai tap l6n (BTL)/D6 an (DA) 45 (Projects) Tu hoc (Self-study) 135 Khac (Others) Tong cöng (Total) 182.5 4 Ty lé danh gia va hinh thúc kiém tra/thi (Evaluation form & ratio) Hinh thuc danh gia Ty le Hinh thurc Thoi gian (Evaluation type) (Ratio) (Format) (Duration) Thao luan (ThL)/Thuc hanh tai lóp (TH) (Tutorial) Thi nghiém (Labs/Practices) Bai tap l6n (BTL)/D6 an (DA) 20% (Projects) Kiém tra 30% Trac nghiém 60 phüt (minutes)", "metadata": { "doc_type": "syllabus", "course_id": "CO1007", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_001.png", "page_index": 0, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:01:25+07:00" } }, "CO1007-syllabus-raw-0001": { "id": "CO1007-syllabus-raw-0001", "text": "Thi nghiém (Labs/Practices) Bai tap l6n (BTL)/D6 an (DA) 20% (Projects) Kiém tra 30% Trac nghiém 60 phüt (minutes) (Midterm Exam) (Multiple choice (MCQ)) Thi 50% Träc nghiém 90 phút (minutes) (Final Exam) (Multiple choice (MCQ)) Tong cöng 100% (Total) 268 Ly Thung Kiet,Phuδng 14,Quan 10,TP.HCM 268 Ly Thuong Kiet St., Ward 14, Dist. 10, Ho Chi Minh City, Vietnam Dién thoai: 028 3864 7256 Phone: 028 3864 7256 www.hcmut.edu.vn www.hcmut.edu.vn 1/8", "metadata": { "doc_type": "syllabus", "course_id": "CO1007", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_001.png", "page_index": 0, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:01:25+07:00" } }, "-syllabus-info-0000": { "id": "-syllabus-info-0000", "text": "Course: Discrete_Structures_for_Computing (ID: )", "metadata": { "doc_type": "syllabus", "course_id": "", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_008.png", "page_index": 7, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:02:37+07:00" } }, "-syllabus-raw-0000": { "id": "-syllabus-raw-0000", "text": "09:44,04/09/2022 DCMH.CO1007 Discrete Structures for Computing HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY BK TP.HCM TRUONG DAI HOC BACH KHOA - DHQG-HCM DCMH.C01007.5.1 Bui Nöi dung (Content) Hoat dong day va hoc (Lecturing) (Session) 11-12 Chuong 6. Cay va thuat toan L.0.2.3 [A.0.1 , A.0.4, A.0.3 ] 6.1. Tinh chat, cay nhi phan o Lec: - Giang ly thuyét - Cho sinh vién lam bai tap trén 6.2. Cac phép duyét trén cay lóp/ online va giai thich 6.3. Cay phu bé nhat trong do thi lién thong c6 trong so (- Lectures on theory - Let students do the exercises in Yeu cau tu hoc d/v sinh vien: 4 gio class/online and explain) Stu: - Sinh vién xem tru6c bai giang tru6c khi lén l6p. - Cau hoi va bai tap tren l6p theo ca nhan/nh6m Chapter 6. Trees and Algorithms - Students preview the lecture before class. 6.1. Properties, binary tree Individual/group class questions and exercises) 6.2. Tree traversals L.0.3.2 [ A.0.1 , A.0.4 , A.0.3 ] 6.3. The smallest covered tree in a weighted connected graph o Lec: - Giang ly thuyét - Cho sinh vién lam bai tap trén Self-study requirements for students: 4 hours) lóp/ online va giai thich - Lectures on theory - Let students do the exercises in class/online and explain) o Stu: - Sinh vién xem tru6c bai giang tru6c khi lén lóp. - Cau hoi va bai tap tren l6p theo cä nhan/nh6m (- Students preview the lecture before class. Individual/group class questions and exercises) L.0.3", "metadata": { "doc_type": "syllabus", "course_id": "", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_008.png", "page_index": 7, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:02:37+07:00" } }, "-syllabus-raw-0001": { "id": "-syllabus-raw-0001", "text": "khi lén lóp. - Cau hoi va bai tap tren l6p theo cä nhan/nh6m (- Students preview the lecture before class. Individual/group class questions and exercises) L.0.3.1 [A.0.1 , A.0.4 , A.0.3 ] Lec: - Giang ly thuyét - Cho sinh vién lam bai tap trén lóp/ online va giäi thich (- Lectures on theory - Let students do the exercises in class/online and explain) Stu: - Sinh vién xem truóc bai giang truóc khi lén lóp. - Cau hoi va bai tap trén l6p theo ca nhan/nh6m - Students preview the lecture before class. Individual/group class questions and exercises) 13-14-15 Chuong 7. Duong di va chu trinh L.0.2.3 [A.0.1 7.1. Dinh nghia Lec: Giäng day va trao dói trén l6p 7.2. Duöng di/chu trinh Hamilton/Euler Lecture and Discussion in class) 7.3. Các giái thuat tim dung di ngan nhát Stu: Bai tap lón 7.4. Ung dung tim dung di ngan nhat trong bai toán döng (Assignment) chay L.0.3.1 [A.0.1 Yéu cau tu hoc d/v sinh vién: 6 gi o Lec: Thuc hanh trén lóp (lab) Chapter 7. The Path and the Cycle o Stu: Thuc hanh 7.1. Define (lab) 7.2. Hamilton/Eulerian path/cycle 7.3. Algorithms to find the shortest path 7.4. The application to find the shortest path in the flow problem Self-study requirements for students: 6 hours 7. Yéu cau khäc vé hoc phan (Other course requirements and expectations) 8. Bién soan va cap nhat dé cuo'ng (Editing information) Dé cuong duoc bién soan vao nam hoc hoc ky (Syllabus edited in year-semester):", "metadata": { "doc_type": "syllabus", "course_id": "", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_008.png", "page_index": 7, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:02:37+07:00" } }, "-syllabus-raw-0002": { "id": "-syllabus-raw-0002", "text": "requirements and expectations) 8. Bién soan va cap nhat dé cuo'ng (Editing information) Dé cuong duoc bién soan vao nam hoc hoc ky (Syllabus edited in year-semester): 20221 Dé cuong duoc chinh sura lan thú (Editing version): DCMH.CO1007.5.1 Nöi dung duoc chinh sira, cap nhat, thay döi lan gan nhat (The latest editing content): -- -- Tp.H6 Chi Minh, ngay 4 thang 9 nam 2022 HCM City, September 4 2022 TRUONG KHOA CHU NHIEM BO MON CB PHU TRACH LAP DE CUONG (Dean) (Head of Department) (Lecturer in-charge) 268 Ly Thuong Kiet,Phuöng 14,Quan 10,TP.HCM 268 Ly Thuong Kiet St., Ward 14, Dist. 10, Ho Chi Minh City, Vietnam Dién thoai: 028 3864 7256 Phone: 028 3864 7256 www.hcmut.edu.vn www.hcmut.edu.vn 8/8", "metadata": { "doc_type": "syllabus", "course_id": "", "source_file": "/mnt/d/Project_and_Assignment/AI_project/data_cvt/CO1007_Discrete_Structures_for _Computing/Syllabus/slide_008.png", "page_index": 7, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-12T20:02:37+07:00" } }, "CO1007-chapter-0-slide-000-0000": { "id": "CO1007-chapter-0-slide-000-0000", "text": "DISCRETE STRUCTURES FOR COMPUTING Tuan Anh Tran CSE - HCMUT", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_001.png", "page_index": 0, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:56:58+07:00" } }, "CO1007-chapter-0-slide-001-0000": { "id": "CO1007-chapter-0-slide-001-0000", "text": "1. Lecturers: (Semester 221) Sakos Nguyen An Khuong, Ph.D Tran Tuan Anh, Ph.D nakhuong@hcmut.edu.vn trtanh@hcmut.edu.vn Nguyen Tien Thinh, Ph.D Mai Xuan Toan, MsC Tran Hong Tai, MsC ntthinh@hcmut.edu.vn mxtoan@hcmut.edu.vn thtai@hcmut.edu.vn", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_002.png", "page_index": 1, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:05+07:00" } }, "CO1007-chapter-0-slide-002-0000": { "id": "CO1007-chapter-0-slide-002-0000", "text": "CONTINUOUS 2. Introduction Why Discrete Mathematics? DISCRETE https://www.youtube.com/watch?v=q4L-wUF3yig", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_003.png", "page_index": 2, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:09+07:00" } }, "CO1007-chapter-0-slide-003-0000": { "id": "CO1007-chapter-0-slide-003-0000", "text": "3. Contents 15 WEEKS & 11 CHAPTERS logic Set Theory FUNCTION f: What is A X B OUTPUT 0 Z IN MATH? C (Easy Concept) one two three 2 3 Graph theory 1 3 2 x five six four 4 5 6 000 5 eight nine PROBABILITY 7 seven 8 9 ten 10 http://e-learning.hcmut.edu.vn/course/view.php?id=67408", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_004.png", "page_index": 3, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:17+07:00" } }, "CO1007-chapter-0-slide-004-0000": { "id": "CO1007-chapter-0-slide-004-0000", "text": "3. Contents Teaching plan Class Week Date Content Kind of teaching Noi dung thi 1 1 2/1-8/1 Chapter 0& 1 - Introduction & Logic Offline 2 2 9/1-15/1 Chapter 2 - Logic (cont) Blended 3 16/1-22/1 4 23/1-29/1 Nghi tét No 3 5 30/1-5/2 Chapter 3-Proof Blended 4 6 6/2-12/2 Overview 1-2-3 & Chapter 4 - Set Offline 5 7 13/2-19/2 Chapter 5 - Function Offline 6 8 20/2-26/2 Chapter 6-Relation Offline Thi giura ki 7 9 27/2-5/3 Chapter 7 - Counting Offline 10 6/3-12/3 11 13/3-19/3 12 20/3-26/3 13 27/3-2/4 Hoc guan su/Thi giua ki No 8 14 3/4-9/4 Chapter 8-Probability Offline 9 15 10/4-16/4 Chapter 8-Probability Offline 10 16 17/4-23/4 Chapter 9-Graph Offline 11 17 24/4-30/4 Chapter 10 - Connectivity Offline 12 18 1/5-7/5 Chapter 10 - Connectivity Offline 13 19 8/5-145 Chapter 11 - Tree Offline 14 20 15/5-21/5 Chapter 11 - Tree Offline 15 21 22/5-28/5 Free Blended Thi Cuói ki", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_005.png", "page_index": 4, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:29+07:00" } }, "CO1007-chapter-0-slide-005-0000": { "id": "CO1007-chapter-0-slide-005-0000", "text": "1. Slides are updated and sent to students. We can print out and use in the class. 2. References [1] Discrete mathematics and applications - Kenneth H. Rosen. [2] Discrete Mathematics and Applications, Kevin Ferland Chapman and Hall/CRC. 4. Resources [3] Graph Theory and Its Applications, Jonathan L. Gross, Jay Yellen & Mark Anderso, Chapman and Hall/CRC [4] The Mathematics of Chip-Firing, Caroline J. Klivans Chapman and Hall/CRC. [5] Our master: GOOGLE 3. Programming languages: Python, Matlab, C++", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_006.png", "page_index": 5, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:34+07:00" } }, "CO1007-chapter-0-slide-006-0000": { "id": "CO1007-chapter-0-slide-006-0000", "text": "Attend every class and ask yourselves Why you are here? What is your goal? Keep in your mind that we will not teach a topic twice. You must review everything you have studied in class Read the textbooks carefully and solve the exercises therein as much as possible 5. Requirements Teamwork and coding skill Respect each other", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_007.png", "page_index": 6, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:38+07:00" } }, "CO1007-chapter-0-slide-007-0000": { "id": "CO1007-chapter-0-slide-007-0000", "text": "1. Midterm Exam (30%) - Multiple choice exam, 60 minutes, closed book - Content: from Logic to Relation 2. F Assignment (20%) - Not decided yet based on your studies (but always 6. Evaluation teamwork): a real problem and need the combination of math and code Duration: 4-5 weeks 3. Final exam (50%) - Multiple choice exam, 80-90 minutes, closed book Content: from Logic to the End 1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_008.png", "page_index": 7, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:43+07:00" } }, "CO1007-chapter-0-slide-008-0000": { "id": "CO1007-chapter-0-slide-008-0000", "text": "Evaluation Given during study = 0.3 *(Midterm + volunteer) + 0.2* (Code + report + present) + 0.5*Final exam Midterm (30% Assignment (30%) Final (50%", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_009.png", "page_index": 8, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:46+07:00" } }, "CO1007-chapter-0-slide-009-0000": { "id": "CO1007-chapter-0-slide-009-0000", "text": "C x C x Q&A X", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_0/slide_010.png", "page_index": 9, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:51+07:00" } }, "CO1007-chapter-1-slide-010-0000": { "id": "CO1007-chapter-1-slide-010-0000", "text": "Logics Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 1 Xuan Toan. Tran Hong Tai Logics BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Propositional Logic Logical Equivalences Exercise Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 1.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_001.png", "page_index": 10, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:54+07:00" } }, "CO1007-chapter-1-slide-011-0000": { "id": "CO1007-chapter-1-slide-011-0000", "text": "Contents Logics Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Propositional Logic Contents Propositional Logic Logical Equivalences Logical Equivalences Exercise Exercise 1.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_002.png", "page_index": 11, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:57:56+07:00" } }, "CO1007-chapter-1-slide-012-0000": { "id": "CO1007-chapter-1-slide-012-0000", "text": "Course outcomes Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Contents Represent and model practical problems with discrete structures L.0.2.1 - Logically describe some problems arising in Computing Propositional Logic L.O.2.2 - Use proving methods: direct, contrapositive, induction Logical Equivalences L.0.2.3 - Explain problem modeling using discrete structures Exercise L.0.3 Understanding of basic probability and random variables L.O.3.1 - Define basic probability theory L.O.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.0.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 1.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_003.png", "page_index": 12, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:01+07:00" } }, "CO1007-chapter-1-slide-013-0000": { "id": "CO1007-chapter-1-slide-013-0000", "text": "Logic Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition (Averroes) BK The tool for distinguishing between the true and the false TP.HCM Contents Definition (Penguin Encyclopedia) Propositional Logic Logical Equivalences The formal systematic study of the principles of valid inference Exercise and correct reasoning Definition (Discrete Mathematics - Rosen) Rules of logic are used to distinguish between valid and invalid mathematical arguments. 1.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_004.png", "page_index": 13, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:05+07:00" } }, "CO1007-chapter-1-slide-014-0000": { "id": "CO1007-chapter-1-slide-014-0000", "text": "Applications in Computer Science Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Design of computer circuits Construction of computer programs Contents Propositional Logic Verification of the correctness of programs Logical Equivalences Constructing proofs automatically Exercise Artificial intelligence Many more... 1.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_005.png", "page_index": 14, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:08+07:00" } }, "CO1007-chapter-1-slide-015-0000": { "id": "CO1007-chapter-1-slide-015-0000", "text": "Propositional Logic Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK Definition TP.HCM A proposition is a declarative sentence that is either true or false but not both. Contents Propositional Logic Examples Logical Equivalences Exercise Hanoi is the capital of Vietnam. New York City is the capital of USA 1+1 = 2 2 + 2 = 3 1.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_006.png", "page_index": 15, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:11+07:00" } }, "CO1007-chapter-1-slide-016-0000": { "id": "CO1007-chapter-1-slide-016-0000", "text": "Examples Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hon Tai Examples (Which of these are propositions?) BK TP.HCM How easy is logic! Read this carefully Contents Propositional Logic H1 building is in Ho Chi Minh City. Logical Equivalences 4> 2 Exercise 2n > 100 The Sun circles the Earth Today is Thursday. : Proposition only when the time is specified 1.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_007.png", "page_index": 16, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:14+07:00" } }, "CO1007-chapter-1-slide-017-0000": { "id": "CO1007-chapter-1-slide-017-0000", "text": "Notations Logics Nguyen An Khuong, Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Propositions are denoted by p, q, : Propositional Logic The truth value (\"chan tri\") is true (T) or false (F) Logical Equiva lences Exercise 1.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_008.png", "page_index": 17, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:16+07:00" } }, "CO1007-chapter-1-slide-018-0000": { "id": "CO1007-chapter-1-slide-018-0000", "text": "Operators Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Negation - \"Phü dinh\": -p Contents Propositional Logic Bäng: Truth Table for Negation Logical Equivalences Exercise p 12 F 1.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_009.png", "page_index": 18, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:18+07:00" } }, "CO1007-chapter-1-slide-019-0000": { "id": "CO1007-chapter-1-slide-019-0000", "text": "Operators Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Conjunction - \"H&i\": p q Disjunction - \"Tuyén\": p V q BK TP.HCM p and q\" \"p or q Contents p p q p q p V q Propositional Logic Logical Equivalences T T T T T T Exercise T F F T F T F T F F T T F F F F F F I'm teaching DM1 and it is We need students who have raining today. experience in Java or C++ Tomorrow, I will eat Pho or Bun bo. 1.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_010.png", "page_index": 19, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:23+07:00" } }, "CO1007-chapter-1-slide-020-0000": { "id": "CO1007-chapter-1-slide-020-0000", "text": "Operators Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Exclusive OR - Tuyén loai: p q Implication - Kéo theo: p -> q TP.HCM p or q (but not both)' \"if p, then q Contents Propositional Logic p p q p q p q Logical Equivalences F Exercise T T T T T T F T T F F F T T F T T F F F F F T If it rains, the pavement will be wet. 1.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_011.png", "page_index": 20, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:28+07:00" } }, "CO1007-chapter-1-slide-021-0000": { "id": "CO1007-chapter-1-slide-021-0000", "text": "More Expressions for Implication p -> q Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK if p, then q TP.HCM p implies q p is sufficient for q Contents Propositional Logic q if p Logical Equivalences p only if q Exercise q unless -p If you get 100% on the final, you will get 10 grade If you feel asleep this afternoon, then 2 + 3 = 5. 1.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_012.png", "page_index": 21, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:30+07:00" } }, "CO1007-chapter-1-slide-022-0000": { "id": "CO1007-chapter-1-slide-022-0000", "text": "Conditional Statements From p -> q Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM q -> p (converse - dao) Contents Propositional Logic -q -> -p (contrapositive - phan dao) Logical Equivalences Prove that only contrapositive have the same truth table with Exercise p-q 1.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_013.png", "page_index": 22, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:33+07:00" } }, "CO1007-chapter-1-slide-023-0000": { "id": "CO1007-chapter-1-slide-023-0000", "text": "Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise BK TP.HCM What are the converse and contrapositive of the following conditional statement Contents \"lf he plays online games too much, his girlfriend leaves him.' Propositional Logic Logical Equivalences Converse: If his girlfriend leaves him, then he plays online Exercise games too much. Contrapositive: If his girlfriend does not leave him, then he does not play online games too much. 1.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_014.png", "page_index": 23, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:38+07:00" } }, "CO1007-chapter-1-slide-024-0000": { "id": "CO1007-chapter-1-slide-024-0000", "text": "Biconditionals Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai p<>q 'p if and only if q' BK TP.HCM p p<>q Contents T T Propositional Logic T Logical Equivalences T F F Exercise F T F F F T \"p is necessary and sufficient for q\". \"if p then q, and conversely\". \"p iff q\". 1.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_015.png", "page_index": 24, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:42+07:00" } }, "CO1007-chapter-1-slide-025-0000": { "id": "CO1007-chapter-1-slide-025-0000", "text": "The order of operators Logics Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK. TP.HCM 1. in the bracket0 Contents 2. negation - Propositional Logic 3.V,, Logical Equivalences Exercise 4. -7 5.! 1.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_016.png", "page_index": 25, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:44+07:00" } }, "CO1007-chapter-1-slide-026-0000": { "id": "CO1007-chapter-1-slide-026-0000", "text": "Translating Natural Sentences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Exercise TP.HCM I will buy a new phone only if I have enough money to buy iPhone 4 or my phone is not working Contents Propositional Logic Logical Equivalences p: I will buy a new phone Exercise q: l have enough money to buy iPhone 4 r: My phone is working p->(qV-r) 1.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_017.png", "page_index": 26, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:47+07:00" } }, "CO1007-chapter-1-slide-027-0000": { "id": "CO1007-chapter-1-slide-027-0000", "text": "Translating Natural Sentences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Exercise Propositional Logic He will not run the red light if he sees the police unless he is too Logical Equivalences risky. Exercise 1.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_018.png", "page_index": 27, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:49+07:00" } }, "CO1007-chapter-1-slide-028-0000": { "id": "CO1007-chapter-1-slide-028-0000", "text": "Construct Truth Table Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise BK TP.HCM Construct the truth table of the compound proposition (pV-q) -(pAq) Contents Propositional Logic Logical Equivalences p q p V -q p q (pV-q) -(pAq) Exercise T T F T T T T F T T F F F T F F F T F F T T F F 1.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_019.png", "page_index": 28, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:58:54+07:00" } }, "CO1007-chapter-1-slide-029-0000": { "id": "CO1007-chapter-1-slide-029-0000", "text": "Exercise - Truth table Logics Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai p -(-q V r) p q r rp q nq V r ip BK T T T F F 1 TP.HCM T T F F F F T T F T F T T T T F F F T T T Contents F 1 T T F T T F T F 1 F F F Propositional Logic F F T T T T T Logical Equivalences F F 1 1 T T Exercise al pAq)--q b (pVr) -(rV-p) (p - q) V(q- p) dl (p V -q) A(p V q) e (p - -q) V(q - p) f -(-p -q) g (p V q) -(p q) h) (p A q) V(r O q 1.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_020.png", "page_index": 29, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:02+07:00" } }, "CO1007-chapter-1-slide-030-0000": { "id": "CO1007-chapter-1-slide-030-0000", "text": "Applications Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM System specifications Contents \"When a user clicked on He/p button, a pop-up will be shown Propositional Logic up\" Logical Equivalences Boolean search Exercise type \"dai hoc bach khoa\"in Google means \"dai AND hoc AND bach AND khoa' 1.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_021.png", "page_index": 30, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:06+07:00" } }, "CO1007-chapter-1-slide-031-0000": { "id": "CO1007-chapter-1-slide-031-0000", "text": "Applications (cont.) Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Logic puzzles - There are two kinds of inhabitants on an island, knights, whc Contents always tell the truth, and their opposites, knaves, who may Propositional Logic lie. You encounter two people A and B. What are A and B if Logical Equivalences A says \"B is a knight\" and B says \"The two of us are Exercise opposite types\"? Bit operations : 101010011 is a bit string of length nine. 1.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_022.png", "page_index": 31, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:09+07:00" } }, "CO1007-chapter-1-slide-032-0000": { "id": "CO1007-chapter-1-slide-032-0000", "text": "Tautology and Contradiction Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition BK TP.HCM A compound proposition that is always true (false) is called a tautology - häng düng (contradiction - hang sai): Contents Tautology: hang düng Propositional Logic Contradiction: mau thu@n Logical Equivalences Exercise Example p V -p (tautology) p -p (contradiction) 1.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_023.png", "page_index": 32, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:12+07:00" } }, "CO1007-chapter-1-slide-033-0000": { "id": "CO1007-chapter-1-slide-033-0000", "text": "Question Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Which of the following is a tautology TP.HCM Hint: Apply truth table. @(pVq)->(pq) Contents Propositional Logic 6)(pAq)->(pVq) Logical Equivalences op-(-q-p) Exercise @p-(p->q) ep-(p-p) @(p-q)-[(p-r)-(q-r) 1.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_024.png", "page_index": 33, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:14+07:00" } }, "CO1007-chapter-1-slide-034-0000": { "id": "CO1007-chapter-1-slide-034-0000", "text": "Proposition? Truth value? Logics Nguyen An Khuong. @ \"Fansipan is the highest mountain in Vietnam.\" Tran Tuan Anh, Mai Xuan Toan. Tran Hong 6 \"Two coprime numbers have the only common divisor of 1.\" Tai c) \"The product of 3 continuous integers is divisible by 3.\" d) \"Stand up!\" e) \"x+1=0\" BK TP.HCM f \"Hexagons have 8 vertices.' g) \"o is a positive number.\" h \"The equation: x2 + 5ε + 6 = 0 has no root.\" Contents \"is 2 a prime number?\" Propositional Logic \"The equation mx2 + 2 - 1 = 0 has a single root if and only if m=-1.\" Logical Equivalences k) \"There is a prime that is even.\" Exercise 0 \"x2+1>0\" n \"When will our class go camping?' n \"Mercury is not a metal.\" 0 '320 > 230.\" p) \"Airplanes are the fastest transport.\" q \"2002 is a leap year.\" r) \"There are infinite prime numbers.' s) \"210 - 1 is divisible by 11.\" t) \"No smoking in public place.\" u) \"All even positive integer is a summation of 2 prime numbers.\" v \"ε is a prime number if it doesn't have any divisor other than 1 and x.' 1.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_025.png", "page_index": 34, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:20+07:00" } }, "CO1007-chapter-1-slide-035-0000": { "id": "CO1007-chapter-1-slide-035-0000", "text": "Logical Equivalences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Definition The compound compositions p and q are called logically equivalent Contents Propositional Logic if p <> q is a tautology, denoted p = q. Logical Equivalences Exercise Example Show that -(p V q) and -p -q are logically equivalent. 1.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_026.png", "page_index": 35, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:23+07:00" } }, "CO1007-chapter-1-slide-036-0000": { "id": "CO1007-chapter-1-slide-036-0000", "text": "Logical Equivalences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK PT p Identity laws TP.HCM p V F p Luat döng nhät Contents p V T T Domination laws Propositional Logic pF F Luat nuót Logical Equivalences p V p Idempotent laws Exercise p pp = Luat lüy däng p -(-p) = p Double negation law Luat phü dinh kép 1.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_027.png", "page_index": 36, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:27+07:00" } }, "CO1007-chapter-1-slide-037-0000": { "id": "CO1007-chapter-1-slide-037-0000", "text": "Logical Equivalences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai p V q q V p Commutative laws BK p q q p Luat giao hoän TP.HCM (p V q) V r p V(q V r) Associative laws Contents (pA q) r p(qr) Luat két hop Propositional Logic pV(qr) Distributive laws Logical Equivalences p V q) A(p V r) Exercise pA(q V r) (pAq)V(pAr) Luat phan phói (p q) p V -q De Morgan's law (p V q) p -q Luat De Morgan pV(pq) p Absorption laws p(pV q) p Luat hüt thu 1.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_028.png", "page_index": 37, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:31+07:00" } }, "CO1007-chapter-1-slide-038-0000": { "id": "CO1007-chapter-1-slide-038-0000", "text": "Logical Equivalences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Equivalence TP.HCM p V -p T p-p F Contents p->q I p V q Propositional Logic (p->q)(p->r) p->(qAr) Logical Equivalences Exercise p-r)A(q-r (pVq)-r (p->q)V(p->r) p-(qV r) (p->r)V(q-r) (pAq)-r p<>q p->q)A(q-p) p<>q (p V q) A(p V -q) 1.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_029.png", "page_index": 38, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:35+07:00" } }, "CO1007-chapter-1-slide-039-0000": { "id": "CO1007-chapter-1-slide-039-0000", "text": "Constructing New Logical Equivalences Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example Show that -(p V (-p q)) and -p -q are logically equivalent by developing a series of logical equivalences. BK TP.HCM Solution Contents Propositional Logic -(p V (p A q)) = -p -(-p q) by the second De Morgan law Logical Equivalences = p A[-(p) v -q] by the first De Morgan law Exercise = p (p V -q) by the double negation law = (-pA p) V(p A-q) by the second distributive law = FV(-p-q) because -p p = F = -p -q by the identity law for F Consequently, -(p V (-p q)) and -p -q are logically equivalent 1.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_030.png", "page_index": 39, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:39+07:00" } }, "CO1007-chapter-1-slide-040-0000": { "id": "CO1007-chapter-1-slide-040-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Negate the following proposition and try to simplify it. Example BK TP.HCM p->(-qr) By using the truth table, we can prove that p -> q and -p V q are Contents logical equivalence. Propositional Logic Negate: -(p -> (-q r) Logical Equivalences =-(-p V (-q Ar)) Exercise =pA-(-q r) =pA(q V-r) @ p A(q V r) A(-p V -q V r) b(pAq)->r @ p V q V (-p A -qA r) @ [[(pAq)Ar] v [(pAr)A-r]]V-q]-s 1.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_031.png", "page_index": 40, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:42+07:00" } }, "CO1007-chapter-1-slide-041-0000": { "id": "CO1007-chapter-1-slide-041-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK Prove the following proposition are logical equivalence TP.HCM Hint: Apply truth table or the series of logical equivalences @ -(p<>q) va-p<>q Contents Propositional Logic 6)(p->q)(p->r) vap->(qr) Logical Equivalences p->r)A(q->r) va(pVq) ->r Exercise (p-q)V(p->r) vap-(qVr) e -p-(q->r) vaq->(pVr) f p>q va(p->q)A(q-p 1.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_032.png", "page_index": 41, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:45+07:00" } }, "CO1007-chapter-1-slide-042-0000": { "id": "CO1007-chapter-1-slide-042-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai The following proposition are logical equivalence? Prove it or give BK TP.HCM an example? @ p(p-q) va pq Contents 6 p-q va -pV(pq) Propositional Logic Logical Equivalences c p->q va -pV-q Exercise @ -p va -(pV q) V(pAq) e[(pH>q) (q<>r)(r<>p)] va [(p->q) (q->r) (r ->p)] @) [(pAq)V(qAr) V(rAp)] va [(pV q) A(q V r)A(rV p)] 1.34", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_033.png", "page_index": 42, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:49+07:00" } }, "CO1007-chapter-1-slide-043-0000": { "id": "CO1007-chapter-1-slide-043-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Determine the truth value and find the contrapositions as well as the contradictions of the following propositions. BK TP.HCM @) \"If ABCD is a rectangle, AB and CD are perpendicular. b) \"If 14 is an odd number, 15 is divisible by 4.\" Contents c) \"Two equal triangles have the same area.\" Propositional Logic \"If the quadratic equation ax2 + bc + c = 0 has a.c < 0, it has root.\" Logical Equivalences \"If two numbers x and y are both divisible by n, (x + y) is also divisible by n.\" Exercise f) \"lf 45 ended with 5, 45 is divisible by 5.\" @) \"lf V2 is an irrational number then /2.V2 is an irrational number.\" h \"If Pythagoras is French, Vietnam belongs to Asia.\" @ \"If 3n + 2 is an odd integer, n is an odd integer.\" \"If 8 < 9, 5 is a prime number.\" k) ) \"A quadrilateral is a rhombus when it has 2 perpendicular diagonals.\" 0 \"If 5 < 3, 7 is a prime number.\" 1.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_034.png", "page_index": 43, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:53+07:00" } }, "CO1007-chapter-1-slide-044-0000": { "id": "CO1007-chapter-1-slide-044-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Let p and q be: Tai p: \"Brandon likes reading\" q:\"Brandon is a good student\" BK TP.HCM The statement that formalize \"If Brandon likes reading, Brandon is a good student, vice versa, If Brandon is a good student, Contents Brandon like reading\" is: Propositional Logic Logical Equivalences Exercise (p A q 77 Bp->q pV q pq p<>q F -p->-q G -pV(pAq) 0 None of the others 1.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_035.png", "page_index": 44, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T15:59:57+07:00" } }, "CO1007-chapter-1-slide-045-0000": { "id": "CO1007-chapter-1-slide-045-0000", "text": "Exercise Logics Nguyen An Khuong. Let P, Q, R be: Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai P: \"Potter is studying Math\". Q: \"Potter is studying Computer science\". BK . R: \"Potter is studying English\". TP.HCM Formalize the following statement using the propositional connectives. Contents Propositional Logic Example Logical Equivalences Potter is studying Math and English but not Computer science: Exercise P R-Q @ Potter is studying Math and Computer science but not Computer science and English at the same time. 6 It is not true that Potter is studying English and not Math. @ It is not true that Potter is studying English or Computer science and not Math. d) Potter is not studying both Computer science and English but is studying Math 1.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_036.png", "page_index": 45, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:00+07:00" } }, "CO1007-chapter-1-slide-046-0000": { "id": "CO1007-chapter-1-slide-046-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Determine the wrong statement among the following @ xE{x} BK TP.HCM 6{x} C{x} G{x} e{x} Contents o{x} e{{x}} Propositional Logic @ 0C{x} Logical Equivalences Exercise A a B 6 C d E) none of the others 1.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_037.png", "page_index": 46, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:03+07:00" } }, "CO1007-chapter-1-slide-047-0000": { "id": "CO1007-chapter-1-slide-047-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Which of the following proposition is a truth. Contents (pV-q) ->q Propositional Logic p-(pAq) Logical Equivalences Exercise -p->(p->q) -(p-q)->q E none of the others 1.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_038.png", "page_index": 47, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:05+07:00" } }, "CO1007-chapter-1-slide-048-0000": { "id": "CO1007-chapter-1-slide-048-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Let's consider a propositional language where: Xuan Toan. Tran Hong Tai p: \"ABC is an isosceles triangle\". q: \"ABC is an equilateral triangle\". BK TP.HCM T: \"ABC has a 60° angle\". Contents Propositional Logic Which of the following compounds formalize the theorem: \"if Logical Equivalences ABC is an isosceles triangle and has a 60° angle then it is an Exercise equilateral triangle\" ? pg yr (pr)->q (pAr) V q q->(pV r) none of the others 1.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_039.png", "page_index": 48, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:09+07:00" } }, "CO1007-chapter-1-slide-049-0000": { "id": "CO1007-chapter-1-slide-049-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai There are 6 soccer teams A. B, C. D, E. F contested in a BK TP.HCM tournament. The following are statements on which two teams are in the grand final: Contents A and C Propositional Logic B and E b. Logical Equivalences B and F Exercise d A and F @ A and D Knowing that there are 4 half true statements and 1 totally false statement. What teams are in the grand final? 1.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_040.png", "page_index": 49, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:12+07:00" } }, "CO1007-chapter-1-slide-050-0000": { "id": "CO1007-chapter-1-slide-050-0000", "text": "Exercise Logics Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Find the truth values of the following statements (with brief Tai explanations) : @ \"Vx E N,x2 + 5x + 6 is not a prime number.\" BK b)\"3x E R,x2+x+1<0\" TP.HCM c \"3n E N, (n3 - n) is not a multiple of 3.\" d)) \"Vn E N*,n2 - 1 is a multiple of 3. Contents ey \"Vx,Vy E R,x2 + y2 > 2xy Propositional Logic f \"3r EQ,3< r< T\" Logical Equivalences g) \"3n E N,n2 + 1 divisible by 8' Exercise hy \"Vx E R,x< 3A x2 <9\" \"3a,b E R,(a+b)2 > 2(a2+b2)\" \"All equilateral triangles are equal.\" k \"There exist quadrilaterals which don't have circumcircles.\" 0) \"There is a natural number m that, for all real numbers , we have f(x) = x2 - 2x + n is not negative.\" m \"For all positive integers ε and y we have x < y.\" \"For all positive integers x, there is a positive integer y so that x y.\" ) \"There is a positive integer x that, for all positive integers y, we have x < y.\" p) \"There exist positive integers x and y so that x y.\" 1.45", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_1/slide_041.png", "page_index": 50, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:17+07:00" } }, "CO1007-chapter-2-slide-051-0000": { "id": "CO1007-chapter-2-slide-051-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 2 Xuan Toan. Tran Hong Tai Predicate logic BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Predicate Logic Exercise Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 2.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_001.png", "page_index": 51, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:20+07:00" } }, "CO1007-chapter-2-slide-052-0000": { "id": "CO1007-chapter-2-slide-052-0000", "text": "Contents Predicate logic Nguyen An Khuong, Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK. TP.HCM 1 Predicate Logic Contents Predicate Logic Exercise Exercise 2.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_002.png", "page_index": 52, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:22+07:00" } }, "CO1007-chapter-2-slide-053-0000": { "id": "CO1007-chapter-2-slide-053-0000", "text": "Course outcomes Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Represent and model practical problems with discrete structures Contents L.O.2.1 - Logically describe some problems arising in Computing Predicate Logic L.O.2.2 - Use proving methods: direct, contrapositive, induction Exercise L.O.2.3 - Explain problem modeling using discrete structures L.0.3 Understanding of basic probability and random variables L.O.3.1 - Define basic probability theory L.O.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.0.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 2.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_003.png", "page_index": 53, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:26+07:00" } }, "CO1007-chapter-2-slide-054-0000": { "id": "CO1007-chapter-2-slide-054-0000", "text": "Limits of Propositional Logic Predicate logic Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents x > 3 Predicate Logic All square numbers are not prime numbers. 100 is a square Exercise number. Therefore 100 is not a prime number. 2.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_004.png", "page_index": 54, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:28+07:00" } }, "CO1007-chapter-2-slide-055-0000": { "id": "CO1007-chapter-2-slide-055-0000", "text": "Predicates Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition BK A predicate (vi tü) is a statement containing one or more TP.HCM variables. If values are assigned to all the variables in a predicate the resulting statement is a proposition (ménh d@) Contents Predicate Logic x>3 -P(x) Exercise 5>3 - P(5) . A predicate with n variables P(ε1, 2, ...,n Example: x > 3 (predicate) 5 > 3 (proposition) 2 > 3 (proposition) 2.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_005.png", "page_index": 55, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:31+07:00" } }, "CO1007-chapter-2-slide-056-0000": { "id": "CO1007-chapter-2-slide-056-0000", "text": "Truth value Predicate logic Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM x > 3 is true or false? Contents 5 > 3 Predicate Logic Exercise For every number c, > 3 holds There is a number x such that x > 3 2.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_006.png", "page_index": 56, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:32+07:00" } }, "CO1007-chapter-2-slide-057-0000": { "id": "CO1007-chapter-2-slide-057-0000", "text": "Quantifiers Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM V: Universal - V6i moi Contents VxP(x) =P(x) is T for al x Predicate Logic 3: Existential - Tön tai Exercise 3xP(x) = There exists an element x such that P(x) is T We need a domain of discourse for variable 2.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_007.png", "page_index": 57, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:35+07:00" } }, "CO1007-chapter-2-slide-058-0000": { "id": "CO1007-chapter-2-slide-058-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Let P(ε) be the statement \"ε < 2\". What is the truth value of the BK quantification VxP(x), where the domain consists of all real TP.HCM number? P(3) = 3< 2 is false Contents Predicate Logic =y VxP(x) is false Exercise 3 is a counterexample (phan vi du) of VxP(x) Example What is the truth value of the quantification 3xP(x), where the domain consists of all real number? 2.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_008.png", "page_index": 58, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:38+07:00" } }, "CO1007-chapter-2-slide-059-0000": { "id": "CO1007-chapter-2-slide-059-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Express the statement \"Some student in this class comes from BK Central Vietnam.\" TP.HCM Solution 1 Contents M() = x comes from Central Vietnam Predicate Logic Domain for is the students in the class Exercise 3xM(x) Solution 2 Domain for : is all people 2.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_009.png", "page_index": 59, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:41+07:00" } }, "CO1007-chapter-2-slide-060-0000": { "id": "CO1007-chapter-2-slide-060-0000", "text": "Negation of Quantifiers Predicate logic Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Statement Negation Equivalent form BK VxP(x) (VxP(x)) 3x-P(x) TP.HCM 3xP(x) (ExP(x)) Vx-P(x) Contents Predicate Logic Example Exercise All CSE students study Discrete Math 1 Let C(x) denote \"x is a CSE student\" Let S(x) denote \"x studies Discrete Math 1\" Vx : C(x) -S(x) 3x :-(C(x) - S(x)) =3x : C(x) -S(x) There is a CSE student who does not study Discrete Math 1 2.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_010.png", "page_index": 60, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:44+07:00" } }, "CO1007-chapter-2-slide-061-0000": { "id": "CO1007-chapter-2-slide-061-0000", "text": "Another Example Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Translate these: BK All lions are fierce. TP.HCM Some lions do not drink coffee Contents Some fierce creatures do not drink coffee Predicate Logic Exercise Solution Let P(x), Q(x) and R(x) be the statements \"x is a lion\", \" is fierce\" and \"ε drinks coffee\", respectively Vx(P(x) -Q(x)) 3x(P(x)-R(x)) 3x(Q(x) R(x) 2.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_011.png", "page_index": 61, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:47+07:00" } }, "CO1007-chapter-2-slide-062-0000": { "id": "CO1007-chapter-2-slide-062-0000", "text": "The Order of Quantifiers Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai The order of quantifiers is important, unless all the quantifiers are universal quantifiers or all are existential quantifiers BK TP.HCM Read from left to right, apply from inner to outer Example Contents Predicate Logic Vx Vy (x +y= y +x Exercise T for all x,y E R Example Vx 3y (x + y = 0) is T, while 3y Vx (x+ y = 0) is F 2.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_012.png", "page_index": 62, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:50+07:00" } }, "CO1007-chapter-2-slide-063-0000": { "id": "CO1007-chapter-2-slide-063-0000", "text": "Translating Nested Quantifiers Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM Vx(C(x) V 3y(C(y)F(x,y Provided that: Contents C(x): : has a computer, Predicate Logic F(x,y): x and y are friends, Exercise x,y E all students in your school. Answer For every student c in your school, has a computer or there is a student y such that y has a computer and : and y are friends 2.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_013.png", "page_index": 63, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:52+07:00" } }, "CO1007-chapter-2-slide-064-0000": { "id": "CO1007-chapter-2-slide-064-0000", "text": "Translating Nested Quantifiers Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM 3xVyVz (((F(x,y) F(x,z)(y z)--F(y,z))) Provided that: Contents . F(x,y): x,y are friends Predicate Logic x,y,z E all students in your school Exercise Answer There is a student ε, so that for every student y, every student z not the same as y, if ε and y are friends, and ε and z are friends then y and z are not friends. 2.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_014.png", "page_index": 64, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:55+07:00" } }, "CO1007-chapter-2-slide-065-0000": { "id": "CO1007-chapter-2-slide-065-0000", "text": "Translating into Logical Expressions Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example \"There is a student in the class has visited Hanoi\". BK TP.HCM \"Every student in the class has visited Nha Trang or Vung Tau\". Contents Predicate Logic Answer Exercise Assume: C(x) : has visited Hanoi D(x) : x has visited Nha Trang E(x) : c has visited Vung Tau We have: 0 3xC(x) @ Vx(D(x) V E(x)) 2.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_015.png", "page_index": 65, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:00:58+07:00" } }, "CO1007-chapter-2-slide-066-0000": { "id": "CO1007-chapter-2-slide-066-0000", "text": "Translating into Logical Expressions Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Example Tai If a person is a woman and a parent, then this person is mother of someone. BK TP.HCM Solution Contents We define: Predicate Logic W(x) : c is woman Exercise : P(x) : x is a parent . M(x,y): x is mother of y We have: Vx((W(x) P(x) -3yM(x,y)) Example \"Every people has only one best friend.\" Assume: . B(x,y) : y is the best friend of x 2.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_016.png", "page_index": 66, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:02+07:00" } }, "CO1007-chapter-2-slide-067-0000": { "id": "CO1007-chapter-2-slide-067-0000", "text": "Translating into Logical Expressions Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Example \"Every people has only one best friend.\" Contents Assume: Predicate Logic . B(x,y) : y is the best friend of x Exercise Solution Vx3yVz(B(x,y) ((y z) ->-B(x,z)) 2.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_017.png", "page_index": 67, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:05+07:00" } }, "CO1007-chapter-2-slide-068-0000": { "id": "CO1007-chapter-2-slide-068-0000", "text": "Inference Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM If I have a girlfriend, I will take her to go shopping Whenever I and my girlfriend go shopping and that day is a Contents special day, I will surely buy her some expensive gift Predicate Logic If I buy my girlfriend expensive gifts, I will eat noodles for a Exercise week. Today is March 8. March 8 is such a special day Therefore, if I have a girlfriend,. I will eat noodles for a week 2.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_018.png", "page_index": 68, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:08+07:00" } }, "CO1007-chapter-2-slide-069-0000": { "id": "CO1007-chapter-2-slide-069-0000", "text": "Propositional Rules of Inferences Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Rule of Inference Name p BK TP.HCM p->q ..q Modus ponens Contents q Predicate Logic p->q Exercise Modus tollens .. p p->q q->r Hypothetical syllogism ..p-?r Tam doan luan gia dinh) p V q Tp Disjunctive syllogism ..q Tam doan luan tuyén) 2.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_019.png", "page_index": 69, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:11+07:00" } }, "CO1007-chapter-2-slide-070-0000": { "id": "CO1007-chapter-2-slide-070-0000", "text": "Propositional Rules of Inferences Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Rule of Inference Name BK p TP.HCM Addition ..p V q (Quy täc cöng) Contents p q Predicate Logic Simplification Exercise ..p (Rüt gon) p q Conjunction ..pq (Két hop) p V q 7p V r Resolution ..qVr (Phan giai) 2.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_020.png", "page_index": 70, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:15+07:00" } }, "CO1007-chapter-2-slide-071-0000": { "id": "CO1007-chapter-2-slide-071-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Example Tai If it rains today, then we will not have a barbecue today. If we do not have a barbecue today, then we will have a barbecue BK tomorrow. Therefore, if it rains today, then we will have a TP.HCM barbecue tomorrow. Contents Solution Predicate Logic Exercise p: It is raining today q: We will not have a barbecue today r: We will have barbecue tomorrow p-q q->r ..p-?r Hypothetical syllogism 2.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_021.png", "page_index": 71, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:19+07:00" } }, "CO1007-chapter-2-slide-072-0000": { "id": "CO1007-chapter-2-slide-072-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example 1.pq Hypothesis BK It is not sunny this afternoon TP.HCM (-p) and it is colder than 2.-p Simplification using (1) yesterday (q) We will go swimming (r) only if 3.r-p Hypothesis Contents it is sunny Predicate Logic 4.-r Modus tollens (2 and 3) Exercise If we do not go swimming, then we will take a canoe trip (s) 5.rs Hypothesis If we take a canoe trip, then we will be home by sunset (t) 6.s Modus ponens (4 and 5) We will be home by sunset (t) 7.s-t Hypothesis 8.t Modus ponens (6 and 7) 2.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_022.png", "page_index": 72, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:22+07:00" } }, "CO1007-chapter-2-slide-073-0000": { "id": "CO1007-chapter-2-slide-073-0000", "text": "Fallacies Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Definition TP.HCM Fallacies (nguy bin) resemble rules of inference but are based on contingencies rather than tautologies Contents Predicate Logic Example Exercise If you do correctly every questions in mid-term exam, you will get 10 grade. You got 10 grade. Therefore, you did correctly every questions in mid-term exam. ls [(p -> q) q] -> p a tautology? 2.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_023.png", "page_index": 73, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:26+07:00" } }, "CO1007-chapter-2-slide-074-0000": { "id": "CO1007-chapter-2-slide-074-0000", "text": "Rules of Inference for Quantified Statements Predicate logic Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Rule of Inference Name BK TP.HCM VxP(x) Universal instantiation ... P(c) (Cu thé h6a ph quät) Contents Predicate Logic P(c)for an arbitrary c Exercise Universal generalization ..VxP(x) (Tng quát hóa ph quát) 3xP(x) Existential instantiation :. P(c)for some element c (Cu thé h6a tön tai) P(c)for some element c Existential generalizatior ..3xP(x) (T6ng quat hóa tn tai) 2.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_024.png", "page_index": 74, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:30+07:00" } }, "CO1007-chapter-2-slide-075-0000": { "id": "CO1007-chapter-2-slide-075-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example A student in this class has not gone to class BK TP.HCM Everyone in this class passed the first exam Someone who passed the first exam has not gone to class Contents Predicate Logic Exercise Hint . C(x): x is in this class B(x): x has gone to class P(x): : passed the first exam Premises??? 2.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_025.png", "page_index": 75, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:33+07:00" } }, "CO1007-chapter-2-slide-076-0000": { "id": "CO1007-chapter-2-slide-076-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 1.3x(C(x) -B(x)) Premise BK 2.C(a) -B(a) Existential instantiation from (1) TP.HCM 3. C(a) Simplification from (2) 4.Vx(C(x)-P(x) Premise Contents 5.C(a) -P(a) Predicate Logic Universal instantiation from (4) 6. P(a) Exercise Modus ponens from (3) and (5) 7.-B(a) Simplification from (2) 8.P(a) -B(a) Conjunction from (6) and (7) 9.3x(P(x) -B(x)) Existential generalization from (8) 2.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_026.png", "page_index": 76, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:37+07:00" } }, "CO1007-chapter-2-slide-077-0000": { "id": "CO1007-chapter-2-slide-077-0000", "text": "Predicate logic Nguyen An K huong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Given the predicate p() :\" x2 - 3x + 2 = 0\". What is the truth BK TP.HCM value (chan tri) of the following propositions: @ p(0) Contents 6 p(1) Predicate Logic @ p(2) Exercise @ 3x,p(x) e Vx,p(x) 2.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_027.png", "page_index": 77, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:40+07:00" } }, "CO1007-chapter-2-slide-078-0000": { "id": "CO1007-chapter-2-slide-078-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Let x,y E Z+, and the predicate: p(x,y): \" is a divisor of y\" Determine the truth value of the following propositions: BK @ p(2,3) TP.HCM 6 p(2,6) Contents D Vy,p(1,y) Predicate Logic @ Vx,p(x,x) Exercise e Vxy;p(x,y) f) 3yVx,p(x,y) @) VxVy,(p(x,y) p(y,x))-(x=y) h VxVyVz(p(x,y)p(y,z) ->(p(x,z)) 2.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_028.png", "page_index": 78, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:43+07:00" } }, "CO1007-chapter-2-slide-079-0000": { "id": "CO1007-chapter-2-slide-079-0000", "text": "Predicate logic Provided that: Nguyen An Khuong. Tran Tuan Anh, Mai : F(x,y) : x is father of y Xuan Toan, Tran Hong Tai M(x,y) : x is mother of y . S(x,y) : x is sister of y, BK B(x,y) : x is brother of y TP.HCM H(x,y) : x is spouse (wife/husband) of y . O(x,y) : x is elder than y. Contents Predicate Logic Exercise Express each of these statements using predicates: @ 'He (a person) has an elder sister and younger brother' by 'All of her brothers are younger than her'. Thuyen has only one husband' (Thuyen is a private name) 'One of his sisters is younger than him'. @ 'Everyone has grandfather, grandmother, maternal grandfather, maternal grandmother'. f 'A father of a person cannot be a mother of other ones'. 2.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_029.png", "page_index": 79, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:47+07:00" } }, "CO1007-chapter-2-slide-080-0000": { "id": "CO1007-chapter-2-slide-080-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Ma Solutions: Xuan Toan, Tran Hong Tai @) 'He (a person) has an elder sister and younger brother' 3x3y(S(x,m) O(x,m) B(y,m) -O(y,m) BK b 'All of her brothers are younger than her'. TP.HCM Vx(B(x,m)--O(x,m)). @ Thuyen has only one husband' (Thuyen is a private name) Contents xVy H(x,Thuyen) H(y,Thuyen) -(x = y) Predicate Logic or 3xVy H(x,Thuyen) (x y) --H(y,Thuyen Exercise @ 'One of his sisters is younger than him'. 3xVy(S(x,m) -O(x,m) S(y,m) (x y) ->O(y,m)) @ Everyone has grandfather, grandmother, maternal grandfather, maternal grandmother'. Vx3y3z3y1y23z1z2 (F(y,x)M(z,x)F(y1,y)M(y2,y)F(z1,z)M(z2,z)) f) 'A father of a person cannot be a mother of other ones'. x3yVz(F(x,y) --M(x,z)). 2.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_030.png", "page_index": 80, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:51+07:00" } }, "CO1007-chapter-2-slide-081-0000": { "id": "CO1007-chapter-2-slide-081-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Translating the following nested quantifiers: TP.HCM @ B(c,m)(O(c,m) VO(m,c))) Contents 6) B(c,m)F(a,m)->O(a,c) F(a,c) Predicate Logic @ VxVy(S(x,m) B(c,y) ->x =y). Exercise d 3x((S(x,m) V H(c,x)) V3x(H(x,m)O(x,m)) @ VxVy(S(x,m) AS(y,m) ->O(x,y) VO(y,x) 2.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_031.png", "page_index": 81, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:01:55+07:00" } }, "CO1007-chapter-2-slide-082-0000": { "id": "CO1007-chapter-2-slide-082-0000", "text": "Predicate logic Nguyen An Khuong. Given a predicate N(x) \"x has been to Da Lat\" with the domain Tran Tuan Anh, Mai Xuan Toan. Tran Hong is the all students in Mathematics class. Translate the following Tai predicates into English @ 3xN(x) BK TP.HCM 6 VxN(x) o -3xN(x) Contents @ 3x-N(x) Predicate Logic -VxN(x) Exercise e 6 Vx-N(x) @) There is a student in this class has been to Da Lat. b) All students in Math class have been to Da Lat. There is no exists a student in Math class has gone to Da Lat d There is a student in this class has never gone to Da Lat. Not all students in Math class have ever been to Da Lat. f) All students in Math class have never been to Da Lat. 2.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_032.png", "page_index": 82, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:00+07:00" } }, "CO1007-chapter-2-slide-083-0000": { "id": "CO1007-chapter-2-slide-083-0000", "text": "Predicate logic Nguyen An K huong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Given the predicate V(c) \"c studies more than 5 hours in class TP.HCM every weekday\" with the domain is the all students in Mathematics class. Express the following predicates Contents @ xN(x) Predicate Logic 6 VxN(x) Exercise @ 3x-N(x) @ Vx-N(x) 2.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_033.png", "page_index": 83, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:02+07:00" } }, "CO1007-chapter-2-slide-084-0000": { "id": "CO1007-chapter-2-slide-084-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai What is the propositional formula for the following pseudo code BK for (i = 0; i3jL(j) 6 VpB(p) ->3jQ(j) @ 3j(Q(j) AL(j)) -3pF(p) @(VpB(p) VjQ(j))-3jL(j 2.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_040.png", "page_index": 90, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:25+07:00" } }, "CO1007-chapter-2-slide-091-0000": { "id": "CO1007-chapter-2-slide-091-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Formalize the following sentences: BK Nobody is perfect. TP.HCM not everyone is perfect All your friends are perfect. Contents d) At least one of your friend is perfect. Predicate Logic Everybody is your friend and they are perfect. Exercise f) Not everybody is your friend or there is somebody not perfect Giving: C(x): x is perfect. D(x): x is your friend. E(x): x is someone else 2.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_041.png", "page_index": 91, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:29+07:00" } }, "CO1007-chapter-2-slide-092-0000": { "id": "CO1007-chapter-2-slide-092-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Giving the following Predicate: - P(x): Program x satisfies ABET standard. BK - Q(x,y): Program x has the same educational goal as program y. TP.HCM - R(x): Educational outcome from program x is verifiable. Which of the following formalize this sentence : \"Every program Contents that has the same educational goal as a ABET satisfied program Predicate Logic and verifiable Educational outcome also satisfies ABET standard' Exercise A Vx(P(x) -Q(x)) ->3x(R(x)) B Vx(Ey(Q(x,y) P(y)R(x))-P(x) Vx(Ey(Q(x,y) P(y) R(x))->P(x) V R(x)) D Vx(Vy(Q(x,y) P(y) V R(x)) -> P(x)) 2.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_042.png", "page_index": 92, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:31+07:00" } }, "CO1007-chapter-2-slide-093-0000": { "id": "CO1007-chapter-2-slide-093-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Let: P(x,y): x is parent of y BK TP.HCM - M(x): x is male . Given: Contents F(v,w)= M(v) A3x3y(P(x,y) AP(x,v)A(y Z v) AP(y,w)) Predicate Logic then F(v,w) means: Exercise A v is brother of w B v is cousin of w v is uncle of w v is grand father of w 2.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_043.png", "page_index": 93, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:34+07:00" } }, "CO1007-chapter-2-slide-094-0000": { "id": "CO1007-chapter-2-slide-094-0000", "text": "Predicate logic Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Formalize the following sentences using predicate logic: Tai @ When a hard drive has less than 30GB free space, a warning will be issued to all the users. BK 6 Do not back up the files if anyone is logging in the system. TP.HCM @ YouTube's videos will be buffered if there are at least 8MB Contents memory and 56kb/s line rate. Predicate Logic @ Few computer student is good at programming Exercise @ No computer student is not hard working. f) Not all computer students are smart. g) All the Pompeians are either loyal to or hate Caesar h Everyone is loyal to someone. @ People only want to assassinate the dictator whom they are not loyal to. 2.44", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_2/slide_044.png", "page_index": 94, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:37+07:00" } }, "CO1007-chapter-3-slide-095-0000": { "id": "CO1007-chapter-3-slide-095-0000", "text": "Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 3 Xuan Toan. Tran Hong Tai Proving methods BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Proving Methods Exercise Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 3.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_001.png", "page_index": 95, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:39+07:00" } }, "CO1007-chapter-3-slide-096-0000": { "id": "CO1007-chapter-3-slide-096-0000", "text": "Contents Proving methods Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK. TP.HCM 1 Proving Methods Contents Proving Methods Exercise Exercise 3.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_002.png", "page_index": 96, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:41+07:00" } }, "CO1007-chapter-3-slide-097-0000": { "id": "CO1007-chapter-3-slide-097-0000", "text": "Course outcomes Proving methods Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Contents Represent and model practical problems with discrete structures L.O.2.1 - Logically describe some problems arising in Computing Proving Methods L.O.2.2 - Use proving methods: direct, contrapositive, induction Exercise L.O.2.3 - Explain problem modeling using discrete structures L.0.3 Understanding of basic probability and random variables L.0.3.1 - Define basic probability theory L.O.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.0.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 3.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_003.png", "page_index": 97, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:45+07:00" } }, "CO1007-chapter-3-slide-098-0000": { "id": "CO1007-chapter-3-slide-098-0000", "text": "ntroduction Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Definition Contents A proof is a sequence of logical deductions from Proving Methods - axioms, and Exercise previously proved theorems that concludes with a new theorem 3.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_004.png", "page_index": 98, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:47+07:00" } }, "CO1007-chapter-3-slide-099-0000": { "id": "CO1007-chapter-3-slide-099-0000", "text": "Terminology Proving methods Nguyen An Khuong. Tran Tuan Anh.Mai Xuan Toan. Tran Hong Tai BK Hypotheses of TP.HCM Proved theorem Atiom theorem Contents Proving Methods Rules of inference Theorem Exercise Theorem (dinh /y) = a statement that can be shown to be true Axiom (tien d@) = a statement we assume to be true Hypothesis (gia thiét) = the premises of the theorem 3.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_005.png", "page_index": 99, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:50+07:00" } }, "CO1007-chapter-3-slide-100-0000": { "id": "CO1007-chapter-3-slide-100-0000", "text": "Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Theorem TP.HCM Lemma Corollary Contents Proving Methods Lemma (b d@) = less important theorem that is helpful in Exercise the proofs of other results Corollary (hé qua) = a theorem that can be established directly from a proved theorem Conjecture (phong doän) = statement being proposed to be true, when it is proved, it becomes theorem 3.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_006.png", "page_index": 100, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:53+07:00" } }, "CO1007-chapter-3-slide-101-0000": { "id": "CO1007-chapter-3-slide-101-0000", "text": "Proving a Theorem Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Many theorem has the form VxP(x) -> Q(x) Contents Goal: Proving Methods * Show that P(c) -> Q(c) is true with arbitrary c of the domain Exercise : Apply universal generalization How to show that conditional statement p -> q is true 3.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_007.png", "page_index": 101, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:55+07:00" } }, "CO1007-chapter-3-slide-102-0000": { "id": "CO1007-chapter-3-slide-102-0000", "text": "Methods of Proof Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Direct proofs (chüng minh truc tiép) Contents Proof by contraposition (chüng minh phan däo) Proving Methods Exercise Proof by contradiction (chüng minh phän chüng) Mathematical induction (quy nap toan hoc) 3.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_008.png", "page_index": 102, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:02:57+07:00" } }, "CO1007-chapter-3-slide-103-0000": { "id": "CO1007-chapter-3-slide-103-0000", "text": "Direct Proofs Proving methods Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Definition TP.HCM A direct proof shows that p -> q is true by showing that if p is true, then q must also be true. Contents Proving Methods Example Exercise Ex.: If m is an odd integer, then m2 is odd. Pr.: Assume that n is odd. By the definition, n = 2k + 1, k E Z n2 =(2k+1)2= 4k2+4k+1= 2(2k2+2k) +1 is an odd number. 3.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_009.png", "page_index": 103, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:00+07:00" } }, "CO1007-chapter-3-slide-104-0000": { "id": "CO1007-chapter-3-slide-104-0000", "text": "Proof by Contraposition Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition BK TP.HCM p -> q can be proved by showing (directly) that its contrapositive -q -> -p, is true Contents Proving Methods Example Exercise Ex.: Given an integer m, show that if 3n + 2 is odd, then n is odd Pr.: Assume that \"n is even\", so n = 2k, k E Z. Substituting 3n + 2 = 3(2k) + 2 = 6k + 2 = 2(3k +1) is even. Because the negation of the conclusion of the conditional statement implies that the hypothesis is false, Q.E.D 3.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_010.png", "page_index": 104, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:03+07:00" } }, "CO1007-chapter-3-slide-105-0000": { "id": "CO1007-chapter-3-slide-105-0000", "text": "Proofs by Contradiction Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition p is true if if can show that -p -> (r -r) is true for some BK TP.HCM proposition r. Contents Example Proving Methods Ex.: Prove that V2 is irrational. Exercise Pr.: Let p is the proposition \"V2 is irrational\". Suppose p is true, which means V2 is rational. If so, 3a,b E Z.V2 = a/b, a,b have no common factors. Squared, 2 = a2/b2, 262 = a2, so a2 is even, and a is even, too. Because of that a = 2c,c E Z. Thus, 2b2 = 4c2,or b2 = 2c2, which means b2 is even and so is b. That means 2 divides both a and b, contradict with the assumption. 3.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_011.png", "page_index": 105, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:07+07:00" } }, "CO1007-chapter-3-slide-106-0000": { "id": "CO1007-chapter-3-slide-106-0000", "text": "Problem Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Proving Methods Exercise Assume that we have an infinite domino string, we want to know whether every dominoes will fall, if we only know two things: 1 We can push the first domino to fall 2 If a domino falls, the next one will be fall We can! Mathematical induction. 3.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_012.png", "page_index": 106, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:09+07:00" } }, "CO1007-chapter-3-slide-107-0000": { "id": "CO1007-chapter-3-slide-107-0000", "text": "Mathematical Induction Proving methods Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition (Induction) BK TP.HCM To prove that P(n) is true for all positive integers n, where P(n is a propositional function, we complete two steps: Contents * Basis Step: Verify that P(1) is true. Proving Methods Exercise : Inductive Step: Show that the conditional statement P(k) -> P(k + 1) is true for all positive integers k Logic form: P(1) AVkP(k) -P(k+1)]-VnP(n What is P(n) in domino string case? 3.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_013.png", "page_index": 107, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:12+07:00" } }, "CO1007-chapter-3-slide-108-0000": { "id": "CO1007-chapter-3-slide-108-0000", "text": "Example on Induction Proving methods Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Show that if n is a positive integer, then n(n+1) 1+2+...+n BK 2 TP.HCM Contents Solution Proving Methods Let P(n) be the proposition that sum of first n is n(n + 1)/2 Exercise 2 Inductive Step: k(k+1) 2 Then: k(k+1) 1+2+...+k+(k+1) (k+1) 2 k(k+1)+2(k+1) 2 (k+1)(k+2) 2 3.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_014.png", "page_index": 108, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:16+07:00" } }, "CO1007-chapter-3-slide-109-0000": { "id": "CO1007-chapter-3-slide-109-0000", "text": "Example on Induction Proving methods Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Example Prove that n < 2n for all positive integers n BK TP.HCM Solution Contents Let P(n) be the proposition that n < 2n. Proving Methods Basis Step: P(1) is true, because 1 < 21 = 2 Exercise Inductive Step: Assume that P(k) is true for the positive k, that is, k < 2k Add 1 to both side of k < 2k, note that 1 2k. k+1<2k+1<2k+2k=2:2k=2k+1 shows that P(k + 1) is true, namely, that k + 1 < 2k+1 based on the assumption that P(k) is true. 3.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_015.png", "page_index": 109, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:19+07:00" } }, "CO1007-chapter-3-slide-110-0000": { "id": "CO1007-chapter-3-slide-110-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Prove that, if n is a non-negative integer and 7n + 9 is an even number, then n is an odd number by three ways: Contents Proving Methods Directed proof. Exercise Contraposition proof(phän däo) Contradiction. 3.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_016.png", "page_index": 110, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:21+07:00" } }, "CO1007-chapter-3-slide-111-0000": { "id": "CO1007-chapter-3-slide-111-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Directed proof Assume: 7r + 9 is the even number. Then, 7n + 9= 2k,(k E Z) So: n = 2k- 6n- 9= 2k - 6n - 10 + 1 = 2(k - 3n - 5) + 1 BK That means m is an odd number. TP.HCM Contrapositive proof (phän däo) Contents To proof the above statement, firstly, we convert it into the logic expression: p - q with Proving Methods p = 7n + 9 is the even number and q = n is the odd number. Its contrapositive: \"If m is not an odd number, then 7n + 9 is not an even number\". We Exercise can prove this statement follows this way: If n is not an odd number, that means n can divisible 2. So that, n = 2k, (k E Z) Weimply: 7n + 9= 7(2k) + 9 = 14k + 9 = 2(7k + 4) + 1 That means: 7n + 9 is not the even number. Totally, we have proved the logic expression: -q -- -p. Therefore p - q is also truth. Contradiction proof Suppose 7n + 9 is an even number and m is not an odd number or n is an even number. Because n is an even number, then n = 2k,(k E Z) Weinfer: 7n +9=7(2k) + 9 = 14k + 9 = 2(7k + 4) +1 Its means: 7n + 9 is the odd number. We can show that, if m is an even number, then 7m + 9 is an odd number. This contradicts with the hypothesis 7r + 9 is an even number. 3.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_017.png", "page_index": 111, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:26+07:00" } }, "CO1007-chapter-3-slide-112-0000": { "id": "CO1007-chapter-3-slide-112-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Which of the method of proof is used for proving the statement BK below: TP.HCM To prove \"If m and n are integers, m x n is an even number, then either m is even or m is even\", we follow these inferences: Contents Assume m and m are odd numbers. Then we can express: m = 2k + 1 Proving Methods and m= 2l +1.So mm=(2k+1)(2l +1) = 2(2kl + k +l) +1 is the Exercise odd number. Contradiction. That means either m is even or n even. 4) Directed proof. B) Contradiction proof or contra-positive proof inductive proof ) All the above answers are incorrect. 3.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_018.png", "page_index": 112, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:30+07:00" } }, "CO1007-chapter-3-slide-113-0000": { "id": "CO1007-chapter-3-slide-113-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Which of the method of proof is used for proving the statement BK below: TP.HCM To prove \"If m and n are integers, m x n is an even number, then either m is even or m is even\", we follow these inferences: Contents Assume m and m are odd numbers. Then we can express: m = 2k + 1 Proving Methods and m= 2l +1.So mm= (2k+1)(2l +1) = 2(2kl + k+l) +1 is the Exercise odd number. Contradiction. That means either m is even or m even. 4) Directed proof. B) Contradiction proof or contra-positive proof correct answer! inductive proof ) All the above answers are incorrect. 3.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_019.png", "page_index": 113, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:33+07:00" } }, "CO1007-chapter-3-slide-114-0000": { "id": "CO1007-chapter-3-slide-114-0000", "text": "Proving methods Nguyen An Khuong. What is wrong in the following induction to prove that all flowers Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai are the same color? Let P(n) be all flowers in a set of n flowers have the same color. BK TP.HCM We can easily infer that, P(1) is a truth 2 Assume that P(n) is correct. Which means that all flowers in Contents the set of n flowers have the same color . Proving Methods 4 Consider a set of n + 1 flowers; numbering as Exercise 1,2,3,...,n,(n+1). Based on the assumption, Sequence of first n flowers has the same color, and seguence of m later flowers also has the same color. Because the 2 set has same n - 1 flowers, all n + 1 flowers should have a same color. Meaning P(n + 1) and the statement is proved by induction 3.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_020.png", "page_index": 114, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:37+07:00" } }, "CO1007-chapter-3-slide-115-0000": { "id": "CO1007-chapter-3-slide-115-0000", "text": "Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Consider a subset D (W x ) which defined in a recursive way: BK TP.HCM @ (n,0 ED. @ If (n,m) e D,(n,n+m) e D Contents Do Proving Methods Exercise 1 Calculate some elements from D prove by induction on k that 'if m = k.n, '(n, m) E D'. 3) prove that if (n,m) E D, we will get m = kn k E W. 3.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_021.png", "page_index": 115, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:39+07:00" } }, "CO1007-chapter-3-slide-116-0000": { "id": "CO1007-chapter-3-slide-116-0000", "text": "Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hon Tai BK @ Prove that Vn E W+ TP.HCM (n+1)2- (n+2)2-(n+3)2 +(n+4)2=4. 6 The infer that for every natural number m, there exists Contents natural number n that can represent m as a sum of squares Proving Methods 12, 22, ..., n2, which mean: Vm E W+, 3n E +, Exercise Hint: Try to display the values of m E {1,2,3,4,5,6} 3.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_022.png", "page_index": 116, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:41+07:00" } }, "CO1007-chapter-3-slide-117-0000": { "id": "CO1007-chapter-3-slide-117-0000", "text": "Proving methods Nguyen An Khuong Tran Tuan Anh, Mai @ Prove that Vn E W+ Xuan Toan, Tran Hong Tai (n+1)2 - (n+2)2- (n+3)2+ (n+4)2= 4. 6) The infer that for every natural number m, there exists BK natural number n that can represent m as a sum of squares TP.HCM 12, 22, ..., n2, which mean: Vm E W+, 3n E W+, 1,...,n E{-1,1}, m=E112+E222+...+nn? Contents Hint: Try to display the values of m E{1,2,3,4,5,6} Proving Methods m = 0: 0 = 12 + 22 - 32 + 42 - 52 - 62 + 72 Exercise m = 1: 1 = 12 m = 2: 2 = -12 - 22 - 32 + 42 m = 3: 3 = -12 + 22 m = 4: 4 = 12 - 22 - 32 + 42 3.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_023.png", "page_index": 117, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:44+07:00" } }, "CO1007-chapter-3-slide-118-0000": { "id": "CO1007-chapter-3-slide-118-0000", "text": "Proving methods Nguyen An Khuong Tran Tuan Anh, Mai @ Prove that Vn E W+ Xuan Toan, Tran Hong Tai (n+1)2-(n+2)2-(n+3)2+(n+4)2 =4. 6) The infer that for every natural number m, there exists BK natural number n that can represent m as a sum of squares TP.HCM 12, 22, .. ., n2, which mean: Vm E W+, n E W+. 1,...,n E{-1,1}, m =&112+£222+...+£nn?. Contents Hint: Try to display the values of m E{1,2,3,4,5,6} Proving Methods m = 0: 0 = 12 + 22 - 32 + 42 - 52 - 62 + 72 Exercise m = 1: 1 = 12 m = 2: 2 = -12 - 22 - 32 + 42 m = 3: 3 = -12 + 22 m = 4: 4 = 12 - 22 - 32 + 42 m = 5: 5 = 12 + (22 - 32 - 42 + 52 3.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_024.png", "page_index": 118, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:47+07:00" } }, "CO1007-chapter-3-slide-119-0000": { "id": "CO1007-chapter-3-slide-119-0000", "text": "Proving methods Nguyen An Khuong Tran Tuan Anh, Mai @ Prove that Vn E W+ Xuan Toan, Tran Hong Tai (n+1)2-(n+2)2-(n+3)2+(n+4)2 =4. b) The infer that for every natural number m, there exists BK natural number n that can represent m as a sum of squares TP.HCM 12, 22, .. ., n2, which mean: Vm e W+, 3n e W+, 1,...,n E{-1,1}, m =E112+£222+...+£nn?. Contents (Hint: Try to display the values of m E {1,2,3,4,5,6}) Proving Methods m = 0: 0 = 12 + 22 - 32 + 42 - 52 - 62 + 72 Exercise m = 1: 1 = 12 m = 2: 2 = -12 - 22 - 32 + 42 m = 3: 3 = -12 + 22 m = 4: 4 = 12 - 22 - 32 + 42 m = 5: 5 = 12 + (22 - 32 - 42 + 52) m = 6: 6 = -12 - 22 - 32 +42 + (52 - 62 - 72 + 82 3.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_025.png", "page_index": 119, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:51+07:00" } }, "CO1007-chapter-3-slide-120-0000": { "id": "CO1007-chapter-3-slide-120-0000", "text": "Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Prove the following: 1.2+2.5+3.8+...+n.(3n-1) =n2(n+1),n 1 Contents 1 2n n+i,n Z1 Proving Methods Exercise 4 log5 (2) is an irrational number. 3.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_026.png", "page_index": 120, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:53+07:00" } }, "CO1007-chapter-3-slide-121-0000": { "id": "CO1007-chapter-3-slide-121-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 1) 1.2+2.5+3.8+...+n.(3n-1)=n2(n+1),n>1(1) BK - With n = 1 we have 1.2 = 12(1 + 1) -> (1) correct. TP.HCM - Assume (1) is correct with n = k, then we have 1.2+2.5+3.8+...+k.(3k-1)=k2(k+1) Contents - We will prove (1) is correct with n = k + 1, meaning Proving Methods 1.2+2.5+3.8+...+k.(3k-1)+(k+1)(3k+2) =(k+1)2(k+2) Exercise In fact, n = k + 1, meaning 1.2+2.5+3.8+...+k.(3k-1)+(k+1)(3k+2) = 1.2+2.5+3.8+...+k.(3k-1]+(k +1(3k +2)= k2(k+1)+(k+1)(3k+2)=(k+1)(k2+3k+2)= k+1)(k+1)(k+2)=(k+1)2(k+2). The expression is correct with n = k + 1.Thus, QED 3.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_027.png", "page_index": 121, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:03:56+07:00" } }, "CO1007-chapter-3-slide-122-0000": { "id": "CO1007-chapter-3-slide-122-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai 2 BK (2) correct TP.HCM 7 1 Assume (2) is correct with n = k, Then we have P(k): 1+1 2k Contents 12 1+2+3 1+2+3+...+k k+1 - We will prove (2) is correct with n = k + 1, meaning Proving Methods P(k+1): Exercise 1 1 1+2+...+k+(k+1 2k 1 2k 2 2k(k+2+2 k+1 + 1+2+...+k+(k+1) k+1 (k+1(k+2) (k+1)(k+2) 2(k2+2k+1) 2(k+1)2 2(k+1) Noted that: (k+1)(k+2) (k+1)(k+2) k+2 1+2+...+k+(k+1)= (k+1)(k+2) The expression is correct with n = k + 1.Thus, QED 3.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_028.png", "page_index": 122, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:01+07:00" } }, "CO1007-chapter-3-slide-123-0000": { "id": "CO1007-chapter-3-slide-123-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai (3) n=12i-1 = 2n -1 (3 - With i =1 we have S(1): Ei=1 2i-1 = 21-1 = 21 - 1 -? BK (3) correct. TP.HCM Assume (3) is correct with n = k, Then we have S(k) Contents - We will prove (3) is correct with n = k + 1, meaning Proving Methods S(k+1): Exercise The expression is correct with m = k + 1. (4) Prove that logs (2) is an irrational number. Assume the opposite log5 (2) is a rational number. Therefore, log5(2) = %,where a,b e Z,b 0,GCD(a,b) = 1 Then, 5% = V5a = 2 = 5a = 2b. Because 5a is always odd and 2b is always even. So, this is a contradiction. Thus, QED 3.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_029.png", "page_index": 123, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:06+07:00" } }, "CO1007-chapter-3-slide-124-0000": { "id": "CO1007-chapter-3-slide-124-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Prove the following statements by induction : Contents Proving Methods For every integer n 1, 6n - 1 is divisible by 5. Exercise 4 For every integer n Z 1, 8n - 1 is divisible by 7. @ For every integer n > 1, 4\" + 15n - 1 is divisible by 9 3.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_030.png", "page_index": 124, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:08+07:00" } }, "CO1007-chapter-3-slide-125-0000": { "id": "CO1007-chapter-3-slide-125-0000", "text": "Solution Proving methods Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai (1) Prove that for every integer n > 1, 32n-1+1 is divisible by 4. BK TP.HCM With n = 1, we have 4:4 - Truth. On the other hand, There is an integer k > 1 so that 32k-1 +1 is divisible by 4. Contents Proving Methods On the other hand, There is an integer m so that 32k-1 +1 = 4m. Exercise We need to prove 32(k+1)-1 +1 is divisible by 4. This is the equivalent to prove 32k+1 +1 is divisible by 4 On the other hand, we have: 32k+1+1=32(32k-1+9-8= 9(32k-1+1)-2.4=4(9m+2 is proved by induction 3.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_031.png", "page_index": 125, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:11+07:00" } }, "CO1007-chapter-3-slide-126-0000": { "id": "CO1007-chapter-3-slide-126-0000", "text": "Solution Proving methods Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai (2) Prove that for every integer n > 1, 6\" - 1 is divisible by 5 BK TP.HCM With n = 1, we have 61 - 1 = 5:5 - Truth. On the other hand, There is an integer n > 1 so that 6n - 1 Contents is divisible by 5. Proving Methods On the other hand, There is an integer m so that Exercise 6n - 1 = 5m. We need to prove 6n+1 - 1 is divisible by 5. On the other hand, we have: 6n+1-1 = 6.6n-6+5 = 6(6n-1)+5 = 30m+5 = 5(6m+1 Therefore, 6n+1 - 1 is a multiple of 5. The above statement is proved by induction. 3.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_032.png", "page_index": 126, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:14+07:00" } }, "CO1007-chapter-3-slide-127-0000": { "id": "CO1007-chapter-3-slide-127-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai (3) Prove that for every integer n > 1, 52n-1 +1 is divisible by 6. BK TP.HCM With n = 1, we have 52-1 + 1 = 4:4 - Truth. Assume there is an integer n 1 so that 52n-1 + 1 is Contents divisible by 6. Proving Methods On the other hand, There is an integer m so that 52n-1 + 1 = 6m. Exercise This is the equivalent to prove 52n+1 + 1 is divisible by 6. On the other hand, we have: 52n+1- 1= 52.52n-1+ 25 - 24= 52(52n-1 +1) -24= of 6. The above statement is proved by induction. 3.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_033.png", "page_index": 127, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:17+07:00" } }, "CO1007-chapter-3-slide-128-0000": { "id": "CO1007-chapter-3-slide-128-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK (4) Prove that for every integer n Z 1, 8n - 1 is divisible by 7. TP.HCM With n = 1, we have 81 - 1 = 7:7 - Truth. Assume there is an integer n > 1 so that 8n - 1 is divisible by Contents 7. Proving Methods In other word, there exists an integer m so that 8\" - 1 = 7m. Exercise On the other hand, We have: 8n+1-1= 8.8n-8+7= 8(8n-1)+7= 56m+7=7(8m+1) Thus, 8n+1 - 1 is also a multiple of 7. The above statement is proved by induction 3.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_034.png", "page_index": 128, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:20+07:00" } }, "CO1007-chapter-3-slide-129-0000": { "id": "CO1007-chapter-3-slide-129-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK (5) Prove that for all integers n > 1, 4\" + 15n -1 are divisible by TP.HCM 9. With n = 1, we has 41 + 15 - 1 = 18:9 - Always true. Contents Proving Methods Assume there is an integer n > 1 so that 4\" + 15n - 1 is Exercise divisible by 9 In other word, There is an integer m so that 4n + 15n - 1 = 9m We need to prove that 4n+1 + 15(n + 1) - 1 is also divisible by 9. 3.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_035.png", "page_index": 129, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:23+07:00" } }, "CO1007-chapter-3-slide-130-0000": { "id": "CO1007-chapter-3-slide-130-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai On the other hand, we have: 4n+1+15(n+1) -1= 4\".41+15n+15 -1= BK 3.4n +4n + 15n + 15 -1 = (4\" + 15n - 1) + 3.4 + 15 TP.HCM In which: (4\" + 15n - 1):9 (based on the assumption) And: For 3.4n + 15 = 3(4n + 5) divisible by 9, we need to prove Contents Proving Methods (4n + 5):3 Exercise Let B(k) = 4k +5.With k = 1,41 +5 = 9:3 -> correct with k = 1 Assume it is also true with k = i -> B(i) = (4 + 5):3.We prove that it is also correct with k = i + 1 -> B(i+1) = 4i+1 +5 = 4i.4 + 5 = 3.4i + 4i + 5 = (4i + 5) + 3.4i. We have: (4i + 5):3 based on the assumption And: 3.4:3. Therefore, (4 + 5):3. thus, QED 3.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_036.png", "page_index": 130, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:26+07:00" } }, "CO1007-chapter-3-slide-131-0000": { "id": "CO1007-chapter-3-slide-131-0000", "text": "Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Prove the following inequalities by induction: Contents Proving Methods For every integer n 4, n!> 2\". Exercise 3.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_037.png", "page_index": 131, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:29+07:00" } }, "CO1007-chapter-3-slide-132-0000": { "id": "CO1007-chapter-3-slide-132-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai with n = 1, we have 3' = 3 va 12 = 1. Therefore, the BK statement is correct for n = 1. TP.HCM With n = 2, we have 32 = 9 and 22 = 4. Therefore, the statement is also correct with n = 2. Contents Assume there is an integer n 2 that 3n > n2. Proving Methods We need to prove 3n+1 is greater than (n + 1)2 Exercise On the other hand, Expanding the right side of the inequality we get: (n +1)2 = n2 +2n +1. Because we only consider n > 2, n2 = n x n > 2n and 3n > 1. Consider the left side of the inequality: 3n+1=3 x 3n=3n+3n +3n > n2+n2 +1 > n2 +2n+1=(n+1)2 (QED) . Thus, The statement is proved by induction 3.34", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_038.png", "page_index": 132, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:32+07:00" } }, "CO1007-chapter-3-slide-133-0000": { "id": "CO1007-chapter-3-slide-133-0000", "text": "Solution Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK (2) Prove that for every integer n > 4, n!> 2n TP.HCM with n = 4, we have 4!= 24 > 24 = 16. Assume there is an integer n > 1 that n! > 2\" Contents Proving Methods On the other hand, we have: Exercise (n + 1)! = (n + 1) x n!. based on the assumption, we have (n +1)!> (n +1) x 2n. Moreover, the expression on the right side is greater than 2 x 2n = 2n+1 (vi n > 4 > 2) because of that,the above statement is proved by induction 3.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_039.png", "page_index": 133, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:35+07:00" } }, "CO1007-chapter-3-slide-134-0000": { "id": "CO1007-chapter-3-slide-134-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Find the following values f(1), f(2) and f(3) knowing that f(n) is defined in the a recursive way with f(0) = 1 and Contents f(n+1)= f(n)2+f(n)+1,n=0,1,2,:: Proving Methods Exercise A 1, 3,13 3 1,3,10 3,10, 111 3, 13,173 E) None of the above 3.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_040.png", "page_index": 134, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:37+07:00" } }, "CO1007-chapter-3-slide-135-0000": { "id": "CO1007-chapter-3-slide-135-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Find the following values f(1), f(2) and f(3) knowing that f(n) is defined in the a recursive way with f(0) = 1 and Contents f(n+1)= f(n)2+f(n)+1, n=0,1,2,:. Proving Methods Exercise A 1, 3,13 3 1,3,10 3,10, 111 3, 13,173 E None of the above correct answer! 3.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_041.png", "page_index": 135, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:40+07:00" } }, "CO1007-chapter-3-slide-136-0000": { "id": "CO1007-chapter-3-slide-136-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai A sequence {t(n)}n is defined by a recursive formula: BK TP.HCM t(n) +t(n-1) -6t(n-2)=0 (t3) Knowing that t(1) = 1, t(2) = 3. The Explicit formula for the Contents Proving Methods sequence is: Exercise 3 1 I5m15 2 15 15 t(n) = 3 :(-2 5 t(n) =-3. 2n 3 E) None of the above 3.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_042.png", "page_index": 136, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:43+07:00" } }, "CO1007-chapter-3-slide-137-0000": { "id": "CO1007-chapter-3-slide-137-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai A sequence {t(n)}n is defined by a recursive formula BK TP.HCM t(n) +t(n-1) -6t(n-2)=0 (t3) Knowing that t(1) = 1, t(2) = 3. The Explicit formula for the Contents sequence is: Proving Methods Exercise 3 1 -3)n correct answer! 15m15 15 15 t(n) =3 :(-2) t(n) =-3. 2n 3 E) None of the above 3.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_043.png", "page_index": 137, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:46+07:00" } }, "CO1007-chapter-3-slide-138-0000": { "id": "CO1007-chapter-3-slide-138-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Find the first six numbers of the sequence defined by the following TP.HCM recursive formula Contents ao =-1,an =-2an-1 Proving Methods Exercise A -1,2, -3, 4, -5. 6 3 1,2, 4,8,16,32 2.-4.8.-16.32.-64 -1, 2, -4, 8, -16, 32 E) None of the above 3.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_044.png", "page_index": 138, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:49+07:00" } }, "CO1007-chapter-3-slide-139-0000": { "id": "CO1007-chapter-3-slide-139-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Find the first six numbers of the sequence defined by the following TP.HCM recursive formula Contents ao =-1,an =-2an-1 Proving Methods Exercise 4 -1, 2, -3, 4, -5, 6 3 1,2, 4,8, 16, 32 2. -4. 8. -16. 32. -64 -1, 2, -4, 8, -16, 32 correct answer! E) None of the above 3.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_045.png", "page_index": 139, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:51+07:00" } }, "CO1007-chapter-3-slide-140-0000": { "id": "CO1007-chapter-3-slide-140-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK Recognize the pattern of the following integer sequence then find TP.HCM the next 4 numbers of the sequence. Contents 1,0.2,2,0,0,4,4,4,0,0,0,:. Proving Methods Exercise Not exist 8,8,8,8 6,6,6,6 Either 8,8,8,8 or 0,8,8,8 None of the above 3.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_046.png", "page_index": 140, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:53+07:00" } }, "CO1007-chapter-3-slide-141-0000": { "id": "CO1007-chapter-3-slide-141-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK Recognize the pattern of the following integer sequence then find TP.HCM the next 4 numbers of the sequence. Contents 1,0.2,2,0,0,4,4,4,0,0,0,:. Proving Methods Exercise Not exist 8,8,8,8 6,6,6,6 Either 8,8,8,8 or 0,8,8,8 correct answer! None of the above 3.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_047.png", "page_index": 141, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:56+07:00" } }, "CO1007-chapter-3-slide-142-0000": { "id": "CO1007-chapter-3-slide-142-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Which of the following statement is correct? Contents A) A number is rational if and only if its square is a rational Proving Methods number. Exercise B integer n is odd if and only if n2 + 2n is odd. number is irrational if and only if its square is an irrational number. A number n is odd if and only if m(n + 1) is even E) None of the above 3.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_048.png", "page_index": 142, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:04:59+07:00" } }, "CO1007-chapter-3-slide-143-0000": { "id": "CO1007-chapter-3-slide-143-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Which of the following statement is correct? Contents A) A number is rational if and only if its square is a rational Proving Methods number. Exercise integer n is odd if and only if n2 + 2n is odd. correct answer! number is irrational if and only if its square is an irrational number. A number n is odd if and only if n(n + 1) is even E) None of the above 3.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_049.png", "page_index": 143, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:01+07:00" } }, "CO1007-chapter-3-slide-144-0000": { "id": "CO1007-chapter-3-slide-144-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai When the sun rises, two models X and Y start walk along the BK TP.HCM seashore. X walk from A to B, and the other from B to A. At 12PM, the two meet the other, and continue walking with the Contents same velocities. The first person arrives at B at 4M, and the other Proving Methods arrives at A at 9PM Exercise When did the sun rise? A) 5:30 AM 6:00 AM 6:30 AM 7:00 AM E) None of the above 3.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_050.png", "page_index": 144, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:04+07:00" } }, "CO1007-chapter-3-slide-145-0000": { "id": "CO1007-chapter-3-slide-145-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai When the sun rises, two models X and Y start walk along the BK TP.HCM seashore. X walk from A to B, and the other from B to A. At 12PM, the two meet the other, and continue walking with the Contents same velocities. The first person arrives at B at 4M, and the other Proving Methods arrives at A at 9PM. Exercise When did the sun rise? A 5:30 AM 6:00 AM correct answer! 6:30 AM 7:00 AM E None of the above 3.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_051.png", "page_index": 145, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:07+07:00" } }, "CO1007-chapter-3-slide-146-0000": { "id": "CO1007-chapter-3-slide-146-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK TP.HCM The Fibonacci numbers form the sequence of 1, 2, 3, 5, 8, 13, 21 34, ..., with the following definition: a1 = 1, a2 = 2, va Contents Proving Methods and ag9 is: Exercise A 1 B 218922995834555169026 None of the above 3.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_052.png", "page_index": 146, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:10+07:00" } }, "CO1007-chapter-3-slide-147-0000": { "id": "CO1007-chapter-3-slide-147-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK TP.HCM The Fibonacci numbers form the sequence of 1, 2, 3, 5, 8, 13, 21 34, ..., with the following definition: a1 = 1, a2 = 2, va Contents Proving Methods and ag9 is: Exercise A) 1 correct answer! 2 218922995834555169026 None of the above 3.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_053.png", "page_index": 147, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:13+07:00" } }, "CO1007-chapter-3-slide-148-0000": { "id": "CO1007-chapter-3-slide-148-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai In the Far Far Away kingdom, money is rarely used. People are BK TP.HCM often trading with their goods or with coins of value 5g or 7g. Is every good that has the price of 29g and above tradable using the above coins? Contents Proving Methods Choose the most correct answer for the above question Exercise A No The question should be \"goods with the price above 25g Should be \"goods with the price above 35g' Should change the question to \"goods with any price' E Yes None of the above 3.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_054.png", "page_index": 148, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:16+07:00" } }, "CO1007-chapter-3-slide-149-0000": { "id": "CO1007-chapter-3-slide-149-0000", "text": "Exercise Proving methods Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai In the Far Far Away kingdom, money is rarely used. People are BK TP.HCM often trading with their goods or with coins of value 5g or 7g. Is every good that has the price of 29g and above tradable using the above coins? Contents Proving Methods Choose the most correct answer for the above question Exercise A No 3 The question should be \"goods with the price above 25g Should be \"goods with the price above 35g' Should change the question to \"goods with any price' Yes correct answer! None of the above 3.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_3/slide_055.png", "page_index": 149, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:19+07:00" } }, "CO1007-chapter-4-slide-150-0000": { "id": "CO1007-chapter-4-slide-150-0000", "text": "Sets Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 4 Xuan Toan, Tran Hong Tai Sets BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Sets Set Operation Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 4.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_001.png", "page_index": 150, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:22+07:00" } }, "CO1007-chapter-4-slide-151-0000": { "id": "CO1007-chapter-4-slide-151-0000", "text": "Contents Sets Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK. TP.HCM Sets Contents Sets Set Operation Set Operation 4.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_002.png", "page_index": 151, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:24+07:00" } }, "CO1007-chapter-4-slide-152-0000": { "id": "CO1007-chapter-4-slide-152-0000", "text": "Course outcomes Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Contents Represent and model practical problems with discrete structures L.O.2.1 - Logically describe some problems arising in Computing Sets L.O.2.2 - Use proving methods: direct, contrapositive, induction Set Operation L.O.2.3 - Explain problem modeling using discrete structures L.0.3 Understanding of basic probability and random variables L.O.3.1 - Define basic probability theory L.O.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.0.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 4.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_003.png", "page_index": 152, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:29+07:00" } }, "CO1007-chapter-4-slide-153-0000": { "id": "CO1007-chapter-4-slide-153-0000", "text": "Set Definition Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Set is a fundamental discrete structure on which all discrete Tai structures are built Sets are used to group objects, which often have the same BK properties TP.HCM Example Contents Sets : Set of all the students who are currently taking Discrete Set Operation Mathematics 1 course. Set of all the subiects that K2011 students have to take in the first semester. Set of natural numbers N Definition A set is an unordered collection of objects The objects in a set are called the elements (phan tü) of the set A set is said to contain (chúa) its elements. 4.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_004.png", "page_index": 153, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:32+07:00" } }, "CO1007-chapter-4-slide-154-0000": { "id": "CO1007-chapter-4-slide-154-0000", "text": "Notations Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Definition BK TP.HCM a E A: a is an element of the set A a A: a is not an element of the set A Contents Sets Definition (Set Description) Set Operation The set V of all vowels in English alphabet, V ={a,e,&,o,u} Set of all real numbers greater than 1??? {xx e R,x > 1} {xx>1} {x :x>1} 4.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_005.png", "page_index": 154, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:35+07:00" } }, "CO1007-chapter-4-slide-155-0000": { "id": "CO1007-chapter-4-slide-155-0000", "text": "Equal Sets Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Definition TP.HCM Two sets are equal iff they have the same elements Contents Sets (A=B)x EB Set Operation Example {1,3,5}={3,5,1} {1,3,5}={1,3,3,3,5,5,5,5} 4.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_006.png", "page_index": 155, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:37+07:00" } }, "CO1007-chapter-4-slide-156-0000": { "id": "CO1007-chapter-4-slide-156-0000", "text": "Venn Diagram Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM John Venn in 1881 Universal set (tap vü tru) is Contents represented by a rectangle AnB Sets B Circles and other AnBnC Set Operation geometrical figures are used AnC BnC to represent sets Points are used to represent particular elements in set 4.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_007.png", "page_index": 156, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:40+07:00" } }, "CO1007-chapter-4-slide-157-0000": { "id": "CO1007-chapter-4-slide-157-0000", "text": "Special Sets Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Empty set (tap röng) has no elements, denoted by 0, or { Contents A set with one element is called a singleton set Sets Set Operation What is{0}? Answer: singleton 4.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_008.png", "page_index": 157, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:42+07:00" } }, "CO1007-chapter-4-slide-158-0000": { "id": "CO1007-chapter-4-slide-158-0000", "text": "Subset Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hon Tai Definition BK The set A is called a subset (tap con) of B iff every element of A TP.HCM is also an element of B, denoted by A C B. lf A B, we write A C B and say A is a proper subset (t@p con Contents Sets thuc su) of B. Set Operation U Vx(x E A-x E B) For every set S, O0CS,(sCS 4.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_009.png", "page_index": 158, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:45+07:00" } }, "CO1007-chapter-4-slide-159-0000": { "id": "CO1007-chapter-4-slide-159-0000", "text": "Cardinality Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Definition Tai If S has exactly m distinct elements where n is non-negative integers, S is finite set (tap hüu han), and m is cardinality (bar BK s6) of S, denoted by [Sl. TP.HCM Example Contents Sets A is the set of odd positive integers less than 10.A = 5. Set Operation S is the letters in Vietnamese alphabet,S = 29 Null set 0=O. Definition A set that is infinite if it is not finite Example : Set of positive integers is infinite 4.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_010.png", "page_index": 159, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:47+07:00" } }, "CO1007-chapter-4-slide-160-0000": { "id": "CO1007-chapter-4-slide-160-0000", "text": "Power Set Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK Given a set S, the power set (tap /üy thüa) of S is the set of all TP.HCM subsets of the set S, denoted by P(S) Contents Example Sets What is the power set of {0,1,2}? Set Operation P({0,1,2}={0,{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} Example What is the power set of the empty set? What is the power set of the set {0} 4.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_011.png", "page_index": 160, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:50+07:00" } }, "CO1007-chapter-4-slide-161-0000": { "id": "CO1007-chapter-4-slide-161-0000", "text": "Power Set Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Theorem Contents Sets If a set has n elements, then its power set has 2\" elements. Set Operation Prove using induction! 4.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_012.png", "page_index": 161, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:52+07:00" } }, "CO1007-chapter-4-slide-162-0000": { "id": "CO1007-chapter-4-slide-162-0000", "text": "Ordered n-tuples Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK The ordered n-tuple (day sap thu tu) (a1,a2,: ..,an) is the TP.HCM ordered collection that has a1 as its first element, a? as its second element, : . ., and an as its nth element. Contents Sets Definition Set Operation Two ordered n-tuples (a1, a2,. . ., an) = (b1, b2,. .., bn) iff ai = bi, for i=1,2, .. .,n. Example 2-tuples, or ordered pairs (cäp), (a,b) and (c,d) are equal iff a = c and b = d 4.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_013.png", "page_index": 162, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:56+07:00" } }, "CO1007-chapter-4-slide-163-0000": { "id": "CO1007-chapter-4-slide-163-0000", "text": "Cartesian Product Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai René Descartes (1596-1650) Definition BK TP.HCM Let A and B be sets. The Cartesian product (tich De-cäc) of A and B, denoted by A x B, is the set of ordered pairs (a,b), where Contents a E A and b E B. Hence, Sets Set Operation AxB={(a,b)aEAbEB} Example Cartesian product of A={1,2} and B ={a,b,c}: Then AxB={(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)} Show that A x B B x A 4.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_014.png", "page_index": 163, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:05:59+07:00" } }, "CO1007-chapter-4-slide-164-0000": { "id": "CO1007-chapter-4-slide-164-0000", "text": "Sets Cartesian Product Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK TP.HCM AxA2x...xAn={(a1,a2,...,an)ai E A; for i=1,2,...,n} Contents Sets Set Operation Example A={0,1},B={1,2},C={0,1,2}.What is AxB x C? A x B x C= {(0,1,0,(0,1,1),(0,1,2),(0,2,0),(0,2,1 0,2,2),(1,1,0,(1,1,1,(1,1,2),(1,2,0 (1,2,1),(1,2,2)} 4.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_015.png", "page_index": 164, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:01+07:00" } }, "CO1007-chapter-4-slide-165-0000": { "id": "CO1007-chapter-4-slide-165-0000", "text": "Union Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Definition The union (hop) of A and B BK TP.HCM AUB={xxEAV xEB} Contents Sets A U B Set Operation A B Example: {1,2,3} U{2,4} ={1,2,3,4} {1,2,3} U0={1,2,3} 4.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_016.png", "page_index": 165, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:03+07:00" } }, "CO1007-chapter-4-slide-166-0000": { "id": "CO1007-chapter-4-slide-166-0000", "text": "Intersection Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition The intersection (giao) of A and B BK TP.HCM ANB={xxEAxEB} Contents Sets An B Set Operation A B Example: {1,2,3}n{2,4}={2} {1,2,3} nN={1,2,3} 4.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_017.png", "page_index": 166, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:06+07:00" } }, "CO1007-chapter-4-slide-167-0000": { "id": "CO1007-chapter-4-slide-167-0000", "text": "Union/1ntersection Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM n Contents JAi=A1 UAz U..UAn={xx E A1 V x E A2 V..V x E A Sets i=1 Set Operation n Ai=A1 NA2 N..NAn={xx E A1Ax E A2 A.Ax E An} i=1 4.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_018.png", "page_index": 167, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:09+07:00" } }, "CO1007-chapter-4-slide-168-0000": { "id": "CO1007-chapter-4-slide-168-0000", "text": "Difference Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition The difference (hi@u) of A and B BK TP.HCM A-B={xxEAx4B} Contents Sets A B Set Operation A B Example: {1,2,3}-{2,4}={1,3} {1,2,3}-N=0 4.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_019.png", "page_index": 168, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:11+07:00" } }, "CO1007-chapter-4-slide-169-0000": { "id": "CO1007-chapter-4-slide-169-0000", "text": "Complement Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition The complement (phan ba) of A BK TP.HCM A={xx4A} Contents Sets Example: Set Operation A={1,2,3} then A=??? Note that A - B = A O B 4.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_020.png", "page_index": 169, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:13+07:00" } }, "CO1007-chapter-4-slide-170-0000": { "id": "CO1007-chapter-4-slide-170-0000", "text": "Set ldentities Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK A U 0 = A Identity laws TP.HCM An U = A Luat dóng nhät Contents A U U = U Domination laws Sets An 0 = 0 Luat nuót Set Operation AUA = A Idempotent laws AA = A Luat l&y d&ng (A) = A Complementation law Luat bü 4.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_021.png", "page_index": 170, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:17+07:00" } }, "CO1007-chapter-4-slide-171-0000": { "id": "CO1007-chapter-4-slide-171-0000", "text": "Set ldentities Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hon Tai BK AUB BU A Commutative laws TP.HCM An B = B n A Luat giao hoán Contents AU(BUC) = AUB)UC Associative laws Sets An(BnC) = AnB)nC Luat két hop Set Operation AU(BNC) = (AUB)N(AUC Distributive laws An(BUC) = AnB)U(AnC) Luat phan phói A U B = A n B De Morgan's laws AN B AU B Luat De Morgan 4.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_022.png", "page_index": 171, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:21+07:00" } }, "CO1007-chapter-4-slide-172-0000": { "id": "CO1007-chapter-4-slide-172-0000", "text": "Method of Proofs of Set Equations Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM To prove A = B,we could use Contents : Venn diagrams Sets Prove that A C B and B C A Set Operation Use membership table Use set builder notation and logical equivalences 4.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_023.png", "page_index": 172, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:23+07:00" } }, "CO1007-chapter-4-slide-173-0000": { "id": "CO1007-chapter-4-slide-173-0000", "text": "Example (1) Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK U TP.HCM 4 P 6 2 8 7 Contents 5 Sets 3 R Set Operation Example Verify the distributive rule P U (QN R) = (P U Q) N(P U R 4.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_024.png", "page_index": 173, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:27+07:00" } }, "CO1007-chapter-4-slide-174-0000": { "id": "CO1007-chapter-4-slide-174-0000", "text": "Example (2) Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM Prove: An B = A U B 1) Show that AN B C A U B Contents Sets Suppose that x E AN B Set Operation By the definition of complement, A B So,x 4 Aor x 4 B Hence, x E A or x E B We conclude,x E AU B Or,AnBCAUB 2) Show that A U B C AN B 4.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_025.png", "page_index": 174, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:30+07:00" } }, "CO1007-chapter-4-slide-175-0000": { "id": "CO1007-chapter-4-slide-175-0000", "text": "Example (3) Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Prove: AO B = AUB A B An B ANB AUB Contents Sets 1 1 1 O Set Operation 1 0 0 1 1 0 1 0 1 1 0 0 1 1 4.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_026.png", "page_index": 175, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:33+07:00" } }, "CO1007-chapter-4-slide-176-0000": { "id": "CO1007-chapter-4-slide-176-0000", "text": "Example (4) Sets Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Prove: AnB = AU B TP.HCM An B ={xx Z ANB} Contents ={x-(x E ANB)} Sets {x-(x E Ax E B} Set Operation {x-(x E A V-(x E B} {xx Z AVx ZB} {xx EAV x EB} ={xx EAUB} 4.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_4/slide_027.png", "page_index": 176, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:35+07:00" } }, "CO1007-chapter-5-slide-177-0000": { "id": "CO1007-chapter-5-slide-177-0000", "text": "Functions Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 5 Xuan Toan. Tran Hong Tai Functions BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Functions One-to-one and Ontc Functions Sequences and Summation Recursion Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 5.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_001.png", "page_index": 177, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:38+07:00" } }, "CO1007-chapter-5-slide-178-0000": { "id": "CO1007-chapter-5-slide-178-0000", "text": "Contents Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Functions TP.HCM Contents One-to-one and Onto Functions Functions One-to-one and Onto Functions Sequences and Summation Seguences and Summation Recursion Recursion 5.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_002.png", "page_index": 178, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:40+07:00" } }, "CO1007-chapter-5-slide-179-0000": { "id": "CO1007-chapter-5-slide-179-0000", "text": "Course outcomes Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Represent and model practical problems with discrete structures Contents L.O.2.1 - Logically describe some problems arising in Computing Functions L.O.2.2 - Use proving methods: direct, contrapositive, induction One-to-one and Onto Functions L.O.2.3 - Explain problem modeling using discrete structures Sequences and Summation L.0.3 Understanding of basic probability and random variables Recursion L.0.3.1 - Define basic probability theory L.0.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.0.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 5.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_003.png", "page_index": 179, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:44+07:00" } }, "CO1007-chapter-5-slide-180-0000": { "id": "CO1007-chapter-5-slide-180-0000", "text": "Introduction Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM Each student is assigned a grade from set {0,0.1,0.2,0.3, -..,9.9,10.0} at the end of semester Contents Function is extremely important in mathematics ang Functions computer science One-to-one and Onto Functions . linear, polynomial, exponential, logarithmic,.. Sequences and Summation Don't worry! For discrete mathematics, we need to Recursion understand functions at a basic set theoretic level 5.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_004.png", "page_index": 180, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:48+07:00" } }, "CO1007-chapter-5-slide-181-0000": { "id": "CO1007-chapter-5-slide-181-0000", "text": "Function Functions Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. BK f:A-B TP.HCM A: domain (mién xac dinh) of f B: codomain (mien gia tri) of f Contents For each a E A,if f(a) =b Functions : b is an image (anh) of a One-to-one and Onto Functions : a is pre-image (nghich anh) of f(a) Sequences and Range of f is the set of all images of elements of A Summation Recursion f maps (anh xa) A to B a A B 5.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_005.png", "page_index": 181, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:51+07:00" } }, "CO1007-chapter-5-slide-182-0000": { "id": "CO1007-chapter-5-slide-182-0000", "text": "Example Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Functions One-to-one and Onto Functions Sequences and Summation Recursion y is an image of d c is a pre-image of z 5.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_006.png", "page_index": 182, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:53+07:00" } }, "CO1007-chapter-5-slide-183-0000": { "id": "CO1007-chapter-5-slide-183-0000", "text": "Example Functions Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK What are domain, codomain, and range of the function that TP.HCM assigns grades to students includes: student A: 5, B: 3.5, C: 9, D: 5.2, E: 4.9? Contents Functions Example One-to-one and Onto Functions Let f : Z - Z assign the the square of an integer to this integer. Sequences and What is f(x)? Domain, codomain, range of f? Summation Recursion : f(x)=x2 Domain: set of all integers Codomain: Set of all integers Range of f :{0,1,4,9,...} 5.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_007.png", "page_index": 183, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:06:56+07:00" } }, "CO1007-chapter-5-slide-184-0000": { "id": "CO1007-chapter-5-slide-184-0000", "text": "Add and multiply real-valued functions Functions Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition Let f1 and f2 be functions from A to R. Then f1 + f2 and f1f2 BK are also functions from A to R defined by TP.HCM (f1+ f2)(x)= f1(x) + fz(x) Contents f1f2)(x) = f1(x)f2(x) Functions One-to-one and Onto Functions Sequences and Example Summation Recursion Let f1(x) = x2 and f2(x) = x - x2. What are the functions f1 + f2 and f1fz? (fi+f2)(x) =f1(x) + f2(x) =x2+x-x2=x (f1f2)(x) = f1(x)f2(x)=x2(x-x2)=x3-x4 5.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_008.png", "page_index": 184, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:00+07:00" } }, "CO1007-chapter-5-slide-185-0000": { "id": "CO1007-chapter-5-slide-185-0000", "text": "mage of a subset Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition Let f : A - B and S C A. The image of S BK f(S)={f(s)sES} TP.HCM Contents Functions One-to-one and Ontc Functions Sequences and Summation Recursion f({a,b,c,d})={x,y,z} 5.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_009.png", "page_index": 185, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:02+07:00" } }, "CO1007-chapter-5-slide-186-0000": { "id": "CO1007-chapter-5-slide-186-0000", "text": "One-to-one Functions Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai A function f is one-to-one or injective (don änh) if and only if VaVb(f(a)= f(b) ->a=b) BK TP.HCM Contents Functions One-to-one and Onto Functions Sequences and Summation Recursion s f:Z-Z,f(x)=x+1 Injective one-to-one? :s f:Z-Z,f(x)=x2 one-to-one? Non-injective 5.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_010.png", "page_index": 186, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:05+07:00" } }, "CO1007-chapter-5-slide-187-0000": { "id": "CO1007-chapter-5-slide-187-0000", "text": "Onto Functions Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai f : A - B is onto or surjective (toan anh) if and only if VbEB,3aEA: f(a)=b BK TP.HCM Contents Functions One-to-one and Onto Functions Sequences and Summation s f:Z-Z,f(x)=x+1 Recursion Surjective onto? s f:Z-Z,f(x)=x2 onto? Non-surjective 5.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_011.png", "page_index": 187, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:08+07:00" } }, "CO1007-chapter-5-slide-188-0000": { "id": "CO1007-chapter-5-slide-188-0000", "text": "One-to-one and onto (bijection) Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Definition TP.HCM f : A -> B is bijective (one-to-one correspondence) (song anh) if and only if f is injective and surjective Contents Functions x Y One-to-one and Onto Functions .D Let f be the function from Sequences and {1,2,3,4} to{D,B,C,A} Summation 2 B withf(1) =D, f(2) =B Recursion 3 *C f(3) =C, f(4) =A. :A Is f a bijection? 5.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_012.png", "page_index": 188, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:11+07:00" } }, "CO1007-chapter-5-slide-189-0000": { "id": "CO1007-chapter-5-slide-189-0000", "text": "Example Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai R A A a a BK >d TP.HCM h 3 Contents Functions One-to-one and Onto Functions Sequences and Summation x Y A Recursion 1 D a B C 3 C c 4 5.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_013.png", "page_index": 189, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:14+07:00" } }, "CO1007-chapter-5-slide-190-0000": { "id": "CO1007-chapter-5-slide-190-0000", "text": "Inverse function (Ham ngu'oc) Functions Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Let f : A -> B be a bijection then the inverse of f is the function Tai f-1 :B -> A defined by BK if f(a) = b then f-1(b) = a TP.HCM A one-to-one correspondence is call invertible (kha nghich) Contents because we can define the inverse of this function. Functions One-to-one and Onto Functions f(a) Sequences and Summation Recursion f-1(b) a= 6 a A f B 5.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_014.png", "page_index": 190, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:17+07:00" } }, "CO1007-chapter-5-slide-191-0000": { "id": "CO1007-chapter-5-slide-191-0000", "text": "Example Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM A={a,b,c} and B={1,2,3} with f(a) =2 f(b) = 3 f(c) =1 Contents Functions f is invertible and its inverse is One-to-one and Onto Functions Sequences and f-1(1) =c f-1(2) = a f-1(3) =b Summation Recursion Example Let f : R-> R with f(x) = x2.If f invertible? 5.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_015.png", "page_index": 191, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:21+07:00" } }, "CO1007-chapter-5-slide-192-0000": { "id": "CO1007-chapter-5-slide-192-0000", "text": "Example Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM f:R-R Contents f(x) =2x+1 Functions One-to-one and Onto Functions Sequences and f-l:R-R Summation Recursion x -1 2 5.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_016.png", "page_index": 192, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:23+07:00" } }, "CO1007-chapter-5-slide-193-0000": { "id": "CO1007-chapter-5-slide-193-0000", "text": "Function Composition Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Definition Given a pair of functions g : A - B and f : B - C. Then the Contents composition (hop thanh) of f and g, denoted f o g is defined by Functions One-to-one and Onto f og:A-C Functions Sequences and Summation f o g(a) = f(g(a)) Recursion 5.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_017.png", "page_index": 193, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:25+07:00" } }, "CO1007-chapter-5-slide-194-0000": { "id": "CO1007-chapter-5-slide-194-0000", "text": "Example Functions Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong X Y Z Tai BK TP.HCM Contents Functions X One-to-one and Onto P Functions Sequences and Summation R Recursion 2 5.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_018.png", "page_index": 194, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:28+07:00" } }, "CO1007-chapter-5-slide-195-0000": { "id": "CO1007-chapter-5-slide-195-0000", "text": "Graphs of Functions Functions Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example The graph of f(x) = x2 from Z to Z BK TP.HCM (-3,9) (3,9) Contents Functions (-2,4) (2,4) One-to-one and Onto Functions Sequences and Summation : (1,1) Recursion 0,0 Definition Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs{(a,b) a E A and f(a) = b} 5.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_019.png", "page_index": 195, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:07:31+07:00" } }, "CO1007-chapter-5-slide-196-0000": { "id": "CO1007-chapter-5-slide-196-0000", "text": "Important Functions Functions Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Floor function (ham sän) of x (x D: the largest integer < x ] = 0, [3.1] = 3,[7] = 7 Ceiling function (ham tran) of x (xD: the smallest integer > x BK TP.HCM 1 =1,[3.1]= 4,[7] = 7 Contents Functions Bang: Properties (n is an integer, x is a real number) One-to-one and Onto Functions (1a) x=n iff n < x < n+1 Sequences and (1b) =n iff n-1< x < n Summation x Recursion (1c) x= n iff c -1< n< x (1d) c]= n iff x < n< x +1 (2) 1. Recurrence Solving H(n) = 2n -1 If one move takes 1 second, for n = 64 264-1 2 x 1019 sec 500 billion years! 5.65", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_5/slide_066.png", "page_index": 242, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:34+07:00" } }, "CO1007-chapter-6-slide-243-0000": { "id": "CO1007-chapter-6-slide-243-0000", "text": "Relations Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 6 Xuan Toan, Tran Hong Tai Relations BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 6.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_001.png", "page_index": 243, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:37+07:00" } }, "CO1007-chapter-6-slide-244-0000": { "id": "CO1007-chapter-6-slide-244-0000", "text": "Contents Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Properties of Relations BK TP.HCM Combining Relations Contents Properties of Relations Combining Relations 3 Representing Relations Representing Relations Closures of Relations Types of Relations Closures of Relations 5) Types of Relations 6.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_002.png", "page_index": 244, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:40+07:00" } }, "CO1007-chapter-6-slide-245-0000": { "id": "CO1007-chapter-6-slide-245-0000", "text": "Course outcomes Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Contents Represent and model practical problems with discrete structures L.O.2.1 - Logically describe some problems arising in Computing Properties of Relations L.O.2.2 - Use proving methods: direct, contrapositive, induction Combining Relations L.O.2.3 - Explain problem modeling using discrete structures Representing Relations Closures of Relations L.0.3 Understanding of basic probability and random variables Types of Relations L.O.3.1 - Define basic probability theory L.0.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.O.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 6.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_003.png", "page_index": 245, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:44+07:00" } }, "CO1007-chapter-6-slide-246-0000": { "id": "CO1007-chapter-6-slide-246-0000", "text": "Introduction Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents O Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations set-maricd WOMAN MAN Function? 6.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_004.png", "page_index": 246, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:46+07:00" } }, "CO1007-chapter-6-slide-247-0000": { "id": "CO1007-chapter-6-slide-247-0000", "text": "Relation Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK Let A and B be sets. A binary relation (quan he hai ngi) from a TP.HCM set A to a set B is a set Contents R C A x B Properties of Relations Combining Relations Representing Relations Notations: Closures of Relations (a,b) ER<>aRb Types of Relations n-ary relations? 6.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_005.png", "page_index": 247, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:49+07:00" } }, "CO1007-chapter-6-slide-248-0000": { "id": "CO1007-chapter-6-slide-248-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Let A={a,b,c} be the set of students, B ={l,c,s,d} be the set BK of the available optional courses. We can have relation R that TP.HCM consists of pairs (a,b), where a is a student enrolled in course b. Contents Properties of Relations Combining Relations a R {(a,l),(a,s),(a,g),(b,c) Representing Relations = Closures of Relations (b,s),(b,g),(c,l),(c,g)} Types of Relations R 1 c S g a x x x b x x x c x x D S 6.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_006.png", "page_index": 248, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:53+07:00" } }, "CO1007-chapter-6-slide-249-0000": { "id": "CO1007-chapter-6-slide-249-0000", "text": "Functions as Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Is a function a relation? Contents Yes! Properties of Relations Combining Relations f: A-B Representing Relations R={(a,b)b=f(a} Closures of Relations Types of Relations 6.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_007.png", "page_index": 249, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:56+07:00" } }, "CO1007-chapter-6-slide-250-0000": { "id": "CO1007-chapter-6-slide-250-0000", "text": "Functions as Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Is a relation a function? BK No TP.HCM Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations A B Relations are a generalization of functions 6.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_008.png", "page_index": 250, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:09:58+07:00" } }, "CO1007-chapter-6-slide-251-0000": { "id": "CO1007-chapter-6-slide-251-0000", "text": "Relations on a Set Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A relation on the set A is a relation from A to A BK TP.HCM Example Let A be the set {1,2,3.4}. Which ordered pairs are in the Contents relation R ={(a,b) a divides b} (a la uóc s cüa b)? Properties of Relations Combining Relations Solution: Representing Relations Closures of Relations R={(1,1),(1,2),(1,3),(1,4,(2,2),(2,4),(3,3),(4,4} Types of Relations R 1 2 3 4 1 x x x x 2 x x 3 x 4 x 6.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_009.png", "page_index": 251, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:02+07:00" } }, "CO1007-chapter-6-slide-252-0000": { "id": "CO1007-chapter-6-slide-252-0000", "text": "Properties of Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Reflexive xRx,Vx E A (phan xa) Symmetric xRy - yRx,Vx,y E A Contents Properties of Relations (dói xung) Combining Relations Antisymmetric (xRyyRx)-> x =y,Vx,y E A Representing Relations (phän dói xúng) Closures of Relations Transitive (xRyyRz)-xRz,Vx,y,z E A Types of Relations (bac cau) 6.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_010.png", "page_index": 252, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:05+07:00" } }, "CO1007-chapter-6-slide-253-0000": { "id": "CO1007-chapter-6-slide-253-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example BK Consider the following relations on {1,2,3,4} : TP.HCM Ri={(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} R2={(1,1),(1,2),(2,1)}, Contents R3={(1,1),(1,2),(1,4),(2,1),(2,2),(3,3),(4,1),(4,4)} Properties of Relations R4={(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} Combining Relations R5 ={(3,4)} Representing Relations Solution: Closures of Relations Types of Relations Reflexive: R3 Symmetric: R2, R3 Antisymmetric: R4, R5 Transitive: R4, R5 6.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_011.png", "page_index": 253, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:08+07:00" } }, "CO1007-chapter-6-slide-254-0000": { "id": "CO1007-chapter-6-slide-254-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM What is the properties of the divides (uöc s) relation on the set of positive integers? Contents Solution: Properties of Relations Combining Relations Va E Z+,aa: reflexive Representing Relations o 1 2, but 2 1: not symmetric Closures of Relations Types of Relations 3a,b E Z+,(a b) (ba) - a = b: antisymmetric ab=3k e Z+,b=ak;bc=3l e Z+,c=bl.Hence, c = a(kl) => ac: transitive 6.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_012.png", "page_index": 254, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:11+07:00" } }, "CO1007-chapter-6-slide-255-0000": { "id": "CO1007-chapter-6-slide-255-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK TP.HCM Example Contents What are the properties of these relations on the set of integers Properties of Relations R1={(a,b)ab} Representing Relations Closures of Relations R3 ={(a,b)a=b or a=-b} Types of Relations 6.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_013.png", "page_index": 255, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:14+07:00" } }, "CO1007-chapter-6-slide-256-0000": { "id": "CO1007-chapter-6-slide-256-0000", "text": "Combining Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Because relations from A to B are subsets of A x B, two relations from A to B can be combined in any way two sets can be combined. BK TP.HCM Example Let A={1,2,3} and B={1,2,3,4}. List the combinations of Contents relations R1 ={(1,1),(2,2),(3,3)} and Properties of Relations R2={(1,1),(1,2),(1,3),(1,4} Combining Relations Representing Relations Solution: R U R2, R N R2, R1 - R2 and R2 - R1 Closures of Relations Types of Relations Example Let A and B be the set of all students and the set of all courses at school, respectively. Suppose R1 = {(a,b) a has taken the course b} and R2 ={(a,b) a requires course b to graduate}. What are the relations R UR2, R1 NR2,R1 O R2, R1 - R2, R2 - R? 6.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_014.png", "page_index": 256, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:18+07:00" } }, "CO1007-chapter-6-slide-257-0000": { "id": "CO1007-chapter-6-slide-257-0000", "text": "Composition of Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK TP.HCM Let R be relations from A to B and S be from B to C. Then the composite (hgp thanh) of S and R is Contents S o R={(a,c) E A x C3b E B(aRbbSc)} Properties of Relations Combining Relations Representing Relations Closures of Relations Example Types of Relations R={(0,0,(0,3,(1,2),(0,1} S={(0,0),(1,0),(2,1),(3,1)} SoR={(0,0,(0,1,(1,1} 6.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_015.png", "page_index": 257, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:21+07:00" } }, "CO1007-chapter-6-slide-258-0000": { "id": "CO1007-chapter-6-slide-258-0000", "text": "Power of Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition Let R be a relation on the set A. The powers (/üy thüa) BK R\",n = 1, 2,3, : . . are defined recursively by TP.HCM Rl =R Rn+1 = R\" . R. and Contents Properties of Relations Combining Relations Example Representing Relations Let R={(1,1),(2,1),(3,2),(4,3)}.Find the powers Closures of Relations R\",n=2,3,4,. Types of Relations Solution: R2={(1,1),(2,1),(3,1),(4,2)} R3 ={(1,1),(2,1),(3,1),(4,1)} R4={(1,1),(2,1),(3,1),(4,1)} 6.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_016.png", "page_index": 258, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:23+07:00" } }, "CO1007-chapter-6-slide-259-0000": { "id": "CO1007-chapter-6-slide-259-0000", "text": "Representing Relations Using Matrices Relations Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai Suppose R is a relation from A ={a1,a2,. .., am} to B ={b1,b2,...,bn}, R can be represented by the matrix MR =mij],where BK TP.HCM if(ai,bj) e R mij 0 if(ai,bj) 4 R Contents Properties of Relations Combining Relations Example Representing Relations Closures of Relations R is relation from A={1,2,3} to B ={1,2} Let Types of Relations R=3(2,1),(3,1).(3,2}the matrix for R is 0 MR = 1 0 1 Determine whether the relation has certain properties (reflexive, symmetric, antisymmetric,... 6.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_017.png", "page_index": 259, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:28+07:00" } }, "CO1007-chapter-6-slide-260-0000": { "id": "CO1007-chapter-6-slide-260-0000", "text": "Representing Relations Using Digraphs Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition BK Suppose R is a relation in A ={a1,a2, TP.HCM .,am}, R can be represented by the digraph (d6 thi c6 huóng) G = (V,E), where Contents V = A Properties of Relations (ai,aj) E E if(ai,aj) E R Combining Relations Representing Relations Closures of Relations Example Types of Relations Given a relation on A ={1,2,3,4} R={(1,1),(1,3),(2,1),(2,3),(2,4),(3,1),(3,2),(4,1)} Draw corresponding digraph 6.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_018.png", "page_index": 260, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:31+07:00" } }, "CO1007-chapter-6-slide-261-0000": { "id": "CO1007-chapter-6-slide-261-0000", "text": "Resulting digraph Relations Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Properties of Relations Combining Relations Representing Relations 2 Closures of Relations Types of Relations 3 6.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_019.png", "page_index": 261, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:33+07:00" } }, "CO1007-chapter-6-slide-262-0000": { "id": "CO1007-chapter-6-slide-262-0000", "text": "Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Definition The closure (bao dóng) of relation R with respect to property P Contents is the relation S that Properties of Relations i. contains R Combining Relations Representing Relations Ii. has property P Closures of Relations iii. is contained in any relation satisfying (i) and (ii) Types of Relations S is the \"smallest\" relation satisfying (i) & (ii) 6.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_020.png", "page_index": 262, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:36+07:00" } }, "CO1007-chapter-6-slide-263-0000": { "id": "CO1007-chapter-6-slide-263-0000", "text": "Reflexive Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example BK TP.HCM LetR={(a,b),(a,c),(b,d),(d,c} The reflexive closure of R Contents {(a,b),(a,c,(b,d),(d,c),(a,a,(b,b,(c,c,(d,d Properties of Relations Combining Relations Representing Relations RuA Clos ures of Relations Types of Relations where ={(a,a)aEA} diagonal relation (quan he dung chéo) 6.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_021.png", "page_index": 263, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:39+07:00" } }, "CO1007-chapter-6-slide-264-0000": { "id": "CO1007-chapter-6-slide-264-0000", "text": "Reflexive Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM a Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations 6.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_022.png", "page_index": 264, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:41+07:00" } }, "CO1007-chapter-6-slide-265-0000": { "id": "CO1007-chapter-6-slide-265-0000", "text": "Symmetric Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example BK TP.HCM Let R={(a,b),(a,c),(b,d),(c,a),(d,e)} The symmetric closure of R Contents {(a,b,(a,c),(b,d,(c,a),(d,e),(b,a),(d,b,(e,d} Properties of Relations Combining Relations Representing Relations RUR-1 Clos ures of Relations Types of Relations where R-1={(b,a)(a,b) e R} inverse relation (quan he nguoc 6.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_023.png", "page_index": 265, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:44+07:00" } }, "CO1007-chapter-6-slide-266-0000": { "id": "CO1007-chapter-6-slide-266-0000", "text": "Symmetric Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM a Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations 6.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_024.png", "page_index": 266, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:46+07:00" } }, "CO1007-chapter-6-slide-267-0000": { "id": "CO1007-chapter-6-slide-267-0000", "text": "Transitive Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Example LetR={(a,b),(a,c,(b,d),(d,e} Contents The transitive closure of R Properties of Relations Combining Relations {(a,b,(a,c,(b,d),(d,e,(a,d,(b,e),(a,e} Representing Relations Closures of Relations Types of Relations 6.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_025.png", "page_index": 267, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:49+07:00" } }, "CO1007-chapter-6-slide-268-0000": { "id": "CO1007-chapter-6-slide-268-0000", "text": "Transitive Closure Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM a Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations 6.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_026.png", "page_index": 268, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:51+07:00" } }, "CO1007-chapter-6-slide-269-0000": { "id": "CO1007-chapter-6-slide-269-0000", "text": "Equivalence Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition A relation on a set A is called an eguivalence relation (quan he BK tuong duong) if it is reflexive, symmetric and transitive. TP.HCM Example (1) Contents Properties of Relations The relation R = {(a,b)a and b are in the same provinces} is an Combining Relations equivalence relation. a is equivalent to b and vice versa, denoted Representing Relations a b. Closures of Relations Types of Relations Example (2) R={(a,b)a=bV a=-b} R is an equivalence relation 6.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_027.png", "page_index": 269, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:54+07:00" } }, "CO1007-chapter-6-slide-270-0000": { "id": "CO1007-chapter-6-slide-270-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example (3) BK TP.HCM R={(x,y)x-y<1} Contents Is R an equivalence relation? Properties of Relations Combining Relations Example (Congruence Modulo m - Döng du modulo m) Representing Relations Closures of Relations Let m be a positive integer with m > 1. Show that the relation Types of Relations R={(a,b)a=b(mod m)} is an equivalence relation on the set of integers 6.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_028.png", "page_index": 270, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:10:57+07:00" } }, "CO1007-chapter-6-slide-271-0000": { "id": "CO1007-chapter-6-slide-271-0000", "text": "Equivalence Classes Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition BK Let R be an eguivalence relation on the set A. The set of all TP.HCM elements that are related to an element a of A is called the equivalence class (lóp tuong duong) of a, denoted by Contents Properties of Relations [a]R={s(a,s)ER} Combining Relations Representing Relations Closures of Relations Types of Relations Example The equivalence class of \"Tan Phü\" for the equivalence relation \"in the same provinces\" is { Tan Phú, G Väp, Binh Thanh, Quan 10,...} 6.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_029.png", "page_index": 271, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:00+07:00" } }, "CO1007-chapter-6-slide-272-0000": { "id": "CO1007-chapter-6-slide-272-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Example TP.HCM What are the equivalence classes of 0, 1, 2, 3 for congruence modulo 4? Contents Properties of Relations Solution: Combining Relations [0]4={...,-8,-4,0,4,8,...} Representing Relations Closures of Relations 14={...,-7,-3,1,5,9,...} Types of Relations 24={...,-6,-2,2,6,10,...} 3]4={...,-5,-1, 3,7,11, ...} 6.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_030.png", "page_index": 272, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:03+07:00" } }, "CO1007-chapter-6-slide-273-0000": { "id": "CO1007-chapter-6-slide-273-0000", "text": "Equivalence Relations and Partitions Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK TP.HCM Theorem Let R be an equivalence relation on a set A. These statements for Contents elements a and b of A are equivalent: Properties of Relations Combining Relations i aRb Representing Relations ii [a]=[b] Closures of Relations ii[a]N[b] 0 Types of Relations 6.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_031.png", "page_index": 273, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:05+07:00" } }, "CO1007-chapter-6-slide-274-0000": { "id": "CO1007-chapter-6-slide-274-0000", "text": "Example 1 Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK Suppose that S ={1,2,3,4,5,6}. The collection of sets TP.HCM A1 ={1,2,3}, A2 ={4,5},and A3={6} forms a partition of S, because these sets are disjoint and their union is S Contents Properties of Relations Combining Relations Representing Relations The equivalence classes of an equivalence relation R on a set S Closures of Relations form a partition of S. Types of Relations Every partition of a set can be used to form an equivalence relation. 6.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_032.png", "page_index": 274, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:08+07:00" } }, "CO1007-chapter-6-slide-275-0000": { "id": "CO1007-chapter-6-slide-275-0000", "text": "Example 2 Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Divides set of all cities and towns in Vietnam into set of 64 BK provinces. We know that: TP.HCM there are no provinces with no cities or towns Contents Properties of Relations no city is in more than one Combining Relations province Representing Relations every city is accounted for Closures of Relations Types of Relations Definition A partition of a Vietnam is a collection of non-overlapping non-empty subsets of Vietnam (provinces) that, together, make up all of Vietnam. TE.LE 6.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_033.png", "page_index": 275, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:12+07:00" } }, "CO1007-chapter-6-slide-276-0000": { "id": "CO1007-chapter-6-slide-276-0000", "text": "Relation in a Partition Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM We divided based on relation R={(a,b)a and b are in the Contents Properties of Relations same provinces } Combining Relations \"Quan 10\" is related Representing Relations (equivalent) to \"G Vap\" Closures of Relations \"Da Lat\" is not related (not Types of Relations equivalent) to \"Long Xuyén\" TEIL 6.34", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_034.png", "page_index": 276, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:15+07:00" } }, "CO1007-chapter-6-slide-277-0000": { "id": "CO1007-chapter-6-slide-277-0000", "text": "Partial Order Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Order words such that x comes before y in the dictionary Schedule projects such that must be completed before y BK . Order set of integers, where x < y TP.HCM Definition Contents A relation R on a set S is called a partial ordering (c6 thu tu b Properties of Relations phan) if it is reflexive, antisymmetric and transitive. A set S Combining Relations together with a partial ordering R is called a partially ordered set, Representing Relations or poset (tap c6 thü tu b phan), and is denoted by (S,R) or Closures of Relations Types of Relations (S,S). Example (Z,>) is a poset Let S a set,(P(S),C) is a poset 6.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_035.png", "page_index": 277, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:18+07:00" } }, "CO1007-chapter-6-slide-278-0000": { "id": "CO1007-chapter-6-slide-278-0000", "text": "Example Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai {x,y,z} BK TP.HCM 4 Contents {x,y} {x,z} {y,z} Properties of Relations 4 Combining Relations Representing Relations Closures of Relations {x} {y} {z} Types of Relations 4 6.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_036.png", "page_index": 278, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:21+07:00" } }, "CO1007-chapter-6-slide-279-0000": { "id": "CO1007-chapter-6-slide-279-0000", "text": "Totally Order Relations Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example BK In the poset (Z+,D, 3 and 9 are comparable (so sanh duoc) TP.HCM because 3 9, but 5 and 7 are not, because 5 7 and 7 5 > That's why we call it partially ordering Contents Properties of Relations Combining Relations Definition Representing Relations lf (S,) is a poset and every two elements of S are comparable, S Closures of Relations is called a totally ordered (c6 thú tu toän phan). A totally Types of Relations ordered set is also called a chain (day xich) Example The poset (Z,<) is totally ordered 6.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_037.png", "page_index": 279, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:24+07:00" } }, "CO1007-chapter-6-slide-280-0000": { "id": "CO1007-chapter-6-slide-280-0000", "text": "Maximal & Minimal Elements Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK TP.HCM a is maximal (cuc dai) in the poset (S,) if there is no b E S such that a - b. Contents a is minimal (cuc tieu) in the poset (S,) if there is no Properties of Relations b E S such that b < a. Combining Relations Representing Relations Closures of Relations Example 12 20 maximal Types of Relations Which elements of the poset 10 25 {2,4,5,10,12,20,25},D are minimal and maximal? minimal 5 6.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_038.png", "page_index": 280, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:28+07:00" } }, "CO1007-chapter-6-slide-281-0000": { "id": "CO1007-chapter-6-slide-281-0000", "text": "Greatest Element& Least Element Relations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK TP.HCM a is the greatest element (/ón nhat) of the poset (S,) if b a for all b E S. Contents a is the least element (nhö nhat) of the poset (S,) if Properties of Relations a S b for all b E S. Combining Relations Representing Relations The greatest and least element are unique if it exists Closures of Relations Types of Relations Example Let S be a set. In the poset (P(S), C), the least element is and the greatest element is S. 6.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_039.png", "page_index": 281, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:31+07:00" } }, "CO1007-chapter-6-slide-282-0000": { "id": "CO1007-chapter-6-slide-282-0000", "text": "Upper Bound & Lower Bound Relations Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Let A C (S,). Tai : If u is an element of S such that a u for all elements a E A, then u is called an upper bound (can trén) of A BK TP.HCM If l is an element of S such that l S a for all elements a E A then l is called a Iower bound (can du6i) of A Contents Properties of Relations 40 Combining Relations Representing Relations Example Closures of Relations Types of Relations 0 Subset A does not have upper bound and lower B bound. 25 The upper bound of B are 20,40 and the lower bounc A is 2. 5 6.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_6/slide_040.png", "page_index": 282, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:35+07:00" } }, "CO1007-chapter-7-slide-283-0000": { "id": "CO1007-chapter-7-slide-283-0000", "text": "Counting Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 7 Xuan Toan, Tran Hong Tai Counting BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 7.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_001.png", "page_index": 283, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:38+07:00" } }, "CO1007-chapter-7-slide-284-0000": { "id": "CO1007-chapter-7-slide-284-0000", "text": "Contents Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Introduction TP.HCM Contents Counting Techniques Introduction Counting Techniques Pigeonhole Principle Permutations & Pigeonhole Principle Combinations Permutations & Combinations 7.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_002.png", "page_index": 284, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:40+07:00" } }, "CO1007-chapter-7-slide-285-0000": { "id": "CO1007-chapter-7-slide-285-0000", "text": "Course outcomes Counting Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Contents Represent and model practical problems with discrete structures L.O.2.1 - Logically describe some problems arising in Computing Introduction L.O.2.2 - Use proving methods: direct, contrapositive, induction Counting Techniques L.O.2.3 - Explain problem modeling using discrete structures Pigeonhole Principle Permutations & L.0.3 Understanding of basic probability and random variables Combinations L.O.3.1 - Define basic probability theory L.0.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.O.4.1 - Operate (compute/ optimize) on discrete structures L.O.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 7.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_003.png", "page_index": 285, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:44+07:00" } }, "CO1007-chapter-7-slide-286-0000": { "id": "CO1007-chapter-7-slide-286-0000", "text": "Introduction Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example BK In games: playing card, gambling, dices,.. TP.HCM How many allowable passwords on a computer system? How many ways to choose a starting line-up for a football Contents Introduction match? Counting Techniques Pigeonhole Principle Permutations & Combinations Combinatorics (tö hop) is the study of arrangements of objects Counting of objects with certain properties is an important part of combinatorics 7.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_004.png", "page_index": 286, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:48+07:00" } }, "CO1007-chapter-7-slide-287-0000": { "id": "CO1007-chapter-7-slide-287-0000", "text": "Applications of Combinatorics Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Number theory Probability Contents Statistics Introduction Counting Techniques Computer science Pigeonhole Principle Game theory Permutations & Combinations Information theory 7.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_005.png", "page_index": 287, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:51+07:00" } }, "CO1007-chapter-7-slide-288-0000": { "id": "CO1007-chapter-7-slide-288-0000", "text": "Problems Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Number of passwords a hacker should try if he wants to use Contents brute force attack Introduction Counting Techniques Number of possible outcomes in experiments Pigeonhole Principle Number of operations used by an algorithm Permutations & Combinations 7.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_006.png", "page_index": 288, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:54+07:00" } }, "CO1007-chapter-7-slide-289-0000": { "id": "CO1007-chapter-7-slide-289-0000", "text": "Product Rule Counting Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai There are 32 routers in a computer center. Each router has 24 ports. How many different ports in the center? BK Solution TP.HCM There are two tasks to choose a port. Contents picking a router Introduction @ picking a port on this router Counting Techniques Because there are 32 ways to choose the router and 24 ways to Pigeonhole Principle choose the port no matter which router has been selected, the Permutations & Combinations number of ports are 32 x 24 = 768 ports Definition (Product Rule (Luat nhan) Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are n1 x n2 ways to do the procedure Can be extended to T, T2, : . ., Tm tasks in sequence. 7.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_007.png", "page_index": 289, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:11:58+07:00" } }, "CO1007-chapter-7-slide-290-0000": { "id": "CO1007-chapter-7-slide-290-0000", "text": "More examples Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example (1) BK Two new students arrive at the dorm and there are 12 rooms TP.HCM available. How many ways are there to assign different rooms to two students? Contents Introduction Counting Techniques Example (2) Pigeonhole Principle How many different bit strings of length seven are there? Permutations & Combinations Example (3) How many one-to-one functions are there from a set with m elements to one with elements? 7.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_008.png", "page_index": 290, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:01+07:00" } }, "CO1007-chapter-7-slide-291-0000": { "id": "CO1007-chapter-7-slide-291-0000", "text": "Sum Rule Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK A student can choose a project from one of three fields: TP.HCM Information system (32 projects), Software Engineering (12 projects) and Computer Science (15 projects). How many ways are Contents there for a student to choose? Introduction Solution: 32 + 12 + 15 Counting Techniques Pigeonhole Principle Definition (Sum Rule (Luat cng)) Permutations & Combinations If a task can be done either in one of m1 ways or in one of n2 ways, there none of the set of n1 ways is the same as any of the set of n2 ways, then there are n1 + n2 ways to do the task Can be extended to n1, n2, . . ., nm disjoint ways 7.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_009.png", "page_index": 291, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:04+07:00" } }, "CO1007-chapter-7-slide-292-0000": { "id": "CO1007-chapter-7-slide-292-0000", "text": "Using Both Rules Counting Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai In a computer language, the name of a variable is a string of one or two alphanumeric characters, where uppercase and lowercase BK letters are not distinguished. Moreover, a variable name must TP.HCM begin with a letter and must be different from the five strings of two characters that are reserved for programming use. How many Contents different variables names are there in this language? Introduction Counting Techniques Solution Pigeonhole Principle Permutations & Let V equal to the number of different variable names. Combinations Let V be the number of these that are one character long, V2 be the number of these that are two characters long. Then, by sum rule, V = V + V2. Note that V = 26, because it must be a letter. Moreover, there are 26 : 36 strings of length two that begin with a letter and end with an alphanumeric character. However, five of these are exc/uded, so V2 = 26:36 - 5 = 931. Hence V = V + V = 957 different names for variables in this language. 7.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_010.png", "page_index": 292, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:08+07:00" } }, "CO1007-chapter-7-slide-293-0000": { "id": "CO1007-chapter-7-slide-293-0000", "text": "nclusion-Exclusion Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK How many bit strings of length eight either start with a 1 bit or TP.HCM end with the two bits 00? Contents Introduction Solution Counting Techniques Pigeonhole Principle Bit string of length eight that begins with a 1 is 27 = 128 Permutations & ways Combinations Bit string of length eight that ends with 00 is 26 = 64 ways Bit string of length eight that begins with 1 and ends with 00: 25 = 32 ways Number of satisfied bit strings are 27 + 26 - 25 = 160 ways. 7.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_011.png", "page_index": 293, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:12+07:00" } }, "CO1007-chapter-7-slide-294-0000": { "id": "CO1007-chapter-7-slide-294-0000", "text": "Inclusion-Exclusion Counting Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM S Contents AUB=A+B-AnB Introduction Counting Techniques A Pigeonhole Principle Permutations & Combinations 7.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_012.png", "page_index": 294, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:14+07:00" } }, "CO1007-chapter-7-slide-295-0000": { "id": "CO1007-chapter-7-slide-295-0000", "text": "Inclusion-Exclusion Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai S AU BUC=??? BK TP.HCM A Contents Introduction Counting Techniques Pigeonhole Principle Example Permutations & Combinations In a certain survey of a group of students, 87 students indicated they liked Arsenal, 91 indicated that they liked Chelsea and 91 indicated that they liked MU. Of the students surveyed, 9 liked only Arsenal, 10 liked only Chelsea, 12 liked only MU and 40 liked all three clubs. How many of the student surveyed liked both MU and Chelsea but not Arsenal? 7.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_013.png", "page_index": 295, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:17+07:00" } }, "CO1007-chapter-7-slide-296-0000": { "id": "CO1007-chapter-7-slide-296-0000", "text": "Pigeonhole Principle Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Introduction Counting Techniques Pigeonhole Principle Permutations & Combinations 7.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_014.png", "page_index": 296, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:19+07:00" } }, "CO1007-chapter-7-slide-297-0000": { "id": "CO1007-chapter-7-slide-297-0000", "text": "Examples Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example (1) BK TP.HCM Among any group of 367 people, there must be at least two with the same birthday. Contents Introduction Because there are only 366 possible birthdays Counting Techniques Pigeonhole Principle Example (2) Permutations & Combinations In any group of 27 English words, there must be at least two that begin with the same letter. Because there are 26 letters in the English alphabet 7.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_015.png", "page_index": 297, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:22+07:00" } }, "CO1007-chapter-7-slide-298-0000": { "id": "CO1007-chapter-7-slide-298-0000", "text": "Exercise Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Prove that if seven distinct numbers are selected from BK TP.HCM {1,2,...,11}, then some two of these numbers sum to 12. Contents Solution Introduction @ Pigeons: seven numbers from{1,2, ...,11} Counting Techniques Pigeonhole Principle Pigeonholes: corresponding to six sets, {1,11}, {2,10} Permutations & {3,9},{4,8},{5,7},{6} Combinations @ Assigning rule: selected number gets placed into the pigeonhole corresponding to the set that contains it. 4 Apply the pigeon hole: seven numbers are se/ected and placed in six pigeonholes, some pigeonhole contains two numbers. 7.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_016.png", "page_index": 298, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:25+07:00" } }, "CO1007-chapter-7-slide-299-0000": { "id": "CO1007-chapter-7-slide-299-0000", "text": "Examples - Permutations Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong How many ways can we arrange three students to stand in line for Tai a picture? BK TP.HCM Contents Introduction Counting Techniques Pigeonhole Principle Number of choices: 6 = 3! Permutations & Combinations 7.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_017.png", "page_index": 299, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:28+07:00" } }, "CO1007-chapter-7-slide-300-0000": { "id": "CO1007-chapter-7-slide-300-0000", "text": "Permutations Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Definition Tai A permutation (hoán vi) of a set of distinct objects is an ordered arrangement of these objects. BK TP.HCM An ordered arrangement of r elements of a set is called an r-permutation (hoan vi chap r) Contents Introduction n! Counting Techniques Pigeonhole Principle Permutations& Combinations Example How many ways are there to select a first-prize winner, a second-prize winner, and a third-prize winner from 100 different people who have entered a contest? P(100,3) = 100. 99. 98 = 970,200 7.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_018.png", "page_index": 300, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:31+07:00" } }, "CO1007-chapter-7-slide-301-0000": { "id": "CO1007-chapter-7-slide-301-0000", "text": "Counting Examples - Combinations Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai How many ways to choose two students from a group of four to offer scholarship? BK TP.HCM 2 : 2 Contents Introduction Counting Techniques 3 3 Pigeonhole Principle Permutations& Combinations 2 Number of choices: 6 7.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_019.png", "page_index": 301, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:34+07:00" } }, "CO1007-chapter-7-slide-302-0000": { "id": "CO1007-chapter-7-slide-302-0000", "text": "Combinations Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition (Combinations) An r-combination (tö hop chap r) of elements of a set is an BK unordered selection of r elements from the set. Thus, an TP.HCM r-combination is simply a subset of the set with r elements Contents n! n Introduction r!(n - r)! Counting Techniques Pigeonhole Principle Permutations & Combinations Example How many ways are there to select eleven players from a 22-member football team to start up? 22! C(22,11) 705432 11!11! 7.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_020.png", "page_index": 302, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:37+07:00" } }, "CO1007-chapter-7-slide-303-0000": { "id": "CO1007-chapter-7-slide-303-0000", "text": "Exercises - Permutations with Repetition Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Suppose that a salesman has to visit eight different cities. She TP.HCM 1 must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible Contents orders can the salesman use when visiting these cities? Introduction Counting Techniques Suppose that there are 9 faculty members in CS department Pigeonhole Principle and 11 in CE department. How many ways are there to select Permutations& a defend committee if the committee is to consist of three Combinations faculty members from the CS and four from the CE department? 7.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_021.png", "page_index": 303, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:41+07:00" } }, "CO1007-chapter-7-slide-304-0000": { "id": "CO1007-chapter-7-slide-304-0000", "text": "Permutations with Repetition Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Theorem TP.HCM The number of r-permutations of a set of n obiects with Contents repetition allowed is nr. Introduction Counting Techniques Example Pigeonhole Principle How many strings of length r can be formed from the English Permutations& Combinations alphabet? By product rule, we see that there are 26\" strings of length r. 7.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_022.png", "page_index": 304, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:43+07:00" } }, "CO1007-chapter-7-slide-305-0000": { "id": "CO1007-chapter-7-slide-305-0000", "text": "Example Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Question: How many ways we can choose 3 students from the TP.HCM faculties of Computer Science, Electrical Engineering and Mechanical Engineering? Contents Introduction CCC CEM Counting Techniques CCE EEE Pigeonhole Principle Permutations& CCM EEM Combinations CEE EMM CMM MMM 7.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_023.png", "page_index": 305, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:46+07:00" } }, "CO1007-chapter-7-slide-306-0000": { "id": "CO1007-chapter-7-slide-306-0000", "text": "Example Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM CCC CEM * * *I **x CCE EEE * * * *** CCM EEM * * * * ** Contents CEE EMM ??? Introduction + * * Counting Techniques CMM MMM * * * ** Pigeonhole Principle Permutations& Combinations How many ways to put * and ??? 7.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_024.png", "page_index": 306, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:50+07:00" } }, "CO1007-chapter-7-slide-307-0000": { "id": "CO1007-chapter-7-slide-307-0000", "text": "Combinations with Repetition Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Theorem TP.HCM There are C(n + r - 1, r) r-combinations from a set with n elements when repetition of elements is allowed. Contents Introduction Example Counting Techniques Pigeonhole Principle How many solutions does the equation Permutations & Combinations a1 + x2 + x3 =11 have, where 1, 2, and 3 are nonnegative integers? 7.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_025.png", "page_index": 307, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:53+07:00" } }, "CO1007-chapter-7-slide-308-0000": { "id": "CO1007-chapter-7-slide-308-0000", "text": "Examples Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Question: How many permutations are there of MISSISSIPPI? Contents Introduction Counting Techniques Pigeonhole Principle MISSISSIPPI = MISSISSIPPI Permutations& Combinations 7.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_026.png", "page_index": 308, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:56+07:00" } }, "CO1007-chapter-7-slide-309-0000": { "id": "CO1007-chapter-7-slide-309-0000", "text": "Permutations with Indistinguishable Objects Counting Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Theorem BK TP.HCM The number of different permutations of n obiects, where there are n1 indistinguishable objects of type 1, n2 indistinguishable Contents objects of type 2, : . : , and nk indistinguishable objects of type k, is Introduction n! Counting Techniques Pigeonhole Principle n!n2!...nk! Permutations& Combinations Example How many permutations are there of MISSISSIPPI? 7.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_7/slide_027.png", "page_index": 309, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:12:58+07:00" } }, "CO1007-chapter-8-slide-310-0000": { "id": "CO1007-chapter-8-slide-310-0000", "text": "Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 8 Xuan Toan. Tran Hong Tai Discrete Probability BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Introduction Probability Randomness Terminology Probability Rules Conditional Probability Dependent and Independent events Concept of Conditional Probability Random Variables Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Probability Models Tai Geomet ric Model Binomial Model Faculty of Computer Science and Engineering PROBLEMS University of Technology - VNUHCM SUMMARY III: trtanh@hcmut.edu.vn Emerging Applications 8.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_001.png", "page_index": 310, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:02+07:00" } }, "CO1007-chapter-8-slide-311-0000": { "id": "CO1007-chapter-8-slide-311-0000", "text": "Contents Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Introduction Xuan Toan. Tran Hong Tai Probability Randomness BK Terminology TP.HCM Probability Rules Contents 4 Conditional Probability Introduction Probability Dependent and Independent events Randomness Concept of Conditional Probability Terminology Probability Rules Random Variables Conditional Probability Dependent and Independent events Probability Models Concept of Conditional Probability Geometric Model Random Variables Binomial Model Probability Models Geometric Model PROBLEMS Binomial Model PROBLEMS SUMMARY Ill: Emerging Applications SUMMARY III: Emerging Applications 8.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_002.png", "page_index": 311, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:06+07:00" } }, "CO1007-chapter-8-slide-312-0000": { "id": "CO1007-chapter-8-slide-312-0000", "text": "Course outcomes Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Represent and model practical problems with discrete structures Contents L.O.2.1 - Logically describe some problems arising in Computing Introduction L.O.2.2 - Use proving methods: direct, contrapositive, induction Probability L.0.2.3 - Explain problem modeling using discrete structures Randomness Terminology L.0.3 Probability Rules Understanding of basic probability and random variables L.O.3.1 - Define basic probability theory Conditional Probability L.0.3.2 - Explain discrete random variables Dependent and Independent events Concept of Conditional Probability L.0.4 Compute quantities of discrete structures and probabilities Random Variables L.O.4.1 - Operate (compute/ optimize) on discrete structures Probability Models L.0.4.2 - Compute probabilities of various events, conditional Geometric Model ones, Bayes theorem Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_003.png", "page_index": 312, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:12+07:00" } }, "CO1007-chapter-8-slide-313-0000": { "id": "CO1007-chapter-8-slide-313-0000", "text": "Motivations Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Gambling Tai .GIALDAC BK That & G 9K3 TP.HCM DA@UOCTE STH818610 Contents 0AIVE 9K3 ngay15-9-2011 Introduction Probability Real life problems Randomness Terminology Probability Rules Conditional Probability Dependent and Independent 12 events Concept of Conditional precip 60% THU Pecip 50% Pecip Probability Random Variables Probability Models Geometric Model Computer Science: cryptology - deals with encrypting codes Binomial Model PROBLEMS or the design of error correcting codes SUMMARY III: Emerging Applications 8.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_004.png", "page_index": 313, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:17+07:00" } }, "CO1007-chapter-8-slide-314-0000": { "id": "CO1007-chapter-8-slide-314-0000", "text": "Learning Objectives of Basic Probability Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK After careful study of basic probability, you should be able to TP.HCM 1 Compute and interpret probability of an event Contents Probability = a chance that an event occurs) Introduction Find probability of mixed events, including conditional Probability probability Randomness Terminology 3 Explain the concepts of random variable, and determine its Probability Rules components Conditional P robability Dependent and Independent 4 Understand expectation and variance of a random variable events Concept of Conditional Probability Random Variables Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_005.png", "page_index": 314, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:21+07:00" } }, "CO1007-chapter-8-slide-315-0000": { "id": "CO1007-chapter-8-slide-315-0000", "text": "Learning Objectives of Basic Probability Discrete Probability Nguyen An Khuong. Probability in practice Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Probabilistic Methods help us to solve realistic problems as follows. 1) In Computing BK Estimate the average computing time of an algorithm TP.HCM Compute the risk of security breach to a computer system by (bad) hackers in cyber-space... Contents Introduction 2) In Finance Science and Insurance Industry Probability Suppose an insurance company B has thousands of Randomness Terminology customers, and each customer is charged $500 a year Probability Rules Since the customer's businesses are risky, from the past Conditional Probability Dependent and Independent experience the company B estimates that about 15% of their events Concept of Conditional customers would get fatal trouble (e.g. fire, accident ...) and Probability as a result they will submit a claim in any given year. Random Variables Probability Models We assume that the claim will always be $3000 for each Geomet ric Model customer. Binomial Model PROBLEMS Our problem: Compute how much average profit can the SUMMARY III: company expect to make per customer in a certain year? Emerging Applications 8.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_006.png", "page_index": 315, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:25+07:00" } }, "CO1007-chapter-8-slide-316-0000": { "id": "CO1007-chapter-8-slide-316-0000", "text": "Randomness Discrete Probability Nguyen An Khuong. Which of these are random phenomena? Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai The number you receive when rolling a fair dice The sequence for lottery special prize (by law!) BK Your blood type (No!) TP.HCM You met the red light on the way to school The traffic light is not random. It has timer. Contents . The pattern of your riding is random. Introduction Probability So what is special about randomness? Randomness Terminology In the long run, they are predictable and have relative frequency Probability Rules (fraction of times that the event occurs over and over and over) Conditional Probability Dependent and Independent events Does randomness exist in the above problem? What is it? Concept of Conditional Probability Could we Random Variables @ Model the amount of money X that the insurance company Probability Models Geomet ric Model believes to obtain from each customer Binomial Model PROBLEMS Find the expectation (Expected Value) of that amount SUMMARY III: denoted by E[X] Emerging Applications 8.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_007.png", "page_index": 316, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:31+07:00" } }, "CO1007-chapter-8-slide-317-0000": { "id": "CO1007-chapter-8-slide-317-0000", "text": "Terminology Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Introduction Probability Randomness Terminology Probability Rules Conditional Probability Experiment (thi nghiem): a procedure that yields one of a Dependent and Independent given set of possible outcomes. events Concept of Conditional Tossing a coin to see the face Probability Random Variables Sample space (khng gian mau): set of possible outcomes Probability Models {Head, Tail} Geometric Model Binomial Model Event (su kien): a subset of sample space. PROBLEMS . You see Head after an experiment. {Head} is an event. SUMMARY III: Emerging Applications 8.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_008.png", "page_index": 317, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:34+07:00" } }, "CO1007-chapter-8-slide-318-0000": { "id": "CO1007-chapter-8-slide-318-0000", "text": "Example Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example (1) Experiment: Rolling a die. What is the sample space? BK Answer:{1,2, 3, 4, 5,6} TP.HCM Contents Example (2) Introduction Probability Experiment: Rolling two dice. What is the sample space? Randomness Terminology Answer: It depends on what we're going to ask! Probability Rules Conditional Probability The total number? Dependent and Independent {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} events Concept of Conditional Probability The number of each die? Random Variables {(1,1), (1,2),(1,3), ...,(6,6)} Probability Models Geometric Model Which is better? Binomial Model The latter one, because they are equally likely outcomes PROBLEMS SUMMARY III: Emerging Applications 8.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_009.png", "page_index": 318, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:39+07:00" } }, "CO1007-chapter-8-slide-319-0000": { "id": "CO1007-chapter-8-slide-319-0000", "text": "The Law of Large Numbers Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK TP.HCM The Law of Large Numbers (Luat só /ón) states that the long-run relative frequency of repeated independent events gets closer and Contents closer to the true relative frequency as the number of trials Introduction Increases. Probability Randomness Terminology Example Probability Rules Do you believe that the true relative frequency of Head when you Conditional Probability Dependent and Independent toss a coin is 50%? events Concept of Conditional Probability Let's try! Random Variables Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_010.png", "page_index": 319, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:43+07:00" } }, "CO1007-chapter-8-slide-320-0000": { "id": "CO1007-chapter-8-slide-320-0000", "text": "Be Careful! Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Don't misunderstand the Law of Large Numbers (LLN). It can lead to money lost and poor business decisions. BK TP.HCM Example Contents I had 8 children, all of them are girls. Thanks to LLN (!?), there Introduction are high possibility that the next one will be a boy. Probability (Overpopulation!!!) Randomness Terminology Probability Rules Conditional Probability Example Dependent and Independent events I'm playing Oysters - Crap - Lobster - Fish game, the fish has not Concept of Conditional Probability appeared in recent 5 games, it will be more likely to be fish next Random Variables game. Thus, I bet all my money in fish. (Sorry, you lose!) Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_011.png", "page_index": 320, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:13:47+07:00" } }, "CO1007-chapter-8-slide-321-0000": { "id": "CO1007-chapter-8-slide-321-0000", "text": "Probability Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Probability of an event E, or a phenomenon ... informally is Xuan Toan. Tran Hong Tai a number P[E] or p(E measuring how much chance that the event E would happen. BK Definition TP.HCM The probability (xäc suät) of an event E of a finite nonempty sample space S of equally likely outcomes is Contents Introduction E Probability P[E] S Randomness Terminology Probability Rules Conditional Probability Note that E C S so 0 [0,1] Randomness Terminology Probability Rules E e Q=>p(E) =P[E] =Prob(E)= Conditional Probability Dependent and Independent events probability or chance that the event E occurs. Concept of Conditional Probability Remarks: Random Variables Probability Models @ S can be a discrete set or continuous set. [DlY: could you Geomet ric Model invent few examples?] Binomial Model PROBLEMS We use 3 notations p(E),P[E] and Prob(E) equivalently SUMMARY III: from now on. Emerging Applications 8.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_015.png", "page_index": 324, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:06+07:00" } }, "CO1007-chapter-8-slide-325-0000": { "id": "CO1007-chapter-8-slide-325-0000", "text": "Event combinations Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong The Union of two events A,B, denoted as A U B Tai consists of all outcomes that are contained in one event or the other. BK The Intersection of two events A.B, denoted as A B TP.HCM consists of all outcomes that are contained in one event and the other. Contents Introduction AU B An B Probability Randomness Terminology Probability Rules A B A B Conditional Probability Dependent and Independent events Concept of Conditional Probability Random Variables Probability Models The Complement of an event B, denoted as Bc. B' or B Geometric Model Binomial Model is the set of outcomes in the sample space that are not PROBLEMS contained in B. SUMMARY III: Emerging Applications 8.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_016.png", "page_index": 325, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:10+07:00" } }, "CO1007-chapter-8-slide-326-0000": { "id": "CO1007-chapter-8-slide-326-0000", "text": "Formal Probability (Axioms, by A. Kolmogorov, 1933 Discrete Probability Nguyen An Khuong. Axiom A1: A probability is a number between 0 and 1. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 0 1) (n = 4 in the above figure Probability Models form a partition of set S2 if Geometric Model Binomial Model PROBLEMS (E;nEj)=0,Vij, and EUEzU..UEn=S. SUMMARY III: Emerging Applications 8.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_022.png", "page_index": 331, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:36+07:00" } }, "CO1007-chapter-8-slide-332-0000": { "id": "CO1007-chapter-8-slide-332-0000", "text": "General addition rule for m events Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM If events E1,E2,... ,En (n > 1) form a partition of set 2 Contents then for any event B, Introduction Probability n Randomness P[B]=>P[BnE;] (1) Terminology Probability Rules Conditional Probability Dependent and Independent events Concept of Conditional Probability Random Variables Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_023.png", "page_index": 332, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:40+07:00" } }, "CO1007-chapter-8-slide-333-0000": { "id": "CO1007-chapter-8-slide-333-0000", "text": "Homework (DIY) Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example (3) BK In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and TP.HCM 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male Contents The sample space is shown here: Introduction Probability Staff Females Male Total Randomness Terminology Nurses 7 1 8 Probability Rules Physicians 3 2 5 Conditional P robability Total 10 3 13 Dependent and Independent events Concept of Conditional Probability The probability is: Random Variables P(nurse or male) = P(nurse) +P(male) - P(male nurse) Probability Models 3 1 10 13 13 13 13 Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_024.png", "page_index": 333, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:45+07:00" } }, "CO1007-chapter-8-slide-334-0000": { "id": "CO1007-chapter-8-slide-334-0000", "text": "Dependent events Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai \"Knowledge\" changes probabilities! Event A and event B are dependent if the appearance of an event BK is related (in a certain way) to the occurrence of the other event TP.HCM Contents Introduction Probability Randomness Terminology BOY OR GIRL Probability Rules Conditional Probability Dependent and Independent events Concept of Conditional Probability We next use event dependency to define Independence (dc /ap) Random Variables then Conditional Probability (Xäc suät c6 dieu kien), given two Probability Models events A,B, where B occurred. Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_025.png", "page_index": 334, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:49+07:00" } }, "CO1007-chapter-8-slide-335-0000": { "id": "CO1007-chapter-8-slide-335-0000", "text": "ndependent events Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition Events A and B are independent whenever the occurrence of A is BK not related to or not influenced by the occurrence of B, and the TP.HCM other way round. Then we write Contents p(AB)=p(A). Introduction Probability Randomness We can say the outcome of one event does not influence the Terminology probability of the other. Probability Rules Conditional Probability Example: p(\"Head\"\"It's raining outside\") = p(\"Head\" Dependent and Independent events Concept of Conditional Probability NOTE: Disjoint Independence Random Variables Probability Models Disjoint events cannot be independent. They have no outcomes in Geometric Model common, so knowing that one occurred means the other did not Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_026.png", "page_index": 335, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:53+07:00" } }, "CO1007-chapter-8-slide-336-0000": { "id": "CO1007-chapter-8-slide-336-0000", "text": "Conditional Probability Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition: The conditional probability p(A B) = Probability of Tai event A given that event B has occurred Two cases of interest: BK TP.HCM 1 When event A is independent of B, meaning the occurrence of A is not influenced by B, then obviously Contents Introduction p(AB)=p(A) Probability Randomness The joint probability of two independent events A, B Terminology then is defined as Probability Rules Conditional Probability p(AnB) =p(B):p(A)=p(A):p(B). (2) Dependent and Independent events Concept of Conditional Probability When A is dependent of B (A is influenced by B) Random Variables Probability Models p(AB) p(A) Geometric Model Binomial Model PROBLEMS How to evaluate p(AB) in the 2nd case? SUMMARY III: Emerging Applications 8.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_027.png", "page_index": 336, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:14:58+07:00" } }, "CO1007-chapter-8-slide-337-0000": { "id": "CO1007-chapter-8-slide-337-0000", "text": "General Multiplication Rule gives Conditional Probability Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK In general, [no matter dependent or independent] we always have TP.HCM p(AnB) = p(B) xp(AB)= p(A) xp(BA) Contents Introduction Probability Therefore, the conditional probability of event A given event B Randomness Terminology Probability Rules p(AnB) (3) Conditional Probability p(AB)=p p(B) Dependent and Independent events Concept of Conditional Probability Random Variables Probability Models Geomet ric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_028.png", "page_index": 337, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:02+07:00" } }, "CO1007-chapter-8-slide-338-0000": { "id": "CO1007-chapter-8-slide-338-0000", "text": "Example Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM What is the probability of drawing a red card and then another red card without replacement (khng hoan lai)? Contents Solution: Let B be the event of drawing the first red card, and Introduction A be the event of drawing the second red card. Clearly. Probability Randomness p(B) = 26/52 = 1/2, the conditional p(AB) = 25/51 Terminology So the event of drawing a red card and then another red card is Probability Rules the joint probability Conditional Probability p(An B) =p(B) x p(AB) = 1/2 x 25/51 = 25/102. Dependent and Independent events Concepl of Conditional Probability Random Variables Probability Models Geomet ric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_029.png", "page_index": 338, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:06+07:00" } }, "CO1007-chapter-8-slide-339-0000": { "id": "CO1007-chapter-8-slide-339-0000", "text": "Example Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example (1) Tai The probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a BK legal parking space and has to park in the no-parking zone is 0.20. TP.HCM On Tuesday, Sam arrives at school and has to park in a no-parking zone. Contents Find the probability that he will get a parking ticket Introduction Probability Randomness Terminology Solution Probability Rules N: parking in a no-parking zone Conditional Probability Dependent and Independent T: get a ticket events 0.06 Concepl of Conditional p(TN) = p(NnT) = 0.3 Probability p(N) 0.2 Random Variables Hence, Sam has a 0.3 probability of getting a parking ticket, given Probability Models that he parked in a no-parking zone. Geomet ric Model Binomial Model REMARK:ln general,p(AB) p(BA) PROBLEMS the conditional probability is not symmetric. SUMMARY III: Emerging Applications 8.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_030.png", "page_index": 339, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:11+07:00" } }, "CO1007-chapter-8-slide-340-0000": { "id": "CO1007-chapter-8-slide-340-0000", "text": "Bayes's Theorem Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Reminder: Tai P(ANB) p(AB) = p(B) BK Note also that TP.HCM p(B)p(AB) =p(AnB) =p(BA)=p(A)p(BA Contents Introduction Probability Randomness Terminology Theorem (Bayes's Theorem- basic version) Probability Rules Conditional Probability We always have the following, for any pair of events A, B, where Dependent and Independent we assume B occurred. events Concept of Conditiona Probability Random Variables P(AB) =P(A).p(BA) (4) p(B) Probability Models Geometric Model Binomial Model PROBLEMS Recall A = Ac = A' = S A, the complement of event A SUMMARY III: Emerging Applications 8.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_031.png", "page_index": 340, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:14+07:00" } }, "CO1007-chapter-8-slide-341-0000": { "id": "CO1007-chapter-8-slide-341-0000", "text": "Bayes's Theorem Il Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Theorem (Bayes's Theorem- full version) Xuan Toan, Tran Hong Tai Given two events B,A where we assume B occurred. p(A).p(BA) p(AB)= BK p(A)p(BA)+p(A)p(BA) TP.HCM Contents Proof: Rule (1) gives Introduction p(B)=p(BnE1)+p(BnEz), with E1 =A,Ez=A Probability Randomness Example Terminology Probability Rules If we know that the probability that a person has tuberculosis (TB) is Conditional P robability p(TB) = 0.0005,also know p(+TB = 0.999 and p(- TB) = 0.99 Dependent and Independen What is p(TB +) and p(TB -)? events Concepl of Conditiona Probability p(+TB)p(TB) Random Variables p(TB+) p(+TB)p(TB)+p(+TB)p(TB) Probability Models Geomet ric Model 0.999 x 0.0005 Binomial Model 0.0476 0.999 x 0.0005 + (1 - 0.99) x (1 - 0.0005) PROBLEMS SUMMARY III: p(TB-) = 0.99 Emerging Applications 8.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_032.png", "page_index": 341, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:20+07:00" } }, "CO1007-chapter-8-slide-342-0000": { "id": "CO1007-chapter-8-slide-342-0000", "text": "Random Variable: Concepts Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai From now on, given a random experiment, we also write 2 for the Xuan Toan. Tran Hong Tai sample space (previously denoted S) of that experiment. Then we perform observation or measurement on elements of 2. BK Definition TP.HCM A random variable X is a map (function or measurement) from Contents Introduction S to the reals R. That is for w E S2 then X(w) E R. Probability The domain of variable X is the sample space S2 Randomness Terminology The range of variable X is the set of all observed values Probability Rules Conditional P robability Rx =Range(X)={X(w)} C R Dependent and Independent events Concept of Conditional Probability For any value b E R, the preimage Random Variables Probability Models A:=X-1(b)={w e 2: X(w)=b} c S Geometric Model Binomial Model PROBLEMS is an event,we define P[X = b] =Prob{X = b} :=Prob(A) SUMMARY III: Emerging Applications 8.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_033.png", "page_index": 342, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:25+07:00" } }, "CO1007-chapter-8-slide-343-0000": { "id": "CO1007-chapter-8-slide-343-0000", "text": "Random Variable: Visualization of a random variable X Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong The sample space Map X: R The observed value set Range(X) Tai ARIN BK TP.HCM b Contents Introduction Probability Randomness Terminology In Entertainment industry, a random experiment can be defined as Probability Rules observing certain students of HCMUT, to get a sample space S2 of Conditional P robability Dependent and Independent students who like music (or art and drama). events Concept of Conditional Next we ask every w E S2 about his/her unique favorite singer, Probability Random Variables to make a random variable \"fan of Probability Models X : S2 -> Range(X) ={x : x is good singer in the world} Geometric Model Binomial Model Here specifically X(w) = b means b is the observed value at w. PROBLEMS Then event {X = b} is A := X-1(b), the violet square... SUMMARY III: Emerging Applications 8.34", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_034.png", "page_index": 343, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:30+07:00" } }, "CO1007-chapter-8-slide-344-0000": { "id": "CO1007-chapter-8-slide-344-0000", "text": "Random Variable: the probability density function Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Now fix another singer with name ARIN, we then view the event Xuan Toan. Tran Hong Tai {X =ARIN} in the meaning of the set A := X-1(ARIN); it is the green oval consisting of fans of ARIN, obviously A C S2 so the probability Prob(A) does exist. BK TP.HCM Definition Contents Introduction In general, for a random variable X : 2 -> Range(X), fix any Probability Randomness value x E Range(X), then A := X-1(x) is an event and the Terminology probability Probability Rules A Conditional P robability P[X = x] := Prob(A S2 Dependent and Independent events Concept of Conditional Probability P[X = x] indicates how much chance observed value x Random Variables occurred, and Probability Models Geometric Model f(x) := P[X = x] called the probability density function Binomial Model PROBLEMS (p.d.f) of variable X at observation (or observed value) SUMMARY III: Emerging Applications 8.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_035.png", "page_index": 344, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:34+07:00" } }, "CO1007-chapter-8-slide-345-0000": { "id": "CO1007-chapter-8-slide-345-0000", "text": "Random Variable: Example revisited with calculation Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Sample space S2 = the set of HCMUT 'musical' students ask each student w E S2 to know his (her) most liked singer x BK in the globe, X(w) = x, [x is the most liked singer of w.] TP.HCM If the university has 30000 students, and 1500 students like - Contents say singer x = ARI' (called observed value or measured Introduction value) then event Probability Randomness A:=X-1(ARIN')={w e S2: X(w)='ARIN'} c S Terminology Probability Rules has cardinality (i.e. the number of students) 1500, and Conditional P robability Dependent and Independent events A 1500 Concept of Conditional Probability P[X ='ARlN'l:= Prob(Aj 1/20. S2 30000 Random Variables Probability Models f(ARIN') := P[X ='ARIN']=1/20 is Geometric Model Binomial Model the probability density or mass at the value 'singer ARIN'. PROBLEMS SUMMARY III: Emerging Applications 8.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_036.png", "page_index": 345, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:39+07:00" } }, "CO1007-chapter-8-slide-346-0000": { "id": "CO1007-chapter-8-slide-346-0000", "text": "Distinguish random variables and observed values Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Notation is used to distinguish between a random variable and the Tai real observed values. : A random variable is denoted by an uppercase letter BK TP.HCM such as X (it is a map). After a measurement (observation) is conducted, the Contents measured va/ue of the random variable is denoted Introduction generally by a lowercase letter x (it is number or Probability specific value), as Randomness Terminology Probability Rules = ADEL in Western entertainment industry Conditional Probability x = DONAL Trump in Politics, Dependent and Independent events Concept of Conditional = 70 milliamperes in Electrical engineering Probability = 1.7 meter in Public Health, Random Variables Probability Models c = success' outcome in Business administration, Geometric Model Binomial Model = fatal accident' in Medical insurance.. PROBLEMS SUMMARY III: Emerging Applications 8.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_037.png", "page_index": 346, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:44+07:00" } }, "CO1007-chapter-8-slide-347-0000": { "id": "CO1007-chapter-8-slide-347-0000", "text": "Discrete random variable and Expected Value Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition (Discrete random variable - bién ngáu nhién ri rac) Tai (1) A rand. variable X(.) is discrete if it has a discrete range, BK meaning that the set of observed values consists of no more than a TP.HCM countable number of elements, i.e. [Sx [N]. Contents In finite set case, [Sx < [N], we usually write set of values Introduction Probability Sx :=Range(X) ={x1,x2,x3,..,xn-1,xn}, n e N Randomness Terminology In countably infinite set case,Sx= N, we write Probability Rules Conditional P robability Sx :=Range(X) ={x1,x2,3,...,xn-1,xn,... Dependent and Independent events Concept of Conditional Probability (2) Expected value (giá tri ky vong) of X Random Variables Probability Models Geometric Model E[X]=Dxesx x`p(X=x) Binomial Model PROBLEMS where p(X = x) := P[X = x] is the density (mass) at x. SUMMARY III: Emerging Applications 8.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_038.png", "page_index": 347, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:49+07:00" } }, "CO1007-chapter-8-slide-348-0000": { "id": "CO1007-chapter-8-slide-348-0000", "text": "The probability distribution table of a discrete variable Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan.Tran Hong The finite Sx ={1,2,3,... ,n} helps us to compute the Tai probability distribution table of X, structured by BK x x1 Xn-1 Xn TP.HCM Pk :=P X =xk P1 Pn-1 Pn. Contents Of course Introduction Z pk=1. Pk > 0 and Probability Randomness xkESx Terminology The probability distribution table in Entertainment industry: Probability Rules Conditional Probability Dependent and Independent A := X-1(ARlN) is the green oval, with A = 1500 events Concept of Conditional E := X-1(b) is the violet square on the left, with E = 21000, Probability Random Variables and C := X-1(c) is the yellow oval, with lC = 7500 students, Probability Models then we can find the densities Pk and finally Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_039.png", "page_index": 348, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:53+07:00" } }, "CO1007-chapter-8-slide-349-0000": { "id": "CO1007-chapter-8-slide-349-0000", "text": "Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong The set of observed values (also called value space) Tai Sx ={x1,x2,x3}={ARIN,b,c} gives BK x X1 = ARIN X2 X3 TP.HCM pi=P[X =xi] p1 20 p2 10 P3 4 Contents Introduction Map X: Q R The observed value set Probability The sample space Q Range(X) Randomness Terminology Probability Rules ARIN Conditional P robability Dependent and Independent events Concept of Conditional Probability b Random Variables Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_040.png", "page_index": 349, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:15:58+07:00" } }, "CO1007-chapter-8-slide-350-0000": { "id": "CO1007-chapter-8-slide-350-0000", "text": "Expectation (Expected Value) to Variability Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Generally, assume that random variable X is discrete, with the Xuan Toan. Tran Hong Tai value space Sx = Range(X ={x1,x2,...,xn}. having the density Pk = p(x), then we knew the expected value expectation or average) u of X is given by BK TP.HCM u=E[X]=Z Xk Pk. Contents xkESx Introduction Probability Definition Randomness Terminology The variability of the random variable X is expressed via two Probability Rules concepts Conditional Probability Dependent and Independent events Variance of X is Concept of Conditional Probability Random Variables V[X]=o2:=E[(X-u)2]= >[xk-u]2pk Probability Models Geometric Model xkESx Binomial Model PROBLEMS Standard deviation of X is ox = o = VV[X] SUMMARY III: Emerging Applications 8.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_041.png", "page_index": 350, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:03+07:00" } }, "CO1007-chapter-8-slide-351-0000": { "id": "CO1007-chapter-8-slide-351-0000", "text": "Expected Value for Central Tendency of X Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Example: An insurance company LifeInsu charges $50 a year. Xuan Toan, Tran Hong Tai Can the company make a profit? Assuming that it made a research on 1000 people and have following table: BK TP.HCM Outcome Payroll (m)Probability p(X = x) = p(M = m) Death 10,000 1000 Contents Disability 5000 Introduction 1000 Good 0 997 Probability 1000 Randomness Terminology Or horizontally Probability Rules Conditional P robability x Death Disability Good Dependent and Independent M (Payroll) 10,000 5000 0 events Concept of Conditional 2 997 Probability pi =P[X = xi] p1 P2 P3 1000 10000 1000 Random Variables Probability Models X is a discrete random variable, so is the payroll M Geometric Model Binomial Model Besides, Pi = P[X = xi]= P[M = mil where m1 = 10,000, PROBLEMS m2 = 5000,and m3 = 0. SUMMARY III: Emerging Applications 8.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_042.png", "page_index": 351, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:10+07:00" } }, "CO1007-chapter-8-slide-352-0000": { "id": "CO1007-chapter-8-slide-352-0000", "text": "Can company make a profit? Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai X is discrete since customer's injury level has three values (outcomes) Death, Disability and Good (nothing happen) The company expects that they have to pay for each BK TP.HCM customer : E[X]=Z x·P(X=x)=> mP(M =m) Contents Introduction 2 997 Probability =$10,000 F500 $0 $20. Randomness 000 Terminology Probability Rules x Death Disability Good Conditional Probability M (Payroll) 10,000 5000 0 Dependent and Independent events 1 2 997 Concept of Conditional pi = P[M = mi] Probability p1 P2 1000 10000 p3 1000 Random Variables Probability Models Expected money amount E[X]= E[M]=u = 20 USD per Geometric Model customer. Well, shall we say LifeInsu gets profit 50 - 20 =$30 Binomial Model PROBLEMS per customer, in average? SUMMARY III: Emerging Applications 8.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_043.png", "page_index": 352, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:15+07:00" } }, "CO1007-chapter-8-slide-353-0000": { "id": "CO1007-chapter-8-slide-353-0000", "text": "Variance: capture the Spreading, Risk or Variability Discrete Probability Nguyen An Khuong. Of course, the expected value $20 will not happen in reality Tran Tuan Anh, Mai Xuan Toan, Tran Hong There will be variability. Let's calculate! Tai Variance (phuong sai) BK TP.HCM V[X] =E(x -E[X]2 :p(X =x V[X]=>(m-E[M])2.p(M=m)=V[M] Contents Introduction Probability Randomness Terminology 149,600 Probability Rules Standard deviation (d lech chuan) Conditional Probability SD(M) = VV[M] Dependent and Independent events SD(M) = v149,600 $386.78 Concept of Conditional Probability Random Variables Probability Models Comment Geomet ric Model Binomial Model The company expects to pay out $20, and make $30. However, PROBLEMS the standard deviation of $386.78 indicates that it's no sure thing SUMMARY III: Emerging Applications That's pretty big spread (and risk) for an average profit of $30 8.44", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_044.png", "page_index": 353, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:20+07:00" } }, "CO1007-chapter-8-slide-354-0000": { "id": "CO1007-chapter-8-slide-354-0000", "text": "Example Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example One thousand tickets are sold at 1 each for a color television BK valued at 350. What is the expected value of the gain if you TP.HCM purchase one ticket? The problem can be set up as follows: Contents Introduction Win Lose Probability Randomness Terminology Gain 349 -1 Probability Rules probability 1 999 Conditional Probability 1000 1000 Dependent and Independent events The solution, then, is Concept of Conditional Probability Random Variables 999 Probability Models E[X] =$349 0.65 L000 Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.45", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_045.png", "page_index": 354, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:24+07:00" } }, "CO1007-chapter-8-slide-355-0000": { "id": "CO1007-chapter-8-slide-355-0000", "text": "Example Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai A person pays $2 to play a certain game by rolling a single die once. If a 1 or a 2 comes up, the person wins nothing. If, however, BK the player rolls a 3,4,5, or 6, he or she wins the difference TP.HCM between the number rolled and 2. Is the game fair? Contents Example Introduction Probability The roulette wheel Randomness Terminology Probability Rules Conditional Probability Dependent and Independent events Concept of Conditional Probability Random Variables 8 Probability Models Geometric Model Red or black $1 0 $35 Binomial Model Od or even S1 00 $35 PROBLEMS 1-18 $1 Any single number $35 SUMMARY III: 19-36 $1 0or 00 $17 erstock.com590802956 Emerging Applications 8.46", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_046.png", "page_index": 355, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:29+07:00" } }, "CO1007-chapter-8-slide-356-0000": { "id": "CO1007-chapter-8-slide-356-0000", "text": "SUMMARY II Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Probability theory: Concepts and operations Tai Random variable X with 5 key components BK The observed value set Range(X) = Sx TP.HCM 21 The probability density function (pdf) f(x) Contents @ The probability cumulative function (or cdf) F(x) Introduction Probability Expectation (or average) E[X] = of X Randomness Terminology @ Variance V[X] = o2 of X, and the Standard deviation . Probability Rules Conditional Probability Dependent and Independent events The last section is spent for Probability Models, which play an Concept of Conditional Probability extremely important role nowadays in modeling of phenomena in Random Variables Science, Engineering and Technology, in particular because of high Probability Models Geometric Model uncertainty of our world (war, epidemic, pandemic, risks in Binomial Model cyberspace, calamity in environment, climate change...) PROBLEMS See SUMMARY Ill of emerging applications at the end for more SUMMARY III: Emerging Applications 8.47", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_047.png", "page_index": 356, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:33+07:00" } }, "CO1007-chapter-8-slide-357-0000": { "id": "CO1007-chapter-8-slide-357-0000", "text": "Bernoulli Trials Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Bernoulli variable or trial describes a random variable B Tai that can take only two possible values, i.e. SB ={0,1} Its probability density function is given by probability of suc BK cess P(B = 1) = p, and TP.HCM P(B =0) =1-p for some p e [0,1] Contents Introduction Expectation and variance E[B] = p; V[B]=p(1-p Probability Randomness Terminology Example Probability Rules Conditional Probability Some people madly drink Coca-Cola, hoping to find a ticket to see Dependent and Independen1 events Big Bang. Let's call tearing a bottle's label trial (phép thü): Concept of Conditional Probability : There are only possible outcomes (congrats or good luck) Random Variables The probability of success, p, is the same on every trial, say Probability Models Geomet ric Model 0.06 So Binomial Model PROBLEMS P(B =1) = p = 0.06,P(B = 0) =1 - p = 0.94 SUMMARY III: Emerging Applications 8.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_048.png", "page_index": 357, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:38+07:00" } }, "CO1007-chapter-8-slide-358-0000": { "id": "CO1007-chapter-8-slide-358-0000", "text": "Geometric Model (M6 hinh hinh hoc) Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Question: How long it will take us to achieve a success, given p. BK the probability of success? TP.HCM Definition (Geometric probability model: Geom(p)) Contents Introduction p = probability of success (q = 1 - p = probability of failure) Probability X = number of trials until the first success occurs Randomness Terminology p(X = x) = qx-1 x p Probability Rules Conditional Probability Dependent and Independent Expected value: = 1 events p Concept of Conditional Probability Standard deviation: o 4 Random Variables D Probability Models Geomet ric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_049.png", "page_index": 358, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:42+07:00" } }, "CO1007-chapter-8-slide-359-0000": { "id": "CO1007-chapter-8-slide-359-0000", "text": "Geometric Model: Example Discrete Probability Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai If the probability of finding a Sound Fest ticket is p = 0.06, how many bottles do you expect to open before you find a ticket? What is the probability that the first ticket is in one of the first BK four bottles? TP.HCM Contents Solution Introduction Let X = number of trials until a ticket is found Probability We can model X with Geom(0.06). Randomness Terminology E(X) = 16.7 0.06 Probability Rules Conditional Probability Dependent and Independent events P(X < 4) - P(X =1) +P(X = 2) +P(X =3) +P(X =4 Concept of Conditional Probability (0.06) + (0.94)(0.06) + (0.94)2(0.06) Random Variables +(0.94)3(0.06) Probability Models Geomet ric Model 0.2193 Binomial Model PROBLEMS Conclusion: We expect to open 16.7 bottles to find a ticket. SUMMARY III: Emerging Applications About 22% of time we'll find one within the first 4 bottles. 8.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_050.png", "page_index": 359, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:47+07:00" } }, "CO1007-chapter-8-slide-360-0000": { "id": "CO1007-chapter-8-slide-360-0000", "text": "Binomial Model (M hinh nhi thüc) Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Previous Question: How long it will take us to achieve a success, given p, the probability of success? BK New Question: You buy 5 Coca-Cola. What's the probability you TP.HCM get exactly 2 Sound Fest tickets? Contents Introduction Definition (Binomial probability model: Binom(n,p)) Probability Randomness m = number of trials Terminology p = probability of success (q = 1 - p = probability of failure) Probability Rules X = number of successes in n trials Conditional P robability Dependent and Independent events p(X=x)=nCx x p x qn Concept of Conditional Probability Random Variables Expected value: = np Probability Models Standard deviation: o = Vnpq Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.51", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_051.png", "page_index": 360, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:51+07:00" } }, "CO1007-chapter-8-slide-361-0000": { "id": "CO1007-chapter-8-slide-361-0000", "text": "Binomial Model: Example Discrete Probability Nguyen An Khuong Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Suppose you buy 20 Coca-Cola bottles. What are the mean and standard deviation of the number of winning bottles among them? What is the probability that there are 2 or 3 tickets? Given that BK the probability of finding a winning bottle is p = 0.06 TP.HCM Contents Solution Introduction Let X = number of tickets among n = 20 bottles Probability We can model X with Binom(20, 0.06). Randomness Terminology E(X) = np = 20(0.06) = 1.2 Probability Rules SD(X) = Vnpq = V20(0.06)(0.94) 1.06 Conditional P robability Dependent and Independent P(X = 2 or 3) P(X = 2) +P(X = 3) events = Concept of Conditional Probability 20C2(0.06)2(0.94)18 + 20C3(0.06)3(0.94)17 = Random Variables 0.2246 + 0.0860 = 0.3106 Probability Models Geometric Model Binomial Model Conclusion: In 20 bottles, we expect to find an average of 1.2 PROBLEMS tickets, with a sd of 1.06. About 31% of the time we'll find 2 or 3 SUMMARY III: Emerging Applications tickets among 20 bott/es 8.52", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_052.png", "page_index": 361, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:16:56+07:00" } }, "CO1007-chapter-8-slide-362-0000": { "id": "CO1007-chapter-8-slide-362-0000", "text": "LECTURE 8's PROBLEMS Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK This part discusses Probability Theory with problem solving skills TP.HCM ASSUME that students have already studied Probability theory at basic level in Lecture 8 at home, we practice the following Contents contents today. Introduction Probability Randomness GROUP A: Experiment and sample space Terminology GROUP B: Compute probability of an event Probability Rules Conditional Probability GROUP C: Independent events Dependent and Independent events GROUP D: Probability Models Concept of Conditional Probability Random Variables Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.53", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_053.png", "page_index": 362, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:00+07:00" } }, "CO1007-chapter-8-slide-363-0000": { "id": "CO1007-chapter-8-slide-363-0000", "text": "GROUP A: Experiment and sample space Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai We consider an experiment in information transmission in banking Xuan Toan. Tran Hong Tai sector. The experiment is to select a sequence of 5 letters for BK transmission of a code in a money transfer operation. TP.HCM Let a1, a2,..., as denote the first, ..., fifth letter chosen. The sample space S2 is the set of all possible sequences of five Contents letters. Formally, Introduction Probability S={(a1,a2,...,a5): aie{a,b,c,...,x,y,z}, i=1,...,5}. Randomness Terminology * This is a finite sample space containing 265 possible sequences Probability Rules Conditional P robability of 5 letters, due to the multiplication rule in Equation ??. Dependent and Independent * A sample point (sample unit or element) is any such sequence events Concept of Conditional Probability (ai, a2, ..., a5) in S2. Random Variables Probability Models Quiz: Let E be the event that all the 5 letters in the se- Geometric Model Binomial Model quence are the same. Describe E and find P[E] PROBLEMS SUMMARY III: Emerging Applications 8.54", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_054.png", "page_index": 363, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:04+07:00" } }, "CO1007-chapter-8-slide-364-0000": { "id": "CO1007-chapter-8-slide-364-0000", "text": "GROUP B: Compute probability of an event Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai We knew Rule 1: P(A) + P(AC) = 1 or P(AC) =1 - P(A) BK Rule 2: If events A and B are mutually exc/usive, then TP.HCM P(Aor B) = P(A) + P(B) Contents Introduction Probability But how to find P(A),for any A C S2? Randomness Terminology Mostly use Counting Techniques: Probability Rules 1. Multiplication rule Conditional Probability 2. Permutation rule Dependent and Independent events 3. Combination rule Concept of Conditional Probability Random Variables Each has its special purpose that must be applied properly - the Probability Models right tool for the right job! Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_055.png", "page_index": 364, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:09+07:00" } }, "CO1007-chapter-8-slide-365-0000": { "id": "CO1007-chapter-8-slide-365-0000", "text": "GROUP C: Independent events Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Five identical departments are designed in a given commercial bank. Let E1, E2, . .. , E5 be the events that these five departments BK TP.HCM comply with the quality specifications (non-defective, non bugs...) Under the model of mutual independence the probability that all the five departments are indeed non-defective is Contents Introduction Probability P(EnE2..nE5)=P(E1) P(E)...P(E5) Randomness Terminology Since these departments come from the same production process Probability Rules (i.e. construction company), we can assume that Conditional Probability P(E)=p, all i=1,... .5. Dependent and Independent events Thus, the probability that all the 5 departments are non-defective Concept of Conditional Probability is p5. Random Variables Probability Models What is the probability that one department is defective and Geometric Model Binomial Model all the other four are non-defective? PROBLEMS SUMMARY III: Emerging Applications 8.56", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_056.png", "page_index": 365, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:16+07:00" } }, "CO1007-chapter-8-slide-366-0000": { "id": "CO1007-chapter-8-slide-366-0000", "text": "Independent events - HINTS Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong What is the probability that one department is defective and Tai all the other four are non-defective? BK Let A1 be the event that one out of five parts is defective TP.HCM In order to simplify the notation, we write the intersection of events as their product. Thus Contents Introduction Probability A =Ec Ez E3 E4 E5 UE1EC E3 E4 E5 UEEzE3 E4 E Randomness Terminology Probability Rules U E1 Ez E3 E E5 U E1 Ez E3 E4 E5 Conditional P robability Dependent and Independent A1 is the union of five disjoint events. Therefore events Concept of Conditional Probability P(A1) =P(Ec E2 E3 E4 E5) +..+ Random Variables Probability Models P(E1 Ez E3 E4 ES) = 5p4(1 -p) Geometric Model Binomial Model Can you explain why? PROBLEMS SUMMARY III: Emerging Applications 8.57", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_057.png", "page_index": 366, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:20+07:00" } }, "CO1007-chapter-8-slide-367-0000": { "id": "CO1007-chapter-8-slide-367-0000", "text": "GROUP D: Probability Models in Production Industry Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Wafer Contamination in PC - Electronic Manufacturing BK Let the random variable X denote the number of wafers TP.HCM that need to be analyzed to detect a large particle of contamination Contents Assume that the probability Introduction that a wafer contains a large particle is p = 0.01, and Probability Randomness that the wafers are independent. Terminology Determine the probability distribution of X. Probability Rules Conditional Probability Dependent and Independent QUESTION events Concept of Conditional Probability How do we choose suitable probability distribution? Random Variables Bernoulli, Binomial or Geometric? Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications 8.58", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_058.png", "page_index": 367, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:24+07:00" } }, "CO1007-chapter-8-slide-368-0000": { "id": "CO1007-chapter-8-slide-368-0000", "text": "Determine the probability distribution Discrete Probability Nguyen An Khuong. Let y denote a wafer in which a large particle is present & Tran Tuan Anh, Mai Xuan Toan. Tran Hong Iet n denote a wafer in which it is absent. Tai The set of sample wafers we took: 2 = {y, ny,nny,nnny,. . .} BK The range of the values of X is: Sx ={1,2,3,4,. ..} = N; TP.HCM Sx a countable set, associated with S2. Contents Hence X Geom(p), the probability distribution table is Introduction Probability Probability Distribution Randomness Terminology P(X =1) = 0.01 0.01 Probability Rules P(X = 2) = (0.99)*0.01 0.0099 Conditional Probability P(X = 3) = (0.99)2*0.01 0.009801 Dependent and Independent events P(X = 4) = (0.99)3*0.01 0.009703 Concept of Conditional Probability Random Variables Probability mass (density) f[i] =P[X = i] Probability Models Geometric Model Binomial Model PROBLEMS SUMMARY III: Emerging Applications THE END OF LECTURE 8 : 8.59", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_059.png", "page_index": 368, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:29+07:00" } }, "CO1007-chapter-8-slide-369-0000": { "id": "CO1007-chapter-8-slide-369-0000", "text": "SUMMARY Ill: Emerging Applications and Reminiscence Discrete Probability Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK 1 Artificial intelligence TP.HCM Climate change with Butterfly effect Contents 3 Game theory Introduction Life game to Cellular Automata Probability Randomness Uncertainty engineering: history, current and future 5 Terminology Probability Rules DISCLAIMER: this summary collects views of many generations of Conditional Probability Dependent and Independent computing students in HCMC- Vietnam to Discrete Mathematics events Concept of Conditional Thank all of beloved students! Probability Random Variables Probability Models Geometric Model Binomial Model PROBLEMS SUMMARYIII: Emerging Applications 8.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_8/slide_060.png", "page_index": 369, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:33+07:00" } }, "CO1007-chapter-9-slide-370-0000": { "id": "CO1007-chapter-9-slide-370-0000", "text": "ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 9 Xuan Toan. Tran Hong Tai Introduction to Graphs BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Isomorphism Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 9.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_001.png", "page_index": 370, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:35+07:00" } }, "CO1007-chapter-9-slide-371-0000": { "id": "CO1007-chapter-9-slide-371-0000", "text": "Contents Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Graph definitions Terminology BK TP.HCM Special Graphs Bipartie graph Contents Graph definitions Terminology Representing Graphs and Graph Isomorphism Special Graphs Bipartie graph Representing Graphs Representing Graphs Graph Isomorphism and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Exercise Graph Graph Isomorphism Isomorphism 9.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_002.png", "page_index": 371, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:39+07:00" } }, "CO1007-chapter-9-slide-372-0000": { "id": "CO1007-chapter-9-slide-372-0000", "text": "Course outcomes ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Represent and model practical problems with discrete structures Contents L.O.2.1 - Logically describe some problems arising in Computing Graph definitions L.O.2.2 - Use proving methods: direct, contrapositive, induction Terminology Special Graphs L.O.2.3 - Explain problem modeling using discrete structures Bipartie graph Representing Graphs L.0.3 Understanding of basic probability and random variables and Graph Isomorphis m L.O.3.1 - Define basic probability theory Representing Gra phs L.0.3.2 - Explain discrete random variables Graph Isomorphism Exercise L.0.4 Compute quantities of discrete structures and probabilities Graph L.O.4.1 - Operate (compute/ optimize) on discrete structures Isomorphism L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 9.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_003.png", "page_index": 372, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:43+07:00" } }, "CO1007-chapter-9-slide-373-0000": { "id": "CO1007-chapter-9-slide-373-0000", "text": "Motivations Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai The need of the graph Its applications BK TP.HCM Representation/Storing Electric circuit/board Searching/sorting Chemical structure Contents Networking Graph definitions : Optimization Terminology Special Graphs Map, geometry, . Bipartie graph Representing Graphs and Graph Isomorphis m Graph theory is useful for analysing \"things that are Representing Gra phs Graph Isomorphism Exercise Some difficult problems become easy when represented using Graph Isomorphism a graph. 9.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_004.png", "page_index": 373, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:47+07:00" } }, "CO1007-chapter-9-slide-374-0000": { "id": "CO1007-chapter-9-slide-374-0000", "text": "Graph ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK TP.HCM A graph (d6 thi) G is a pair of (V,E), which are: V - nonempty set of vertices (nodes) (dinh) Contents E - set of edges (canh) Graph definitions Terminology A graph captures abstract relationships between vertices Special Graphs Bipartie graph 2 Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Undirected graph Directed graph Isomorphism 9.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_005.png", "page_index": 374, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:50+07:00" } }, "CO1007-chapter-9-slide-375-0000": { "id": "CO1007-chapter-9-slide-375-0000", "text": "Undirected Graph (Dö thi vö hu6ng) ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Definition (Simple graph (do'n d thi)) TP.HCM Each edge connects two different vertices, and Contents No two edges connect the same pair of vertices Graph definitions An edge between two vertices u and v is denoted as {u,} Terminology Special Graphs Bipartie graph Representing Graphs Lang Son Da Nang Ca Mau and Graph Isomorphis m Representing Gra phs Graph Isomorphism Ho Chi Minh City Exercise Graph Isomorphism 9.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_006.png", "page_index": 375, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:53+07:00" } }, "CO1007-chapter-9-slide-376-0000": { "id": "CO1007-chapter-9-slide-376-0000", "text": "Undirected Graph Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh.Mai Xuan Toan. Tran Hong Tai Definition (Multigraph (da dó thi)) BK TP.HCM Graphs that may have multiple edges connecting the same vertices. An unordered pair of vertices {u,v} are called multiplicity m (b Contents m) if it has m different edges between. Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs Lang Son Da Nang a Mau and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Ho Chi Minh City Graph Isomorphism 9.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_007.png", "page_index": 376, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:17:56+07:00" } }, "CO1007-chapter-9-slide-377-0000": { "id": "CO1007-chapter-9-slide-377-0000", "text": "Undirected Graph Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition (Pseudograph (giá dö thi)) BK TP.HCM Are multigraphs that have loops (khuyén)- edges that connect a vertex to itself Contents Graph definitions Terminology Special Graphs Da Nang Bipartie graph Lang Son Ca Mau Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Ho Chi Minh City Exercise Graph Isomorphism 9.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_008.png", "page_index": 377, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:00+07:00" } }, "CO1007-chapter-9-slide-378-0000": { "id": "CO1007-chapter-9-slide-378-0000", "text": "Terminologies For Undirected Graph Int roduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Neighborhood In an undirected graph G = (V,E) BK two vertices u and v E V are called adjacent (lien ke) if they TP.HCM are end-points (diém dau müt) of edge e E E, and Contents e is incident with (canh /ien thuc) u and Graph definitions e is said to connect (canh ni) u and v; Terminology Special Graphs Bipartie graph Representing Graphs The degree of a vertex and Graph Isomorphis m The degree of a vertex (bac cüa mt dinh), denoted by deg(v) is Representing Gra phs Graph Isomorphism the number of edges incident with it, except that a loop Exercise contributes twice to the degree of that vertex. Graph Isomorphism isolated vertex (dinh c6 lap): vertex of degree 0 pendant vertex (dinh treo): vertex of degree 1 9.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_009.png", "page_index": 378, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:04+07:00" } }, "CO1007-chapter-9-slide-379-0000": { "id": "CO1007-chapter-9-slide-379-0000", "text": "Example ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai What are the degrees and neighborhoods of the vertices in these graphs? BK b c d a 6 c TP.HCM Contents Graph definitions Terminology a f e g e d Special Graphs Bipartie graph G H Representing Graphs and Graph Isomorphis m Representing Gra phs Solution Graph Isomorphism In G, deg(a) = 2, deg(b) = deg(c) = deg(f) = 4, deg(d) = 1, : Exercise Graph Neiborhoods of these vertices are Isomorphism N(a)={b,f},N(b={a,c,e,f},. In H, deg(a) = 4, deg(b) = deg(e) = 6, deg(c) = 1, : Neiborhoods of these vertices are N(a)={b,d,e},N(b={a,b,c,d,e},. 9.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_010.png", "page_index": 379, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:08+07:00" } }, "CO1007-chapter-9-slide-380-0000": { "id": "CO1007-chapter-9-slide-380-0000", "text": "Basic Theorems ntroduction to Graph Nguyen An Khuong. Theorem (The Handshaking Theorem) Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Let G = (V, E) be an undirected graph with m edges. Ther 2m = L deg(v) BK TP.HCM VEV Note that this applies even if multiple edges and loops are Contents present.) Graph definitions Terminology Special Graphs Bipartie graph Example Representing Graphs What are the degrees and neighborhoods of the vertices in these and Graph Isomorphis m graphs? Representing Gra phs Graph Isomorphism 6 c d a b C Exercise Graph somorphism a f g d G H 9.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_011.png", "page_index": 380, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:12+07:00" } }, "CO1007-chapter-9-slide-381-0000": { "id": "CO1007-chapter-9-slide-381-0000", "text": "Prove that .- Int roduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Theorem An undirected graph has an even number of odd-degree vertices BK TP.HCM If the number of vertices in an undirected graph is an odd number, Contents Graph definitions then there exists an even-degree vertex. Terminology Special Graphs Bipartie graph Representing Graphs If the number of vertices in an undirected graph is an odd number, and Graph Isomorphism then the number of vertices with even degree is odd. Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism If the number of vertices in an undirected graph is an even number, then the number of vertices with even degree is even 9.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_012.png", "page_index": 381, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:16+07:00" } }, "CO1007-chapter-9-slide-382-0000": { "id": "CO1007-chapter-9-slide-382-0000", "text": "Exercise Introduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Exercise (1) Tai Is there any undirected simple graph including four vertices that their degrees are respectively 1, 1, 2, 2 ? BK TP.HCM Exercise (2) Contents Is there any undirected simple graph including six vertices that Graph definitions their degree are respectively 2, 3, 3, 3, 3, 3 ? Terminology Special Graphs Bipartie graph Exercise (3) Representing Graphs and Graph Isomorphis m An undirected simple graph G has 15 edges, 3 vertices of degree 4 Representing Gra phs and other vertices having degree 3. What is the number of vertices Graph Isomorphism Exercise of the graph G? Graph Isomorphism Exercise (4) Is it possible that each person has exactly 5 friends in the same group of 9 people ? 9.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_013.png", "page_index": 382, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:20+07:00" } }, "CO1007-chapter-9-slide-383-0000": { "id": "CO1007-chapter-9-slide-383-0000", "text": "Exercise Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM Exercise (6) Give an undirected simple graph G = (V,E) with V= n,show Contents that Graph definitions Terminology @ Vv E V, deg(v) < n, Special Graphs Bipartie graph b) there does not exist simultaneously both a vertex of degree 0 Representing Graphs and a vertex of degree (n - 1), and Graph Isomorphis m c) deduce that there are at least two vertices of the same degree. Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_014.png", "page_index": 383, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:23+07:00" } }, "CO1007-chapter-9-slide-384-0000": { "id": "CO1007-chapter-9-slide-384-0000", "text": "Directed Graph Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Definition (Directed Graph (dó thi c6 hu6ng)) A directed graph G is a pair of (V,E), in which: BK TP.HCM - V - nonempty set of vertices E - set of directed edges (canh c6 huóng, arcs) Contents A directed edge start at u and end at v is denoted as (u, v) Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs Lang Son Da Nang and Graph Qa Mau Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Ho Chi Minh City Isomorphism 9.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_015.png", "page_index": 384, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:28+07:00" } }, "CO1007-chapter-9-slide-385-0000": { "id": "CO1007-chapter-9-slide-385-0000", "text": "Terminologies for Directed Graph Int roduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Neighborhood Xuan Toan. Tran Hong Tai In an directed graph G = (V,E) u is said to be adjacent to (nói tói) v and v is said to be BK adjacent from (duoc nói tü) u if (u,v) is an arc of G, and TP.HCM u is called initial vertex (dinh dau) of (u,u) v is called terminal (dinh cui) or end vertex of (u,) Contents Graph definitions the initial vertex and terminal vertex of a loop are the same Terminology Special Graphs Bipartie graph Representing Graphs The degree of a vertex and Graph Isomorphism In a graph G with directed edges: Representing Graphs Graph Isomorphism : in-degree (bac väo) of a vertex v, denoted by deg(u), is Exercise the number of edges with y as their terminal vertex. Graph Isomorphism out-degree (bac ra) of a vertex v, denoted by deg+(u), is the number of edges with y as their initial vertex. Note: a loop at a vertex contributes 1 to both the in-degree and the out-degree of this vertex. 9.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_016.png", "page_index": 385, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:32+07:00" } }, "CO1007-chapter-9-slide-386-0000": { "id": "CO1007-chapter-9-slide-386-0000", "text": "Basic Theorem Introduction to Graph Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan, Tran Hong Tai Theorem Let G = (V,E) be a graph with directed edges. Then BK TP.HCM deg(v) =C deg+(v) =El. VEV VEV Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs Lang Son Da Nang and Graph Mau Isomorphis m Representing Gra phs Graph Isomorphism Exercise Ho Chi Minh City Graph Isomorphism 9.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_017.png", "page_index": 386, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:36+07:00" } }, "CO1007-chapter-9-slide-387-0000": { "id": "CO1007-chapter-9-slide-387-0000", "text": "Complete Graphs Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong A complete graph (d thi day dü) on n vertices, Kn, is a simple Tai graph that contains exactly one edge between each pair of distinct vertices. BK TP.HCM Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism K5 K4 Exercise What is the largest number of edges a undirected simple graph with 10 vertices can have? Kn can have? 9.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_018.png", "page_index": 387, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:39+07:00" } }, "CO1007-chapter-9-slide-388-0000": { "id": "CO1007-chapter-9-slide-388-0000", "text": "Complement Graphs Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Given G = (V,E) is a simple undirected graph,V= n. A complement graph (D thi bu) of G is Gc = (V,F) satisfies: BK G UGc = Kn and En F = 0. TP.HCM Note: Some documents denote a complement graph is G Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_019.png", "page_index": 388, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:42+07:00" } }, "CO1007-chapter-9-slide-389-0000": { "id": "CO1007-chapter-9-slide-389-0000", "text": "Cycles Introduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai A cycle (d thi vng) Cn n > 3, consists of m vertices V1,V2,...,Vn and edges{v1,V2},{v2,V3},...,{vn-1,Vn}, and BK TP.HCM {vn,V1}. Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism C4 9.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_020.png", "page_index": 389, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:45+07:00" } }, "CO1007-chapter-9-slide-390-0000": { "id": "CO1007-chapter-9-slide-390-0000", "text": "Wheels Introduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai We obtain a wheel (d6 thi hinh banh xe) Wn when we add an additional vertex to a cycle Cn for n > 3, and connect this new BK TP.HCM vertex to each of the n vertices in Cn. Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism W5 W4 9.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_021.png", "page_index": 390, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:48+07:00" } }, "CO1007-chapter-9-slide-391-0000": { "id": "CO1007-chapter-9-slide-391-0000", "text": "n-cube Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai An n-dimensional hypercube (khói n chiêu), Qn, is a graph that has vertices representing the 2\" bit strings of length n. Two vertices are adiacent iff the bit strings that they represent differ in BK TP.HCM exactly one bit position. 110 111 Contents Graph definitions Terminology 10 11 100 101 Special Graphs Bipartie graph Representing Graphs and Graph 0 0 010 011 Isomorphis m O 1 Representing Gra phs Graph Isomorphism Exercise 00 01 000 001 Graph Isomorphism Q1 Q2 Q3 What's about Q4? 9.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_022.png", "page_index": 391, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:52+07:00" } }, "CO1007-chapter-9-slide-392-0000": { "id": "CO1007-chapter-9-slide-392-0000", "text": "Applications of Special Graphs Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Local networks topologies Contents . Star, ring, hybrid Graph definitions Terminology : Parallel processing Special Graphs Bipartie graph Linear array Representing Graphs Mesh network and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_023.png", "page_index": 392, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:18:55+07:00" } }, "CO1007-chapter-9-slide-393-0000": { "id": "CO1007-chapter-9-slide-393-0000", "text": "Graph Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai One goat, a cabbage and a wolf are on a side of river; a boatman wishes to transport them to the other side but, his BK boat being too small, he could transport only one of them at TP.HCM once. How does he proceed not to leave them together without Contents surveillance: the wolf and the goat, as well as the goat and Graph definitions Terminology the cabbage? Special Graphs Bipartie graph Representing Graphs and Graph (BGCW,-) (BGC,W) (BGW,C) (BCW,G) (BG,CW) Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism (CW, BG) (G,BCW) (C,BGW) (W,BGC) (-,BGCW) 9.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_024.png", "page_index": 393, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:00+07:00" } }, "CO1007-chapter-9-slide-394-0000": { "id": "CO1007-chapter-9-slide-394-0000", "text": "Bipartite Graphs Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Definition Tai A simple graph G is called bipartite (d thi phan di) if its vertex set V can be partitioned into two disjoint sets V and V such that BK every edge in the graph connects a vertex in V1 and a vertex in V2 TP.HCM (so that no edge in G connects either two vertices in V or two vertices in V2) Contents Graph definitions Terminology Example Special Graphs Bipartie graph C6 is bipartite Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_025.png", "page_index": 394, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:03+07:00" } }, "CO1007-chapter-9-slide-395-0000": { "id": "CO1007-chapter-9-slide-395-0000", "text": "Complete Bipartite Graphs ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A complete bipartite Km,n is a graph that BK has its vertex set partitioned into two subsets of m and n TP.HCM vertices, respectively, with an edge between two vertices iff one vertex is in the first Contents subset and the other is in the second one Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism K3,3 9.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_026.png", "page_index": 395, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:06+07:00" } }, "CO1007-chapter-9-slide-396-0000": { "id": "CO1007-chapter-9-slide-396-0000", "text": "Bipartite graphs Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan, Tran Hong Tai Example (Bipartite graphs? C6 BK TP.HCM K3 Contents Kn Graph definitions the following graph Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism Wtex($V_1$} wtcx{SV_2$ 9.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_027.png", "page_index": 396, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:09+07:00" } }, "CO1007-chapter-9-slide-397-0000": { "id": "CO1007-chapter-9-slide-397-0000", "text": "Bipartie graph Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphism Representing Graphs Graph Isomorphism Exercise Graph Isomorphism 9.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_028.png", "page_index": 397, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:11+07:00" } }, "CO1007-chapter-9-slide-398-0000": { "id": "CO1007-chapter-9-slide-398-0000", "text": "New Graph From OId Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A subgraph (d6 thi con) of a graph G = (V,E) is a graph BK H =(W,F) where W C V and F C E TP.HCM Definition Contents The union (hop) of two simple graphs G1 = (V1,E1) and Graph definitions Terminology G2 = (V2,E2) is a simple graph with vertex set V1 U V2 and edge Special Graphs set E1 U E2. The union of G1 and G2 is denoted by G1 UG2. Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism G1 G2 G1 U G2 9.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_029.png", "page_index": 398, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:14+07:00" } }, "CO1007-chapter-9-slide-399-0000": { "id": "CO1007-chapter-9-slide-399-0000", "text": "Planar Graphs Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs +5V OV signal and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_030.png", "page_index": 399, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:18+07:00" } }, "CO1007-chapter-9-slide-400-0000": { "id": "CO1007-chapter-9-slide-400-0000", "text": "Planar Graphs Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A graph is called planar (phäng) if it can be drawn in the BK plane without any edges crossing. TP.HCM Such a drawing is called planar representation (bieu di@n phäng) of the graph. Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism K4 K4 with no crossing 9.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_031.png", "page_index": 400, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:21+07:00" } }, "CO1007-chapter-9-slide-401-0000": { "id": "CO1007-chapter-9-slide-401-0000", "text": "Example Introduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example . Is K5 planar? BK Is Q3 planar? TP.HCM 110 111 Contents Graph definitions 100 101 Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m 010 011 Representing Gra phs Graph Isomorphism Exercise Graph 000 001 Isomorphism K5 9.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_032.png", "page_index": 401, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:25+07:00" } }, "CO1007-chapter-9-slide-402-0000": { "id": "CO1007-chapter-9-slide-402-0000", "text": "mportant Corollaries ntroduction to Graph Nguyen An Khuong. Corollary Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai If G is a connected planar simple graph with e edges and vertices where v > 3, then e < 3u - 6. BK If G is a connected planar simple graph with e edges and TP.HCM vertices where v > 3, and no circuits of length 3, then e < 2v - 4. Contents Graph definitions Terminology Example Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph somorphism 9.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_033.png", "page_index": 402, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:28+07:00" } }, "CO1007-chapter-9-slide-403-0000": { "id": "CO1007-chapter-9-slide-403-0000", "text": "Elementary Subdivision ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition Given a planar graph G, an elementary subdivision (phan chia BK TP.HCM so cap) is removing an edge {u,v} and adding a new vertex w together with edges {u,w} and {w,v}. Contents Graphs G1 = (V1,E1) and G2 =(V2,Ez) are called Graph definitions homeomorphic (d6ng phi) if they can obtained from the Terminology Special Graphs same graph by a sequence of elementary subdivisions Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.34", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_034.png", "page_index": 403, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:32+07:00" } }, "CO1007-chapter-9-slide-404-0000": { "id": "CO1007-chapter-9-slide-404-0000", "text": "Kuratowski's Theorem Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Theorem A graph is nonplanar iff it contains a subgraph homeomorphic to BK K3.3 or K5. TP.HCM Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m Representing Gra phs K3,3 Graph Isomorphism Exercise Non-planar Graph K5 Isomorphism Non-planar 9.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_035.png", "page_index": 404, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:35+07:00" } }, "CO1007-chapter-9-slide-405-0000": { "id": "CO1007-chapter-9-slide-405-0000", "text": "Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example A graph is nonplanar iff it contains a subgraph homeomorphic to BK K3.3 TP.HCM Contents a Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph a Isomorphis m Representing Gra phs Graph Isomorphism h Exercise d Graph Isomorphism 9.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_036.png", "page_index": 405, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:38+07:00" } }, "CO1007-chapter-9-slide-406-0000": { "id": "CO1007-chapter-9-slide-406-0000", "text": "Adjacency Lists (Danh säch k@ Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Vertex Adjacent vertices nitial vertex Terminal vertices BK a b,c,e a b, c, d,e TP.HCM b a b b,d c a,d, e c a,c,e Contents d c, e d c, e Graph definitions e a, c,d e b, c,d Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphism Representing Gra phs Graph Isomorphism Exercise Graph Isomorphism 9.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_037.png", "page_index": 406, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:43+07:00" } }, "CO1007-chapter-9-slide-407-0000": { "id": "CO1007-chapter-9-slide-407-0000", "text": "Adjacency Matrices ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition Tai Adjacency matrix (Ma tran k@) AG of G = (V,E) DimensionV xV] BK TP.HCM Matrix elements 1 if(vi,Vj) E E aii 0 otherwise Contents Graph definitions Terminology Special Graphs a b Bipartie graph Representing Graphs and Graph 6 c d Isomorphis m a Representing Gra phs a 0 1 1 1 Graph Isomorphism b 1 0 1 0 Exercise Graph c 1 1 0 0 Isomorphism d 1 0 0 0 c d 9.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_038.png", "page_index": 407, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:48+07:00" } }, "CO1007-chapter-9-slide-408-0000": { "id": "CO1007-chapter-9-slide-408-0000", "text": "Examples Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM Give the graph defined by the following adjacency matrix Contents A B C D E Graph definitions Terminology Special Graphs A 0 0 1 1 0 Bipartie graph B 0 0 0 1 0 Representing Graphs and Graph C 1 0 0 1 0 Isomorphis m Representing Gra phs D 1 1 1 0 1 Graph Isomorphism E 0 0 0 1 0 Exercise Graph Isomorphism 9.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_039.png", "page_index": 408, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:19:54+07:00" } }, "CO1007-chapter-9-slide-409-0000": { "id": "CO1007-chapter-9-slide-409-0000", "text": "Adjacency Matrices ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM Give the directed graph defined by the following adjacency matrix Contents A B C D E Graph definitions Terminology Special Graphs A 1 0 1 1 0 Bipartie graph B 0 0 0 0 Representing Graphs and Graph C 1 0 0 0 0 Isomorphis m Representing Graphs D 1 1 1 0 1 Graph Isomorphism E 1 0 0 0 0 Exercise Graph Isomorphism 9.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_040.png", "page_index": 409, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:00+07:00" } }, "CO1007-chapter-9-slide-410-0000": { "id": "CO1007-chapter-9-slide-410-0000", "text": "ncidence Matrices ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition Tai Incidence matrix (ma tran lien thuc) MG of G = (V,E Dimension VxE BK TP.HCM Matrix elements 1 if ej is incident with vi mij 0 otherwise Contents Graph definitions Terminology Special Graphs a e2 b Bipartie graph Representing Graphs and Graph e1 Isomorphism e2 e3 e4 Representing Graphs a 1 1 1 0 Graph Isomorphism e1 b 0 1 0 1 Exercise Graph c 1 0 0 1 Isomorphism d 0 0 1 0 C d 9.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_041.png", "page_index": 410, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:05+07:00" } }, "CO1007-chapter-9-slide-411-0000": { "id": "CO1007-chapter-9-slide-411-0000", "text": "Examples ntroduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Example BK TP.HCM Give incidence matrix according to the following graph Contents F Graph definitions Terminology B Special Graphs Bipartie graph Representing Graphs and Graph A Isomorphism Representing Graphs Graph Isomorphism E D Exercise Graph G Isomorphism 9.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_042.png", "page_index": 411, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:08+07:00" } }, "CO1007-chapter-9-slide-412-0000": { "id": "CO1007-chapter-9-slide-412-0000", "text": "Graph Isomorphism Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition G1 =(V1,E) and G2 =(V2,E2) are isomorphic (dang cäu) if BK there is a one-to-one function f from V to V2 with the property TP.HCM that a and b are adjacent in G1 iif f(a) and f(b) are adjacent in G2, for all a and b in V. Such a function f is called an Contents isomorphism (möt däng cäu). Graph definitions Terminology (i.e. there is a one-to-one correspondence between vertices of the Special Graphs two graphs that preserves the adjacency relationship.) Bipartie graph Representing Graphs and Graph Isomorphis m u1 u2 V1 V2 Representing Gra phs Isomorphism function f : U -? Graph Isomorphism V with Exercise f(u1) =v1 f(u2) = v4 Graph Isomorphism f(u3) = V3 f(u4) = v2 u3 u4 V3 V4 9.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_043.png", "page_index": 412, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:13+07:00" } }, "CO1007-chapter-9-slide-413-0000": { "id": "CO1007-chapter-9-slide-413-0000", "text": "Isomorphism ? Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Graph definitions Terminology Special Graphs Bipartie graph Representing Graphs and Graph Isomorphism Representing Graphs Graph Isomorphism Exercise Graph Isomorphism 9.44", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_044.png", "page_index": 413, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:15+07:00" } }, "CO1007-chapter-9-slide-414-0000": { "id": "CO1007-chapter-9-slide-414-0000", "text": "Isomorphism? ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK A TP.HCM A B Contents B Graph definitions Terminology Special Graphs c F Bipartie graph Representing Graphs and Graph Isomorphis m D Representing Gra phs E D Graph Isomorphism G2 Exercise F E Graph G1 Isomorphism 9.45", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_045.png", "page_index": 414, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:18+07:00" } }, "CO1007-chapter-9-slide-415-0000": { "id": "CO1007-chapter-9-slide-415-0000", "text": "Isomorphism? Introduction to Graphs Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai A1 B2 BK TP.HCM Contents Graph definitions Terminology D1 E1 Special Graphs Bipartie graph F1 Representing Graphs and Graph Isomorphis m Representing Gra phs B1 Graph Isomorphism D2 C2 G1 G2 Exercise Graph Isomorphism 9.46", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_046.png", "page_index": 415, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:21+07:00" } }, "CO1007-chapter-9-slide-416-0000": { "id": "CO1007-chapter-9-slide-416-0000", "text": "Isomorphism Int roduction to Graphs Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Are the simple graphs with the following adjacency matrices BK isomorphic ? TP.HCM 0 0 1 Contents 0 0 1 1 Graph definitions 1 0 J 0 0 Terminology Special Graphs 0 1 0 1 0 1 1 1 Bipartie graph 1 0 0 1 1 0 0 Representing Graphs and Graph 0 0 0 1 0 0 Isomorphis m Representing Graphs 1 1 0 1 1 0 Graph Isomorphism 1 0 1 0 Exercise 0 1 0 1 Graph 1 0 0 1 1 0 0 0 Isomorphism 3 1 0 0 1 0 0 0 1 0 1 1 0 1 0 1 0 9.47", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_047.png", "page_index": 416, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:30+07:00" } }, "CO1007-chapter-9-slide-417-0000": { "id": "CO1007-chapter-9-slide-417-0000", "text": "Isomorphism ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Determine whether the graphs (without loops) with the incidence BK TP.HCM matrices are isomorphic. 0 1 1 0 Contents 0 1 1 1 0 Graph definitions Terminology 1 0 0 1 Special Graphs Bipartie graph 1 0 0 0 0 1 0 0 1 Representing Graphs 1 0 1 0 1 0 1 1 1 0 and Graph Isomorphis m 0 0 0 1 1 1 0 0 1 0 Representing Gra phs 0 1 1 1 0 0 1 0 Graph Isomorphism 1 1 Exercise Extend the definition of isomorphism of simple graphs to Graph undirected graphs containing loops and multiple edges Isomorphism Define isomorphism of directed graphs 9.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_048.png", "page_index": 417, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:36+07:00" } }, "CO1007-chapter-9-slide-418-0000": { "id": "CO1007-chapter-9-slide-418-0000", "text": "Revision Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai Given an adjacency matrix of G and an incidence matrix of BK (G1) A B C D E (G2) TP.HCM e1 C2 C3 c4 C5 C6 C7 A 0 1 1 1 1 A 1 0 0 0 1 1 0 B 1 0 0 1 0 B 1 1 0 1 0 0 0 C 1 0 0 1 0 c 0 1 1 0 1 0 0 Contents D 1 1 1 0 1 D 0 0 1 1 0 0 1 Graph definitions E 1 0 0 1 0 E 0 0 0 0 0 1 1 Terminology What is the relation between G1 and G2 Special Graphs Bipartie graph A Isomorphism Representing Graphs and Graph Isomorphis m Non-isomorphism. Representing Gra phs Order. Graph Isomorphism Exercise Equivalence Graph Isomorphism 9.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_049.png", "page_index": 418, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:45+07:00" } }, "CO1007-chapter-9-slide-419-0000": { "id": "CO1007-chapter-9-slide-419-0000", "text": "Revision ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Given a complete graph K5 and a complete bi-partite graph K3,2 Contents Which of the following assessment is/are true? Graph definitions Terminology A) K3.2 and K5 are non-isomorphic Special Graphs Bipartie graph K3,2 and K5 have the same number of vertex. Representing Graphs and Graph K3,2 and K5 have the same number of edge Isomorphis m Representing Gra phs K3,2 and K5 are isomorphic Graph Isomorphism Exercise Graph Isomorphism 9.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_050.png", "page_index": 419, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:48+07:00" } }, "CO1007-chapter-9-slide-420-0000": { "id": "CO1007-chapter-9-slide-420-0000", "text": "Revision ntroduction to Graph Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Choose the correct statement of a (undirected simp/e graph) has n vertices. Contents Graph definitions A Bac cüa möt dinh bät ky trong dö thi nh6 hon n - 2. Terminology Special Graphs B Tön tai mt dinh trong d6 thi c6 bac la 1. Bipartie graph Representing Graphs Khng thé chúa dinh c lap. and Graph Isomorphis m Tn tai hai dinh trong dö thi c6 cüng s bac Representing Gra phs Graph Isomorphism E) Cac dap an khäc déu sai. Exercise Graph Isomorphism 9.51", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_051.png", "page_index": 420, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:52+07:00" } }, "CO1007-chapter-9-slide-421-0000": { "id": "CO1007-chapter-9-slide-421-0000", "text": "Prove that .. Int roduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai There are 101 invited people in a party. BK Suppose that A knows B =7 B knows A TP.HCM Prove that 1 at least one people knows an even number of other people Contents at least two people who know the same number of people Graph definitions Terminology (but not considering himself) Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m A chess tournament of n persons plays according to the circle Representing Gra phs Graph Isomorphism competition. Prove that at any moment of the tournament there Exercise are always two players having identical number of games played. Graph And if n > 4, at any intermediate moment of the tournament, Isomorphism there are always two players having identical number of games that they are the winner. 9.52", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_052.png", "page_index": 421, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:20:56+07:00" } }, "CO1007-chapter-9-slide-422-0000": { "id": "CO1007-chapter-9-slide-422-0000", "text": "Revision Introduction to Graph: Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai In a tournament with n teams participated (n > 4), n + 1 competition games were happening. Prove that there exists a team that has played at least three matches. BK TP.HCM With any four of the n people (n > 4), there exists a person who Contents Graph definitions knows the three others. Prove that there exists a person who Terminology knows all n - 1 others. Special Graphs Bipartie graph Representing Graphs and Graph Isomorphis m In a party of 6 people, prove that there are 3 people who know Representing Gra phs Graph Isomorphism each other or 3 people who do not know each other. Exercise Graph Isomorphism During a summer vacation, 7 friends are vacationing away. They promised each other that during the holidays each person must write to exactly three of them. Prove that there is someone who does not write back to the his sender. 9.53", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_9/slide_053.png", "page_index": 422, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:00+07:00" } }, "CO1007-chapter-10-slide-423-0000": { "id": "CO1007-chapter-10-slide-423-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 10 Xuan Toan. Tran Hong Tai Graph connectivity BK TP.HCM Discrete Structures for Computing on June 16, 2024 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Others Graph Coloring Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 10.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_001.png", "page_index": 423, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:03+07:00" } }, "CO1007-chapter-10-slide-424-0000": { "id": "CO1007-chapter-10-slide-424-0000", "text": "Acknowledgement Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Some slides about Euler and Hamilton circuits are created by Connectivity Paths and Circuits Chung Ki-hong and Hur Joon-seok from KAlST. Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorit hn Ford's algorithm Others Graph Coloring 10.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_002.png", "page_index": 424, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:06+07:00" } }, "CO1007-chapter-10-slide-425-0000": { "id": "CO1007-chapter-10-slide-425-0000", "text": "Course outcomes Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Represent and model practical problems with discrete structures Contents L.O.2.1 - Logically describe some problems arising in Computing Connectivity L.O.2.2 - Use proving methods: direct, contrapositive, induction Paths and Circuits L.O.2.3 - Explain problem modeling using discrete structures Euler and Hamilton Paths Euler Paths and Circuits L.0.3 Understanding of basic probability and random variables Hamilton Paths and Circuits L.O.3.1 - Define basic probability theory Shortest Path Problem L.O.3.2 - Explain discrete random variables Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm L.0.4 Compute quantities of discrete structures and probabilities Ford's algorith m L.O.4.1 - Operate (compute/ optimize) on discrete structures Others L.0.4.2 - Compute probabilities of various events, conditional Graph Coloring ones, Bayes theorem 10.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_003.png", "page_index": 425, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:11+07:00" } }, "CO1007-chapter-10-slide-426-0000": { "id": "CO1007-chapter-10-slide-426-0000", "text": "Contents Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Connectivity Paths and Circuits BK TP.HCM Euler and Hamilton Paths Euler Paths and Circuits Contents Hamilton Paths and Circuits Connectivity Paths and Circuits Euler and Hamilton 3) Shortest Path Problem Paths Dijkstra's Algorithm Euler Paths and Circuits Hamilton Paths and Circuits Bellman-Ford Algorithm Shortest Path Problem Floyd-Warshall Algorithm Dijkstra's Algorith m Bellman-Ford Algorith m Ford`s algorithm Floyd-Warshall Algorithm Others Ford's algorith m Others Graph Coloring Graph Coloring 10.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_004.png", "page_index": 426, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:15+07:00" } }, "CO1007-chapter-10-slide-427-0000": { "id": "CO1007-chapter-10-slide-427-0000", "text": "Path and Circuits Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Definition (in undirected graph) Xuan Toan. Tran Hong Tai Path (dung di) of length n from u to v: a sequence of n edges{xo,X1},{x1,x2},...,{n-1,xn}, where xo= u and BK TP.HCM Xn A path is a circuit (chu trinh) if it begins and ends at the Contents same vertex, u - v. Connectivity A path or circuit is simple (don) if it does not contain the Paths and Circuits same edge more than once. Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits a a c Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring d e d e Simple path Not simple path 10.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_005.png", "page_index": 427, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:20+07:00" } }, "CO1007-chapter-10-slide-428-0000": { "id": "CO1007-chapter-10-slide-428-0000", "text": "Paths and Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai b Xuan Toan. Tran Hong C Tai BK TP.HCM Original Graph a b Contents Simple path Connectivity (length 1) Paths and Circuits Euler and Hamilton Paths d e Euler Paths and Circuits b b Hamilton Paths and Circuits a C a Shortest Path Problem simple path Simple Circuit Dijkstra's Algorith m (length 3) (length 3) Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m d e f d € Others a b C a b C Graph Coloring Simple Circuit Simple path (length 4) (length 4) d e d e + 10.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_006.png", "page_index": 428, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:24+07:00" } }, "CO1007-chapter-10-slide-429-0000": { "id": "CO1007-chapter-10-slide-429-0000", "text": "Path and Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition (in directed graphs) BK Path is a sequence of (o,1,(1,2), TP.HCM ,(n-1,n), where xo = u and xn = v. Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_007.png", "page_index": 429, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:27+07:00" } }, "CO1007-chapter-10-slide-430-0000": { "id": "CO1007-chapter-10-slide-430-0000", "text": "Connectedness in Undirected Graphs Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition Tai An undirected graph is called connected (lién thng) if there is a path between every pair of distinct vertices of the graph BK TP.HCM There is a simple path between every pair of distinct vertices of a connected undirected graph Contents Connectivity Paths and Circuits d Euler and Hamilton e Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others a c h g Graph Coloring Connected graph 10.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_008.png", "page_index": 430, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:31+07:00" } }, "CO1007-chapter-10-slide-431-0000": { "id": "CO1007-chapter-10-slide-431-0000", "text": "Connectedness in Undirected Graphs Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition Tai An undirected graph is called connected (lién thng) if there is a path between every pair of distinct vertices of the graph BK TP.HCM There is a simple path between every pair of distinct vertices of a connected undirected graph Contents Connectivity Paths and Circuits d Euler and Hamilton e Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others a C h g Graph Coloring Disconnected graph 10.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_009.png", "page_index": 431, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:35+07:00" } }, "CO1007-chapter-10-slide-432-0000": { "id": "CO1007-chapter-10-slide-432-0000", "text": "Connectedness in Undirected Graphs Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Definition Tai An undirected graph is called connected (lién thng) if there is a path between every pair of distinct vertices of the graph BK TP.HCM There is a simple path between every pair of distinct vertices of a connected undirected graph Contents Connectivity Paths and Circuits d Euler and Hamilton Paths h Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorit hm Ford's algorith m Others a Graph Coloring Connected components (thänh phan lien thöng) 10.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_010.png", "page_index": 432, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:40+07:00" } }, "CO1007-chapter-10-slide-433-0000": { "id": "CO1007-chapter-10-slide-433-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai a d g BK TP.HCM Contents Connectivity Paths and Circuits b C e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem b is a cut vertex (dinh cät) or articulation point (diem khóp) Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_011.png", "page_index": 433, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:43+07:00" } }, "CO1007-chapter-10-slide-434-0000": { "id": "CO1007-chapter-10-slide-434-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai a d g BK TP.HCM Contents Connectivity Paths and Circuits C e h Euler and Hamilton b Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem b is a cut vertex (dinh cät) or articulation point (diem khóp) Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_012.png", "page_index": 434, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:46+07:00" } }, "CO1007-chapter-10-slide-435-0000": { "id": "CO1007-chapter-10-slide-435-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai a a g BK TP.HCM Contents Connectivity Paths and Circuits e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem b is a cut vertex (dinh cat) or articulation point (diem khöp) Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_013.png", "page_index": 435, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:50+07:00" } }, "CO1007-chapter-10-slide-436-0000": { "id": "CO1007-chapter-10-slide-436-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai a d g BK TP.HCM Contents Connectivity Paths and Circuits b C e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem b is a cut vertex (dinh cät) or articulation point (diem khöp) Dijkstra's Algorithm Bellman-Ford Algorith m What else? Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_014.png", "page_index": 436, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:53+07:00" } }, "CO1007-chapter-10-slide-437-0000": { "id": "CO1007-chapter-10-slide-437-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai a a g BK TP.HCM Contents Connectivity Paths and Circuits b C e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem : b is a cut vertex (dinh cät) or articulation point (diem khóp) Dijkstra's Algorithm Bellman-Ford Algorith m What else? Floyd- Warshall Algorithm Ford's algorith m {a,b} is a cut edge (canh cät) or bridge (cau) Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_015.png", "page_index": 437, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:21:56+07:00" } }, "CO1007-chapter-10-slide-438-0000": { "id": "CO1007-chapter-10-slide-438-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai a a g BK TP.HCM Contents Connectivity Paths and Circuits b C e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem : b is a cut vertex (dinh cät) or articulation point (diem khóp) Dijkstra's Algorithm Bellman-Ford Algorith m What else? Floyd- Warshall Algorithm Ford's algorith m {a,b} is a cut edge (canh cät) or bridge (cau) Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_016.png", "page_index": 438, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:00+07:00" } }, "CO1007-chapter-10-slide-439-0000": { "id": "CO1007-chapter-10-slide-439-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai a a g BK TP.HCM Contents Connectivity Paths and Circuits b C e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem b is a cut vertex (dinh cat) or articulation point (diem khöp) Dijkstra's Algorithm Bellman-Ford Algorith m What else? Floyd- Warshall Algorithm Ford's algorith m {a,b} is a cut edge (canh cät) or bridge (cau) Others Graph Coloring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_017.png", "page_index": 439, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:03+07:00" } }, "CO1007-chapter-10-slide-440-0000": { "id": "CO1007-chapter-10-slide-440-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai a a g BK TP.HCM Contents Connectivity Paths and Circuits b c e h Euler and Hamilton Paths Euler Paths and Circuits Definition Hamilton Paths and Circuits Shortest Path Problem b is a cut vertex (dinh cät) or articulation point (diem khöp) Dijkstra's Algorithm Bellman-Ford Algorith m What else? Floyd- Warshall Algorithm Ford's algorith m {a,b} is a cut edge (canh cat) or bridge (cau). What else? Others Graph Colring 10.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_018.png", "page_index": 440, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:07+07:00" } }, "CO1007-chapter-10-slide-441-0000": { "id": "CO1007-chapter-10-slide-441-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. b c d Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM g a e Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithn Ford's algorithm Others Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_019.png", "page_index": 441, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:10+07:00" } }, "CO1007-chapter-10-slide-442-0000": { "id": "CO1007-chapter-10-slide-442-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. b c d Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM g a C Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khng thé phán täch) Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_020.png", "page_index": 442, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:13+07:00" } }, "CO1007-chapter-10-slide-443-0000": { "id": "CO1007-chapter-10-slide-443-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. b d Tran Tuan Anh, Mai C Xuan Toan,Tran Hon Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khng thé phán täch) Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_021.png", "page_index": 443, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:17+07:00" } }, "CO1007-chapter-10-slide-444-0000": { "id": "CO1007-chapter-10-slide-444-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. b d Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi kh0ng the phan tach Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithn Ford's algorith m Others Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_022.png", "page_index": 444, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:20+07:00" } }, "CO1007-chapter-10-slide-445-0000": { "id": "CO1007-chapter-10-slide-445-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong b d Tran Tuan Anh, Mai C Xuan Toan. Tran Hong Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khöng the phan tach) Shortest Path Problem Dijkstra's Algorithm The vertex cut is {c, f}, so the minimum number of vertices Bellman-Ford Algorith m Floyd- Warshall Algorithm in a vertex cut, vertex connectivity (lién thng dinh) Ford's algorith m Others k(G) =2. Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_023.png", "page_index": 445, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:23+07:00" } }, "CO1007-chapter-10-slide-446-0000": { "id": "CO1007-chapter-10-slide-446-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong b c d Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khöng thé phan tách) Shortest Path Problem Dijkstra's Algorithm The vertex cut is {c, f}, so the minimum number of vertices Bellman-Ford Algorith m Floyd- Warshall Algorithm in a vertex cut, vertex connectivity (lién thóng dinh Ford's algorith m Others k(G) =2. Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_024.png", "page_index": 446, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:27+07:00" } }, "CO1007-chapter-10-slide-447-0000": { "id": "CO1007-chapter-10-slide-447-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong. b c d Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khöng thé phan tách) Shortest Path Problem Dijkstra's Algorithm The vertex cut is {c, f}, so the minimum number of vertices Bellman-Ford Algorith m Floyd- Warshall Algorithm in a vertex cut, vertex connectivity (lién thóng dinh Ford's algorith m Others k(G) =2. Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_025.png", "page_index": 447, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:30+07:00" } }, "CO1007-chapter-10-slide-448-0000": { "id": "CO1007-chapter-10-slide-448-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong b c d Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khöng thé phan tách) Shortest Path Problem Dijkstra's Algorithm The vertex cut is {c, f}, so the minimum number of vertices Bellman-Ford Algorith m Floyd- Warshall Algorithm in a vertex cut, vertex connectivity (lién thóng dinh Ford's algorith m Others k(G) =2. Graph Coloring 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_026.png", "page_index": 448, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:33+07:00" } }, "CO1007-chapter-10-slide-449-0000": { "id": "CO1007-chapter-10-slide-449-0000", "text": "How Connected is a Graph? Graph connectivity Nguyen An Khuong b c d Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM g a Contents Connectivity Paths and Circuits Euler and Hamilton Definition Paths Euler Paths and Circuits This graph doesn't have cut vertices: nonseparable graph (dó Hamilton Paths and Circuits thi khóng thé phán tách) Shortest Path Problem Dijkstra's Algorithm The vertex cut is {c, f}, so the minimum number of vertices Bellman-Ford Algorith m Floyd- Warshall Algorit hm in a vertex cut, vertex connectivity (lien thng dinh) Ford's algorith m Others k(G) =2. Graph Coloring The edge cut is{{b,c},{a, f},{f,g}}, the minimum number of edges in an edge cut, edge connectivity (lien thng canh) (G) = 3. 10.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_027.png", "page_index": 449, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:39+07:00" } }, "CO1007-chapter-10-slide-450-0000": { "id": "CO1007-chapter-10-slide-450-0000", "text": "Applications of Vertex and Edge Connectivity Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Reliability of networks Minimum number of routers that disconnect the network Contents Minimum number of fiber optic links that can be down to Connectivity Paths and Circuits disconnect the network Euler and Hamilton Highway network Paths Euler Paths and Circuits Minimum number of intersections that can be closed Hamilton Paths and Circuits Minimum number of roads that can be closed Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_028.png", "page_index": 450, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:42+07:00" } }, "CO1007-chapter-10-slide-451-0000": { "id": "CO1007-chapter-10-slide-451-0000", "text": "Applications Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Determine whether the graphs below are isomorphic u1 V1 BK TP.HCM u2 V6 V2 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits u5 V5 V3 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm U4 Others G H Graph Coloring 10.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_029.png", "page_index": 451, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:46+07:00" } }, "CO1007-chapter-10-slide-452-0000": { "id": "CO1007-chapter-10-slide-452-0000", "text": "Applications Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Determine whether the graphs below are isomorphic u1 V1 BK TP.HCM u2 V6 V2 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits u5 V5 V3 Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m U4 Others G H Graph Coloring Solution H has a simple circuit of length three, not G. 10.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_030.png", "page_index": 452, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:51+07:00" } }, "CO1007-chapter-10-slide-453-0000": { "id": "CO1007-chapter-10-slide-453-0000", "text": "Applications Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Determine whether the graphs below are isomorphic BK u2 22 TP.HCM Contents Connectivity V1 V3 Paths and Circuits Euler and Hamilton Paths V5 V4 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_031.png", "page_index": 453, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:54+07:00" } }, "CO1007-chapter-10-slide-454-0000": { "id": "CO1007-chapter-10-slide-454-0000", "text": "Applications Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Determine whether the graphs below are isomorphic BK u2 V2 TP.HCM Contents Connectivity V1 23 Paths and Circuits Euler and Hamilton Paths u5 u4 V5 U4 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Solution Dijkstra's Algorithm Bellman-Ford Algorith m Both graphs have the same vertices, edges, degrees, circuits. They Floyd-Warshall Algorit hm Ford's algorith m may be isomorphic. Others To find a possible isomorphism, we can follow paths that go Graph Coloring through all vertices so that the corresponding vertices in the two graphs have the same degrees. 10.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_032.png", "page_index": 454, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:22:58+07:00" } }, "CO1007-chapter-10-slide-455-0000": { "id": "CO1007-chapter-10-slide-455-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Find all the cut vertices, cut edges of the graphs Tai @) Cn; where n > 3 b Wn where n > 3 BK TP.HCM @ Km,n where m Z 2,n Z 2 a d e a Contents Connectivity Paths and Circuits Euler and Hamilton Paths b c Euler Paths and Circuits b Hamilton Paths and Circuits a Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m g Others Graph Coloring a 10.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_033.png", "page_index": 455, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:02+07:00" } }, "CO1007-chapter-10-slide-456-0000": { "id": "CO1007-chapter-10-slide-456-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai For each of these graphs, find x(G),(G) a b a BK TP.HCM b) g h Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_034.png", "page_index": 456, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:05+07:00" } }, "CO1007-chapter-10-slide-457-0000": { "id": "CO1007-chapter-10-slide-457-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai For each of these graphs,find x(G),(G a b a BK TP.HCM b) g Contents Connectivity Paths and Circuits Construct a graph G with k(G) = 1,A(G) = 2, and Euler and Hamilton Paths minvey deg(v) = 3. Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_035.png", "page_index": 457, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:10+07:00" } }, "CO1007-chapter-10-slide-458-0000": { "id": "CO1007-chapter-10-slide-458-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai For each of these graphs,find x(G),(G a 6 BK TP.HCM b) g Contents Connectivity Paths and Circuits Construct a graph G with k(G) = 1,A(G) = 2, and Euler and Hamilton Paths minvev deg(v) = 3. Euler Paths and Circuits b f Hamilton Paths and Circuits c 9 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring a d h 10.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_036.png", "page_index": 458, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:13+07:00" } }, "CO1007-chapter-10-slide-459-0000": { "id": "CO1007-chapter-10-slide-459-0000", "text": "Connectedness in Directed Graphs Graph connectivity Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai An directed graph is strongly connected (lien thng manh) if there is a path between any two vertices in the graph (for BK both directions). TP.HCM An directed graph is weakly connected (lien thng yéu) if there is a path between any two vertices in the underlying Contents undirected graph. Connectivity Paths and Circuits Euler and Hamilton a b a Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring e d e d Strongly connected Weakly connected 10.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_037.png", "page_index": 459, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:18+07:00" } }, "CO1007-chapter-10-slide-460-0000": { "id": "CO1007-chapter-10-slide-460-0000", "text": "The Famous Problem of Seven Bridges of Königsberg Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Is there a route that a person crosses all the seven bridges Bellman-Ford Algorith m once? Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_038.png", "page_index": 460, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:22+07:00" } }, "CO1007-chapter-10-slide-461-0000": { "id": "CO1007-chapter-10-slide-461-0000", "text": "Euler Solution Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Euler gave the solution: It is not possible to cross all the Shortest Path Problem Dijkstra's Algorith m bridges exactly once Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_039.png", "page_index": 461, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:24+07:00" } }, "CO1007-chapter-10-slide-462-0000": { "id": "CO1007-chapter-10-slide-462-0000", "text": "What is Euler Path and Circuit? Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Euler Path (dung di Eu/er) is a path in the graph that passes each edge only once. Contents The problem of Seven Bridges of Königsberg can be also Connectivity stated: Does Euler Path exist in the graph? Paths and Circuits Euler and Hamilton Euler Circuit (chu trinh Eu/er) is a path in the graph that Paths Euler Paths and Circuits passes each edge only once and return back to its original Hamilton Paths and Circuits position. Shortest Path Problem Dijkstra's Algorith m From Definition, Euler Circuit is a subset of Euler Path Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_040.png", "page_index": 462, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:29+07:00" } }, "CO1007-chapter-10-slide-463-0000": { "id": "CO1007-chapter-10-slide-463-0000", "text": "Examples of Euler Path and Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh.Mai Xuan Toan, Tran Hong END Tai 1 BK TP.HCM ESTART Contents Connectivity Euler path Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others STAKT 4 Graph Coloring END Euler circuit 10.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_041.png", "page_index": 463, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:33+07:00" } }, "CO1007-chapter-10-slide-464-0000": { "id": "CO1007-chapter-10-slide-464-0000", "text": "Conditions for Existence Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK TP.HCM In a connected multigraph Contents Euler Circuit existence: no odd-degree nodes exist in the Connectivity graph. Paths and Circuits Euler and Hamilton Euler Path existence: 2 or no odd-degree nodes exist in the Paths Euler Paths and Circuits graph. Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_042.png", "page_index": 464, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:36+07:00" } }, "CO1007-chapter-10-slide-465-0000": { "id": "CO1007-chapter-10-slide-465-0000", "text": "Back to the Seven Bridges Problem Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Four vertices of odd degree Hamilton Paths and Circuits Shortest Path Problem No Euler circuit - cannot cross each bridge exactly once Dijkstra's Algorith m and return to starting point Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm No Euler path, either Others Graph Coloring 10.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_043.png", "page_index": 465, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:39+07:00" } }, "CO1007-chapter-10-slide-466-0000": { "id": "CO1007-chapter-10-slide-466-0000", "text": "Searching Euler Circuits and Paths - Fleury's Algorithm Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Choose a random vertex (if circuit) or an odd degree vertex (if path) Contents Connectivity Pick an edge joined to another vertex so that it is not a cut Paths and Circuits edge unless there is no alternative. Euler and Hamilton Paths Remove the chosen edge. The above procedure is repeated Euler Paths and Circuits Hamilton Paths and Circuits until all edges are covered. Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_044.png", "page_index": 466, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:43+07:00" } }, "CO1007-chapter-10-slide-467-0000": { "id": "CO1007-chapter-10-slide-467-0000", "text": "Searching Euler Circuits and Paths - Hierholzer's Algorithm Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM Choose a starting vertex and find a circuit Contents As long as there exists a vertex that belongs to the current Connectivity tour but that has adjacent edges not part of the tour, start Paths and Circuits Euler and Hamilton another circuit from v Paths Euler Paths and Circuits More efficient algorithm, O(n) Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_045.png", "page_index": 467, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:46+07:00" } }, "CO1007-chapter-10-slide-468-0000": { "id": "CO1007-chapter-10-slide-468-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Are these following graph Euler path (circuit)? If yes, find one a g e BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths C d Euler Paths and Circuits Hamilton Paths and Circuits a Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring e d 10.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_046.png", "page_index": 468, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:50+07:00" } }, "CO1007-chapter-10-slide-469-0000": { "id": "CO1007-chapter-10-slide-469-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Determine whether the picture shown can be drawn with a pencil TP.HCM in a continuous motion without lifting the pencil or retracing part of the picture. Contents Connectivity Paths and Circuits Euler and Hamilton Paths b) Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_047.png", "page_index": 469, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:53+07:00" } }, "CO1007-chapter-10-slide-470-0000": { "id": "CO1007-chapter-10-slide-470-0000", "text": "Exercise - Euler path & circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai b 6 Xuan Toan. Tran Hong a Tai BK TP.HCM g 7 a Contents Connectivity Paths and Circuits d Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m a 6 Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring F g 10.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_048.png", "page_index": 470, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:23:56+07:00" } }, "CO1007-chapter-10-slide-471-0000": { "id": "CO1007-chapter-10-slide-471-0000", "text": "Traveling Salesman Problem Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths KIEN GIANG Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m ONUAC Bellman-Ford Algorith m (BA RA-VUNG TAU Floyd-Warshall Algorithm Ford's algorithm Is there the possible tour that visits each city exactly once? Others Graph Coloring 10.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_049.png", "page_index": 471, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:00+07:00" } }, "CO1007-chapter-10-slide-472-0000": { "id": "CO1007-chapter-10-slide-472-0000", "text": "What Is A Hamilton Circuit? Graph connectivity Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong The circuit that visit each vertex in a graph once Tai Example BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Original Graphs Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorit hm Ford's algorithm Others Graph Coloring Hamilton Circuit 10.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_050.png", "page_index": 472, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:02+07:00" } }, "CO1007-chapter-10-slide-473-0000": { "id": "CO1007-chapter-10-slide-473-0000", "text": "Rules of Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai deg(v) = 2 for Vv in Hamilton circuit! BK TP.HCM Rule 1 if deg() = 2, both edge must be used. Contents Connectivity Paths and Circuits Euler and Hamilton Paths Rule 2 No subcircuit (chu trinh con) can be formed Euler Paths and Circuits Hamilton Paths and Circuits Rule 3 Once two edges at a vertex v is determined, all Shortest Path Problem Dijkstra's Algorith m other edges incident at u must be removed. Bellman-Ford Algorith m Floyd-Warshall Algorithm V Ford's algorith m Others Graph Coloring 10.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_051.png", "page_index": 473, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:06+07:00" } }, "CO1007-chapter-10-slide-474-0000": { "id": "CO1007-chapter-10-slide-474-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m O thers Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_052.png", "page_index": 474, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:09+07:00" } }, "CO1007-chapter-10-slide-475-0000": { "id": "CO1007-chapter-10-slide-475-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m O thers Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_053.png", "page_index": 475, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:13+07:00" } }, "CO1007-chapter-10-slide-476-0000": { "id": "CO1007-chapter-10-slide-476-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_054.png", "page_index": 476, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:16+07:00" } }, "CO1007-chapter-10-slide-477-0000": { "id": "CO1007-chapter-10-slide-477-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits 0 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_055.png", "page_index": 477, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:19+07:00" } }, "CO1007-chapter-10-slide-478-0000": { "id": "CO1007-chapter-10-slide-478-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_056.png", "page_index": 478, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:22+07:00" } }, "CO1007-chapter-10-slide-479-0000": { "id": "CO1007-chapter-10-slide-479-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm - k - Ford's algorith m Others Graph Coloring - 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_057.png", "page_index": 479, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:25+07:00" } }, "CO1007-chapter-10-slide-480-0000": { "id": "CO1007-chapter-10-slide-480-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm k - Ford's algorith m Others Graph Coloring - 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_058.png", "page_index": 480, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:29+07:00" } }, "CO1007-chapter-10-slide-481-0000": { "id": "CO1007-chapter-10-slide-481-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_059.png", "page_index": 481, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:32+07:00" } }, "CO1007-chapter-10-slide-482-0000": { "id": "CO1007-chapter-10-slide-482-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_060.png", "page_index": 482, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:36+07:00" } }, "CO1007-chapter-10-slide-483-0000": { "id": "CO1007-chapter-10-slide-483-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_061.png", "page_index": 483, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:41+07:00" } }, "CO1007-chapter-10-slide-484-0000": { "id": "CO1007-chapter-10-slide-484-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. 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Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd- Warshall Algorit hm K Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_063.png", "page_index": 485, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:49+07:00" } }, "CO1007-chapter-10-slide-486-0000": { "id": "CO1007-chapter-10-slide-486-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_064.png", "page_index": 486, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:53+07:00" } }, "CO1007-chapter-10-slide-487-0000": { "id": "CO1007-chapter-10-slide-487-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd- Warshall Algorit hm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_065.png", "page_index": 487, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:24:57+07:00" } }, "CO1007-chapter-10-slide-488-0000": { "id": "CO1007-chapter-10-slide-488-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd- Warshall Algorit hm 7 k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_066.png", "page_index": 488, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:01+07:00" } }, "CO1007-chapter-10-slide-489-0000": { "id": "CO1007-chapter-10-slide-489-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm 7 k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_067.png", "page_index": 489, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:05+07:00" } }, "CO1007-chapter-10-slide-490-0000": { "id": "CO1007-chapter-10-slide-490-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits g Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_068.png", "page_index": 490, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:09+07:00" } }, "CO1007-chapter-10-slide-491-0000": { "id": "CO1007-chapter-10-slide-491-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits 9 Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_069.png", "page_index": 491, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:13+07:00" } }, "CO1007-chapter-10-slide-492-0000": { "id": "CO1007-chapter-10-slide-492-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. 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Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes a Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_072.png", "page_index": 494, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:24+07:00" } }, "CO1007-chapter-10-slide-495-0000": { "id": "CO1007-chapter-10-slide-495-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorithm Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_073.png", "page_index": 495, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:29+07:00" } }, "CO1007-chapter-10-slide-496-0000": { "id": "CO1007-chapter-10-slide-496-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_074.png", "page_index": 496, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:32+07:00" } }, "CO1007-chapter-10-slide-497-0000": { "id": "CO1007-chapter-10-slide-497-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits 9 Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_075.png", "page_index": 497, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:36+07:00" } }, "CO1007-chapter-10-slide-498-0000": { "id": "CO1007-chapter-10-slide-498-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. 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Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_077.png", "page_index": 499, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:43+07:00" } }, "CO1007-chapter-10-slide-500-0000": { "id": "CO1007-chapter-10-slide-500-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_078.png", "page_index": 500, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:47+07:00" } }, "CO1007-chapter-10-slide-501-0000": { "id": "CO1007-chapter-10-slide-501-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths Euler Paths and Circuits 9 Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others We get Hamilton circuit! Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_079.png", "page_index": 501, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:50+07:00" } }, "CO1007-chapter-10-slide-502-0000": { "id": "CO1007-chapter-10-slide-502-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes a Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_080.png", "page_index": 502, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:54+07:00" } }, "CO1007-chapter-10-slide-503-0000": { "id": "CO1007-chapter-10-slide-503-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_081.png", "page_index": 503, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:25:58+07:00" } }, "CO1007-chapter-10-slide-504-0000": { "id": "CO1007-chapter-10-slide-504-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_082.png", "page_index": 504, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:02+07:00" } }, "CO1007-chapter-10-slide-505-0000": { "id": "CO1007-chapter-10-slide-505-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_083.png", "page_index": 505, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:06+07:00" } }, "CO1007-chapter-10-slide-506-0000": { "id": "CO1007-chapter-10-slide-506-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes d Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_084.png", "page_index": 506, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:10+07:00" } }, "CO1007-chapter-10-slide-507-0000": { "id": "CO1007-chapter-10-slide-507-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths h Euler Paths and Circuits Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm k Ford's algorith m Others Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_085.png", "page_index": 507, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:13+07:00" } }, "CO1007-chapter-10-slide-508-0000": { "id": "CO1007-chapter-10-slide-508-0000", "text": "Finding Hamilton Circuits Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK a Vertices : cities TP.HCM b Edges : possible routes Contents Rule 1 Connectivity deg(v) = 2 Paths and Circuits e Euler and Hamilton Rule 3 Paths Euler Paths and Circuits q Once two edges are determined Hamilton Paths and Circuits other edges must be removed Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others We get Hamilton circuit! Graph Coloring 10.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_086.png", "page_index": 508, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:17+07:00" } }, "CO1007-chapter-10-slide-509-0000": { "id": "CO1007-chapter-10-slide-509-0000", "text": "Existence of Hamilton Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Hamilton circuit does not exist for all graph. But, there is no TP.HCM specific way to find whether Hamilton circuit exists or not. Simple check by rules of Hamilton circuit Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Pat hs and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_087.png", "page_index": 509, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:20+07:00" } }, "CO1007-chapter-10-slide-510-0000": { "id": "CO1007-chapter-10-slide-510-0000", "text": "Existence of Hamilton Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Hamilton circuit does not exist for all graph. But, there is no TP.HCM specific way to find whether Hamilton circuit exists or not. Simple check by rules of Hamilton circuit Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Pat hs and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_088.png", "page_index": 510, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:23+07:00" } }, "CO1007-chapter-10-slide-511-0000": { "id": "CO1007-chapter-10-slide-511-0000", "text": "Existence of Hamilton Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Hamilton circuit does not exist for all graph. But, there is no TP.HCM specific way to find whether Hamilton circuit exists or not. Simple check by rules of Hamilton circuit Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Pat hs and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_089.png", "page_index": 511, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:26+07:00" } }, "CO1007-chapter-10-slide-512-0000": { "id": "CO1007-chapter-10-slide-512-0000", "text": "Existence of Hamilton Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Hamilton circuit does not exist for all graph. But, there is no TP.HCM specific way to find whether Hamilton circuit exists or not. Simple check by rules of Hamilton circuit Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Pat hs and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_090.png", "page_index": 512, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:30+07:00" } }, "CO1007-chapter-10-slide-513-0000": { "id": "CO1007-chapter-10-slide-513-0000", "text": "Existence of Hamilton Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Hamilton circuit does not exist for all graph. But, there is no TP.HCM specific way to find whether Hamilton circuit exists or not. Simple check by rules of Hamilton circuit Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Pat hs and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_091.png", "page_index": 513, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:33+07:00" } }, "CO1007-chapter-10-slide-514-0000": { "id": "CO1007-chapter-10-slide-514-0000", "text": "Existence of Hamilton Circuit Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK Hamilton circuit does not exist for all graph. But, there is no TP.HCM specific way to find whether Hamilton circuit exists or not. Simple check by rules of Hamilton circuit Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Violates Rule 2! (No subcircuit Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_092.png", "page_index": 514, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:37+07:00" } }, "CO1007-chapter-10-slide-515-0000": { "id": "CO1007-chapter-10-slide-515-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_093.png", "page_index": 515, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:40+07:00" } }, "CO1007-chapter-10-slide-516-0000": { "id": "CO1007-chapter-10-slide-516-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM 6 d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_094.png", "page_index": 516, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:43+07:00" } }, "CO1007-chapter-10-slide-517-0000": { "id": "CO1007-chapter-10-slide-517-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_095.png", "page_index": 517, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:46+07:00" } }, "CO1007-chapter-10-slide-518-0000": { "id": "CO1007-chapter-10-slide-518-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. 1 BK TP.HCM 6 d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_096.png", "page_index": 518, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:49+07:00" } }, "CO1007-chapter-10-slide-519-0000": { "id": "CO1007-chapter-10-slide-519-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. 1 BK TP.HCM 6 d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_097.png", "page_index": 519, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:52+07:00" } }, "CO1007-chapter-10-slide-520-0000": { "id": "CO1007-chapter-10-slide-520-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a 1 BK TP.HCM - Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m - k Floyd-Warshall Algorithm - Ford's algorith m - O thers - Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_098.png", "page_index": 520, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:55+07:00" } }, "CO1007-chapter-10-slide-521-0000": { "id": "CO1007-chapter-10-slide-521-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a 1 BK TP.HCM - Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m 7 - Floyd-Warshall Algorithm - Ford's algorith m - O thers - Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_099.png", "page_index": 521, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:26:59+07:00" } }, "CO1007-chapter-10-slide-522-0000": { "id": "CO1007-chapter-10-slide-522-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a 1 BK TP.HCM - Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m 7 - k Floyd-Warshall Algorithm - Ford's algorith m - O thers - Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_100.png", "page_index": 522, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:03+07:00" } }, "CO1007-chapter-10-slide-523-0000": { "id": "CO1007-chapter-10-slide-523-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_101.png", "page_index": 523, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:06+07:00" } }, "CO1007-chapter-10-slide-524-0000": { "id": "CO1007-chapter-10-slide-524-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_102.png", "page_index": 524, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:10+07:00" } }, "CO1007-chapter-10-slide-525-0000": { "id": "CO1007-chapter-10-slide-525-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_103.png", "page_index": 525, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:14+07:00" } }, "CO1007-chapter-10-slide-526-0000": { "id": "CO1007-chapter-10-slide-526-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_104.png", "page_index": 526, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:18+07:00" } }, "CO1007-chapter-10-slide-527-0000": { "id": "CO1007-chapter-10-slide-527-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_105.png", "page_index": 527, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:21+07:00" } }, "CO1007-chapter-10-slide-528-0000": { "id": "CO1007-chapter-10-slide-528-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_106.png", "page_index": 528, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:24+07:00" } }, "CO1007-chapter-10-slide-529-0000": { "id": "CO1007-chapter-10-slide-529-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_107.png", "page_index": 529, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:27+07:00" } }, "CO1007-chapter-10-slide-530-0000": { "id": "CO1007-chapter-10-slide-530-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_108.png", "page_index": 530, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:31+07:00" } }, "CO1007-chapter-10-slide-531-0000": { "id": "CO1007-chapter-10-slide-531-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_109.png", "page_index": 531, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:34+07:00" } }, "CO1007-chapter-10-slide-532-0000": { "id": "CO1007-chapter-10-slide-532-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_110.png", "page_index": 532, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:37+07:00" } }, "CO1007-chapter-10-slide-533-0000": { "id": "CO1007-chapter-10-slide-533-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_111.png", "page_index": 533, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:40+07:00" } }, "CO1007-chapter-10-slide-534-0000": { "id": "CO1007-chapter-10-slide-534-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_112.png", "page_index": 534, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:43+07:00" } }, "CO1007-chapter-10-slide-535-0000": { "id": "CO1007-chapter-10-slide-535-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM d Contents Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_113.png", "page_index": 535, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:46+07:00" } }, "CO1007-chapter-10-slide-536-0000": { "id": "CO1007-chapter-10-slide-536-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM Contents Contradict with Rule 1l Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m k Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_114.png", "page_index": 536, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:50+07:00" } }, "CO1007-chapter-10-slide-537-0000": { "id": "CO1007-chapter-10-slide-537-0000", "text": "Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai We can verify nonexistence of the graph during find Hamilton Xuan Toan. Tran Hong Tai circuit. a BK TP.HCM Contents Contradict with Rule 1l Connectivity Paths and Circuits e Euler and Hamilton Paths h Euler Paths and Circuits g Hamilton Paths and Circuits Shortest Path Problem 2 Dijkstra's Algorith m Bellman-Ford Algorith m .7 k Floyd-Warshall Algorithm Ford's algorithm Others Hamilton circuit doesn't exist! Graph Coloring 10.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_115.png", "page_index": 537, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:53+07:00" } }, "CO1007-chapter-10-slide-538-0000": { "id": "CO1007-chapter-10-slide-538-0000", "text": "Application - Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition The binary sequence that express consecutive numbers by differing BK TP.HCM just one position of sequence. Contents Decimal number Binary number Gray code Connectivity 1 001 000 Paths and Circuits Euler and Hamilton 2 010 100 - Paths 3 011 110 Euler Paths and Circuits Hamilton Paths and Circuits 4 100 010 Shortest Path Problem 5 101 011 Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Used at digital communication for reduce the effect of noise: it Graph Coloring prevents serious changes of information by noise 10.34", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_116.png", "page_index": 538, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:27:58+07:00" } }, "CO1007-chapter-10-slide-539-0000": { "id": "CO1007-chapter-10-slide-539-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_117.png", "page_index": 539, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:02+07:00" } }, "CO1007-chapter-10-slide-540-0000": { "id": "CO1007-chapter-10-slide-540-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 011 111 Contents 001 101 Connectivity Paths and Circuits Euler and Hamilton Paths 010 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem 000 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_118.png", "page_index": 540, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:05+07:00" } }, "CO1007-chapter-10-slide-541-0000": { "id": "CO1007-chapter-10-slide-541-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 011 111 Contents 001 101 Connectivity Paths and Circuits Euler and Hamilton Paths 010 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem 000 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_119.png", "page_index": 541, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:08+07:00" } }, "CO1007-chapter-10-slide-542-0000": { "id": "CO1007-chapter-10-slide-542-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 5 = 011 111 Contents 8 = 001 101 Connectivity Paths and Circuits Euler and Hamilton Paths 4 = 010 3 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem = 000 2 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_120.png", "page_index": 542, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:13+07:00" } }, "CO1007-chapter-10-slide-543-0000": { "id": "CO1007-chapter-10-slide-543-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 5 = 011 111 Contents 8 = 001 101 Connectivity Paths and Circuits Euler and Hamilton Paths 4 = 010 3 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem = 000 2 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_121.png", "page_index": 543, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:17+07:00" } }, "CO1007-chapter-10-slide-544-0000": { "id": "CO1007-chapter-10-slide-544-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 011 111 Contents 001 101 Connectivity Paths and Circuits Euler and Hamilton Paths 010 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem 000 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_122.png", "page_index": 544, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:21+07:00" } }, "CO1007-chapter-10-slide-545-0000": { "id": "CO1007-chapter-10-slide-545-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 7 = 011 111 Contents 8 = 001 3 101 Connectivity Paths and Circuits Euler and Hamilton Paths 6 = 010 5 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem = 000 2 - 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_123.png", "page_index": 545, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:25+07:00" } }, "CO1007-chapter-10-slide-546-0000": { "id": "CO1007-chapter-10-slide-546-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 7 = 011 111 Contents 8 = 001 3 101 Connectivity Paths and Circuits Euler and Hamilton Paths 6 = 010 5 = 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem = 000 2 100 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_124.png", "page_index": 546, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:31+07:00" } }, "CO1007-chapter-10-slide-547-0000": { "id": "CO1007-chapter-10-slide-547-0000", "text": "Gray Code Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai n-digit gray code can be generated by finding Hamilton circuits of n-dimensional hypercube! Consider the case n = 3. BK TP.HCM 011 111 Contents 001 101 Connectivity Paths and Circuits Euler and Hamilton Paths 010 110 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem 000 100 Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Coordinate of each vertex is 3-digit binary sequences. Ford's algorith m Others Coordinates of adjacent vertices differ in just on place. Graph Coloring Hamilton circuits of a cubic graph makes the order of binary sequences! 10.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_125.png", "page_index": 547, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:34+07:00" } }, "CO1007-chapter-10-slide-548-0000": { "id": "CO1007-chapter-10-slide-548-0000", "text": "Exercise - Hamilton path & circuit Graph connectivity Nguyen An Khuong. d Tran Tuan Anh. Mai a a b Xuan Toan, Tran Hong Tai b BK b d TP.HCM a 6 b Q 50 Contents d) Connectivity Paths and Circuits Euler and Hamilton d e e C a Paths Euler Paths and Circuits a Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m a 6 Bellman-Ford Algorith m Floyd-Warshall Algorit hm 0 q Ford's algorithm h Others Graph Coloring n m e g 10.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_126.png", "page_index": 548, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:40+07:00" } }, "CO1007-chapter-10-slide-549-0000": { "id": "CO1007-chapter-10-slide-549-0000", "text": "Weighted Graphs Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Dien Bien Tai 296 Hai Phong BK TP.HCM Vinh Contents 1146 Connectivity Paths and Circuits 886 Euler and Hamilton Da Nang Paths 436 Euler Paths and Circuits Hamilton Paths and Circuits 019 Shortest Path Problem Dijkstra's Algorith m Nha Trang Bellman-Ford Algorith m Floyd-Warshall Algorit hm Ford's algorith m 322 Others 126 2200 HO CHI MINH Graph Coloring Can Tho Con Dao 10.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_127.png", "page_index": 549, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:43+07:00" } }, "CO1007-chapter-10-slide-550-0000": { "id": "CO1007-chapter-10-slide-550-0000", "text": "Problem Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK The problem is also sometimes called the single-pair shortest path problem, to TP.HCM distinguish it from the following generalizations: : The single-source shortest path problem, in which we have to find Contents shortest paths from a source vertex to all other vertices in the graph Connectivity The single-destination shortest path problem, in which we have to find Paths and Circuits shortest paths from all vertices in the graph to a single destination Euler and Hamilton vertex v. This can be reduced to the single-source shortest path problem Paths by reversing the edges in the graph. Euler Paths and Circuits Hamilton Paths and Circuits The all-pairs shortest path problem, in which we have to find shortest Shortest Path Problem paths between every pair of vertices u, u' in the graph. Dijkstra's Algorithm Bellman-Ford Algorith m These generalizations have significantly more efficient algorithms than the Floyd-Warshall Algorithm simplistic approach of running a single-pair shortest path algorithm on all Ford's algorith m relevant pairs of vertices. Others Graph Coloring 10.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_128.png", "page_index": 550, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:48+07:00" } }, "CO1007-chapter-10-slide-551-0000": { "id": "CO1007-chapter-10-slide-551-0000", "text": "Dijkstra's Algorithm Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai procedure Dijkstra(G,a) // Initialization Step forall vertices v BK Label[v] := * TP.HCM Prev[v] := -1 endfor Label(a := 0 // a is the source node Contents S := 0 Connectivity Paths and Circuits // Iteration Step Euler and Hamilton Paths while z S Euler Paths and Circuits u := a vertex not in S with minimal Label Hamilton Paths and Circuits S := S U {u} Shortest Path Problem forall vertices v not in S Dijkstra's Algorith m if (Label[u] + Wt(u,v)) < Label(v) Bellman-Ford Algorith m then begin Floyd-Warshall Algorithm Ford's algorith m Label[v] := Label[u] + Wt(u,v Others Pred[v] := u Graph Coloring end endwhile 10.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_129.png", "page_index": 551, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:53+07:00" } }, "CO1007-chapter-10-slide-552-0000": { "id": "CO1007-chapter-10-slide-552-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths s a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 8 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_130.png", "page_index": 552, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:28:58+07:00" } }, "CO1007-chapter-10-slide-553-0000": { "id": "CO1007-chapter-10-slide-553-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths s a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 8 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_131.png", "page_index": 553, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:03+07:00" } }, "CO1007-chapter-10-slide-554-0000": { "id": "CO1007-chapter-10-slide-554-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 8 Shortest Path Problem a 0 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_132.png", "page_index": 554, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:08+07:00" } }, "CO1007-chapter-10-slide-555-0000": { "id": "CO1007-chapter-10-slide-555-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 8 Shortest Path Problem a 0 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_133.png", "page_index": 555, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:13+07:00" } }, "CO1007-chapter-10-slide-556-0000": { "id": "CO1007-chapter-10-slide-556-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 8 Shortest Path Problem a 0 4 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_134.png", "page_index": 556, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:18+07:00" } }, "CO1007-chapter-10-slide-557-0000": { "id": "CO1007-chapter-10-slide-557-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_135.png", "page_index": 557, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:23+07:00" } }, "CO1007-chapter-10-slide-558-0000": { "id": "CO1007-chapter-10-slide-558-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 2 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_136.png", "page_index": 558, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:28+07:00" } }, "CO1007-chapter-10-slide-559-0000": { "id": "CO1007-chapter-10-slide-559-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem 0 4 2 a 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_137.png", "page_index": 559, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:33+07:00" } }, "CO1007-chapter-10-slide-560-0000": { "id": "CO1007-chapter-10-slide-560-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem 0 4 2 a 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_138.png", "page_index": 560, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:38+07:00" } }, "CO1007-chapter-10-slide-561-0000": { "id": "CO1007-chapter-10-slide-561-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 2 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m c O 2 Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_139.png", "page_index": 561, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:29:43+07:00" } }, "CO1007-chapter-10-slide-562-0000": { "id": "CO1007-chapter-10-slide-562-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. 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Tran Hong Tai 5 d 4 6 BK TP.HCM 8 a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 2 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m c 0 3 2 10 12 Floyd-Warshall Algorithm 3 2 8 12 Ford's algorith m b 0 Others d 0 3 2 8 10 Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_156.png", "page_index": 578, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:31:21+07:00" } }, "CO1007-chapter-10-slide-579-0000": { "id": "CO1007-chapter-10-slide-579-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM 8 a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths S a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 2 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m c 0 3 2 10 12 8 Floyd-Warshall Algorithm 3 2 8 12 Ford's algorith m b 0 Others d 0 3 2 8 10 14 Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_157.png", "page_index": 579, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:31:27+07:00" } }, "CO1007-chapter-10-slide-580-0000": { "id": "CO1007-chapter-10-slide-580-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM 8 a - 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths s a b d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 2 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m c 0 3 2 10 12 Floyd-Warshall Algorithm 0 3 2 8 12 Ford's algorith m Others d 0 3 2 8 10 14 Graph Coloring 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_158.png", "page_index": 580, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:31:33+07:00" } }, "CO1007-chapter-10-slide-581-0000": { "id": "CO1007-chapter-10-slide-581-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 5 d 4 6 BK TP.HCM 8 a - 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths s a b d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 8 8 Shortest Path Problem a 0 4 2 8 8 Dijkstra's Algorith m Bellman-Ford Algorith m c 0 3 2 10 12 Floyd-Warshall Algorithm 3 2 8 12 Ford's algorith m b 0 Others d 0 3 2 8 10 14 Graph Coloring e 0 3 2 8 10 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_159.png", "page_index": 581, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:31:40+07:00" } }, "CO1007-chapter-10-slide-582-0000": { "id": "CO1007-chapter-10-slide-582-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. 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Tran Hong Tai 5 d 4 6 BK TP.HCM 8 a 2 2 Contents 2 3 Connectivity 10 Paths and Circuits c e Euler and Hamilton Paths s a d Euler Paths and Circuits c e Hamilton Paths and Circuits 0 0 8 8 Shortest Path Problem a 0 4 2 8 Dijkstra's Algorith m Bellman-Ford Algorith m c 0 3 2 10 12 Floyd-Warshall Algorithn 3 2 8 12 Ford's algorith m b 0 Others d 0 3 2 8 10 14 Graph Coloring e 0 3 2 8 10 13 10.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_162.png", "page_index": 584, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:03+07:00" } }, "CO1007-chapter-10-slide-585-0000": { "id": "CO1007-chapter-10-slide-585-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 1 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths Euler Paths and Circuits c 10 e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_163.png", "page_index": 585, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:10+07:00" } }, "CO1007-chapter-10-slide-586-0000": { "id": "CO1007-chapter-10-slide-586-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 1 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths Euler Paths and Circuits c 10 e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_164.png", "page_index": 586, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:17+07:00" } }, "CO1007-chapter-10-slide-587-0000": { "id": "CO1007-chapter-10-slide-587-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 1 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths Euler Paths and Circuits c 10 e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_165.png", "page_index": 587, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:25+07:00" } }, "CO1007-chapter-10-slide-588-0000": { "id": "CO1007-chapter-10-slide-588-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 1 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths Euler Paths and Circuits c 10 e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_166.png", "page_index": 588, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:32+07:00" } }, "CO1007-chapter-10-slide-589-0000": { "id": "CO1007-chapter-10-slide-589-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths 10 Euler Paths and Circuits c e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_167.png", "page_index": 589, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:39+07:00" } }, "CO1007-chapter-10-slide-590-0000": { "id": "CO1007-chapter-10-slide-590-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths 10 Euler Paths and Circuits c e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_168.png", "page_index": 590, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:47+07:00" } }, "CO1007-chapter-10-slide-591-0000": { "id": "CO1007-chapter-10-slide-591-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths 10 Euler Paths and Circuits c e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_169.png", "page_index": 591, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:32:54+07:00" } }, "CO1007-chapter-10-slide-592-0000": { "id": "CO1007-chapter-10-slide-592-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 1 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths 10 Euler Paths and Circuits c e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_170.png", "page_index": 592, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:01+07:00" } }, "CO1007-chapter-10-slide-593-0000": { "id": "CO1007-chapter-10-slide-593-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 1 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths 10 Euler Paths and Circuits c e Hamilton Paths and Circuits Shortest Path Problem S b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 Others 0 3 2 c 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_171.png", "page_index": 593, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:09+07:00" } }, "CO1007-chapter-10-slide-594-0000": { "id": "CO1007-chapter-10-slide-594-0000", "text": "Back tracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai How to determine shortest path from a to d according to Dijkstra's algorithm? BK b 5 d TP.HCM 4 6 Contents 8 2 Connectivity a 2 Paths and Circuits Euler and Hamilton 2 3 Paths 10 Euler Paths and Circuits c e Hamilton Paths and Circuits Shortest Path Problem s b d Dijkstra's Algorith m a c e 2 Bellman-Ford Algorith m 0 0 8 8 8 Floyd-Warshall Algorithm Ford's algorith m a 0 4 2 8 8 Others c 0 3 2 10 12 Graph Coloring b 0 3 2 8 12 d 0 3 2 8 10 14 e 0 3 2 8 10 13 10.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_172.png", "page_index": 594, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:17+07:00" } }, "CO1007-chapter-10-slide-595-0000": { "id": "CO1007-chapter-10-slide-595-0000", "text": "Dijkstra's Algorithm Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Property Connectivity Applicable for any G, any length ((vi) 0, Vi; one-to-all: Paths and Circuits Euler and Hamilton complexity O(V2). Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorit hn Ford's algorith m Others Graph Colring 10.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_173.png", "page_index": 595, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:20+07:00" } }, "CO1007-chapter-10-slide-596-0000": { "id": "CO1007-chapter-10-slide-596-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai Find the shortest path from a to other vertices using Dijkstra's algorithm BK TP.HCM 6 5 Contents 3 9 Connectivity 4 Paths and Circuits 4 Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits 6 Shortest Path Problem Dijkstra's Algorith m a Bellman-Ford Algorith m 8 Floyd-Warshall Algorithm 1 3 Ford's algorith m 1 2 Others Graph Coloring d 2 5 h f 10.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_174.png", "page_index": 596, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:25+07:00" } }, "CO1007-chapter-10-slide-597-0000": { "id": "CO1007-chapter-10-slide-597-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Find the shortest path from e to other vertices using Dijkstra's algorithm. BK TP.HCM a 4 b 5 c Contents Connectivity Paths and Circuits 4 1 7 Euler and Hamilton 10 Paths Euler Paths and Circuits Hamilton Paths and Circuits d e Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m 5 10 Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring h f 10 9 10.44", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_175.png", "page_index": 597, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:29+07:00" } }, "CO1007-chapter-10-slide-598-0000": { "id": "CO1007-chapter-10-slide-598-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example BK TP.HCM Find the shortest path from a to other vertices using Dijkstra's algorithm. Contents Connectivity b 5 d Paths and Circuits Euler and Hamilton 5 7 Paths Euler Paths and Circuits 8 3 Hamilton Paths and Circuits a 3 Shortest Path Problem Dijkstra's Algorith m 2 Bellman-Ford Algorith m Floyd-Warshall Algorithm c 10 e Ford's algorithm Others Graph Coloring 10.45", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_176.png", "page_index": 598, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:33+07:00" } }, "CO1007-chapter-10-slide-599-0000": { "id": "CO1007-chapter-10-slide-599-0000", "text": "Dijkstra's Algorithm Flaw Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Can Dijkstra's Algorithm be used on.. BK . ...digraph? TP.HCM Yes! ...negative weighted graph? Contents No! Why? Connectivity Paths and Circuits Euler and Hamilton b Paths Euler Paths and Circuits Hamilton Paths and Circuits 2 Shortest Path Problem Dijkstra's Algorith m a -4 Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others Graph Coloring c 10.46", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_177.png", "page_index": 599, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:36+07:00" } }, "CO1007-chapter-10-slide-600-0000": { "id": "CO1007-chapter-10-slide-600-0000", "text": "Bellman-Ford Algorithm Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai procedure BellmanFord(G,a) Xuan Toan. Tran Hong Tai // Initialization Step forall vertices v Label[v] := co Prev[v] := -1 BK TP.HCM Label(a) := 0 // a is the source node // Iteration Step Contents for i from 1 to size(vertices)-1 Connectivity forall vertices v Paths and Circuits if (Label[u] + Wt(u,v)) < Label[v] Euler and Hamilton then Paths Label[v] := Label[u] + Wt(u,v Euler Paths and Circuits Prev[v] := u Hamilton Paths and Circuits Shortest Path Problem // Check circuit of negative weight Dijkstra's Algorith m Bellman-Ford Algorith m forall vertices v Floyd-Warshall Algorithm if (Label[u] + Wt(u,v)) < Label(v) Ford's algorithm Others error \"Contains circuit of negative weight\" Graph Coloring Property any G, any weighted; one-to-all; detect whether there exists a circle of negative length; complexity O(V x ED). 10.47", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_178.png", "page_index": 600, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:41+07:00" } }, "CO1007-chapter-10-slide-601-0000": { "id": "CO1007-chapter-10-slide-601-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a b c d e f BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_179.png", "page_index": 601, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:44+07:00" } }, "CO1007-chapter-10-slide-602-0000": { "id": "CO1007-chapter-10-slide-602-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a c d e f 0 0 8 8 8 8 BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_180.png", "page_index": 602, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:49+07:00" } }, "CO1007-chapter-10-slide-603-0000": { "id": "CO1007-chapter-10-slide-603-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a b c d f O 0 8 8 BK 1 0 -2a 3a TP.HCM X Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 3 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_181.png", "page_index": 603, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:54+07:00" } }, "CO1007-chapter-10-slide-604-0000": { "id": "CO1007-chapter-10-slide-604-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a c d f 0 0 8 8 BK 1 0 -2a 8 8 3a TP.HCM 2 0 -2 3b -5b 3 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 3 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_182.png", "page_index": 604, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:33:59+07:00" } }, "CO1007-chapter-10-slide-605-0000": { "id": "CO1007-chapter-10-slide-605-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a c d f 0 0 8 8 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b 8 -5b 3 3 0 -2 3 -4e -5 3 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 3 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_183.png", "page_index": 605, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:04+07:00" } }, "CO1007-chapter-10-slide-606-0000": { "id": "CO1007-chapter-10-slide-606-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a c d f 0 0 8 8 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits b Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 3 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_184.png", "page_index": 606, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:10+07:00" } }, "CO1007-chapter-10-slide-607-0000": { "id": "CO1007-chapter-10-slide-607-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Step a b c d f 0 0 8 8 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Stop since Step 4 = Step 3 Euler and Hamilton Paths Euler Paths and Circuits b Hamilton Paths and Circuits 5 c Shortest Path Problem -2 -5 Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm a 4 3 d Ford's algorith m Others Graph Coloring 3 + 1 e 10.48", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_185.png", "page_index": 607, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:16+07:00" } }, "CO1007-chapter-10-slide-608-0000": { "id": "CO1007-chapter-10-slide-608-0000", "text": "Backtracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai Step a b c d e f 0 0 8 BK 1 0 -2a 3a TP.HCM 2 0 -2 3b -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Stop since Step 4 = Step 3 Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits b 5 c Shortest Path Problem Dijkstra's Algorithm -2 -5 Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m a 4 3 d Others Graph Coloring 3 XY 1 e 10.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_186.png", "page_index": 608, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:23+07:00" } }, "CO1007-chapter-10-slide-609-0000": { "id": "CO1007-chapter-10-slide-609-0000", "text": "Backtracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai Step a b c d e f 0 0 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Stop since Step 4 = Step 3. Euler and Hamilton How to find shortest path from a to d? a -? d Paths Euler Paths and Circuits Hamilton Paths and Circuits b 5 c Shortest Path Problem Dijkstra's Algorith m -2 -5 Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m a 4 3 d Others Graph Coloring 1 e 10.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_187.png", "page_index": 609, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:30+07:00" } }, "CO1007-chapter-10-slide-610-0000": { "id": "CO1007-chapter-10-slide-610-0000", "text": "Backtracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai Step a b c d e f 0 0 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b 8 -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Stop since Step 4 = Step 3. Euler and Hamilton How to find shortest path from a to d? a -? e->d Paths Euler Paths and Circuits Hamilton Paths and Circuits b 5 c Shortest Path Problem Dijkstra's Algorith m -2 -5 Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm a 4 3 d Others Graph Coloring 1 e 10.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_188.png", "page_index": 610, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:36+07:00" } }, "CO1007-chapter-10-slide-611-0000": { "id": "CO1007-chapter-10-slide-611-0000", "text": "Backtracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai Step a b c d e f 0 0 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Stop since Step 4 = Step 3. Euler and Hamilton How to find shortest path from a to d? a -> b -> e - d Paths Euler Paths and Circuits Hamilton Paths and Circuits b 5 c Shortest Path Problem Dijkstra's Algorith m -2 -5 Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm 4 3 d Others Graph Coloring 1 e 10.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_189.png", "page_index": 611, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:43+07:00" } }, "CO1007-chapter-10-slide-612-0000": { "id": "CO1007-chapter-10-slide-612-0000", "text": "Backtracking procedure Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai Step a b c d e f 0 0 BK 1 0 -2a 8 3a TP.HCM 2 0 -2 3b -5b 3 3 0 -2 3 -4e -5 3 Contents 4 0 -2 3 -4 -5 3 Connectivity Paths and Circuits Stop since Step 4 = Step 3. Euler and Hamilton How to find shortest path from a to d? a -> b -> e - d Paths Euler Paths and Circuits Hamilton Paths and Circuits b 5 c Shortest Path Problem Dijkstra's Algorith m 2 -5 Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm 4 3 d Others Graph Coloring 1 e 10.49", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_190.png", "page_index": 612, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:50+07:00" } }, "CO1007-chapter-10-slide-613-0000": { "id": "CO1007-chapter-10-slide-613-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step b d Tai a c e f BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m b 1 c Bellman-Ford Algorith m -2 Floyd- Warshall Algorit hm -5 Ford's algorith m Others 4 ) Graph Coloring a d 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_191.png", "page_index": 613, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:53+07:00" } }, "CO1007-chapter-10-slide-614-0000": { "id": "CO1007-chapter-10-slide-614-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step a b d f Tai c e 0 8 8 BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m b 1 c Bellman-Ford Algorith m Floyd- Warshall Algorit hm -2 -5 Ford's algorith m Others 4 ) Graph Coloring a d 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_192.png", "page_index": 614, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:34:58+07:00" } }, "CO1007-chapter-10-slide-615-0000": { "id": "CO1007-chapter-10-slide-615-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step a b d f Tai c e O 0 8 1 0 -2a 8 8 BK TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m b 1 c Bellman-Ford Algorith m Floyd- Warshall Algorit hm -2 -5 Ford's algorith m Others 4 3 Graph Coloring a d 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_193.png", "page_index": 615, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:02+07:00" } }, "CO1007-chapter-10-slide-616-0000": { "id": "CO1007-chapter-10-slide-616-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step b d f Tai a c e O 0 8 1 0 -2a 8 8 BK 2 0 -2 -1b -1b TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m b 1 c Bellman-Ford Algorith m Floyd-Warshall Algorithm -2 -5 Ford's algorith m Others 4 3 Graph Coloring a d f 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_194.png", "page_index": 616, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:07+07:00" } }, "CO1007-chapter-10-slide-617-0000": { "id": "CO1007-chapter-10-slide-617-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step b d f Tai a c e O 8 1 0 -2a 8 BK 2 0 -2 -1b -1b TP.HCM 3 O -2 -1 -6c -1 -4c Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m b 1 c Bellman-Ford Algorith m Floyd- Warshall Algorit hm -2 -5 Ford's algorith m Others 4 3 Graph Coloring a d 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_195.png", "page_index": 617, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:13+07:00" } }, "CO1007-chapter-10-slide-618-0000": { "id": "CO1007-chapter-10-slide-618-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step b d f Tai a c e O 1 0 -2a 8 BK 2 0 -2 -1b -1b TP.HCM 3 0 -2 -1 -6c -1 -4c 4 -1f -2 -1 -6 -3f -4 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm b 1 c Bellman-Ford Algorith m Floyd- Warshall Algorithn -2 -5 Ford's algorith m Others 4 3 Graph Coloring a 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_196.png", "page_index": 618, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:19+07:00" } }, "CO1007-chapter-10-slide-619-0000": { "id": "CO1007-chapter-10-slide-619-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan, Tran Hong Step b d f Tai a c e O 1 O -2a BK 2 0 -2 -1b -1b TP.HCM 3 0 -2 -1 -6c -1 -4c 4 -1f -2 -1 -6 -3f -4 Contents 5 -1 -3a -1 -6 -3 -4 Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm b 1 c Bellman-Ford Algorith m Floyd-Warshall Algorithn -2 -5 Ford's algorith m Others 4 3 Graph Coloring a 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_197.png", "page_index": 619, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:25+07:00" } }, "CO1007-chapter-10-slide-620-0000": { "id": "CO1007-chapter-10-slide-620-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan, Tran Hong Step a d f Tai c e O O 1 O -2a BK 2 -2 -1b -1b TP.HCM 3 0 -2 -1 -6c -1 -4c 4 -1f -2 -1 -6 -3f -4 Contents 5 -1 -3a -1 -6 -3 -4 Connectivity 6 -3 -2b -6 -3 -4 Paths and Circuits -1 Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m 1 c Bellman-Ford Algorith Floyd-Warshall Algorithn -2 -5 Ford's algorith m Others 4 3 Graph Coloring a 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_198.png", "page_index": 620, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:33+07:00" } }, "CO1007-chapter-10-slide-621-0000": { "id": "CO1007-chapter-10-slide-621-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step b d f Tai a c e 0 8 1 0 -2a BK 2 0 -2 -1b -1b TP.HCM 3 0 -2 -1 -6c -1 -4c 4 -1f -2 -1 -6 -3f -4 Contents 5 -1 -3a -1 -6 -3 -4 Connectivity 6 -1 -3 -2b -6 -3 -4 Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits There exists a circle of negative length since Step 6 Step 5 Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm b 1 c Bellman-Ford Algorith m Floyd- Warshall Algorit hn -2 -5 Ford's algorith m Others 3 Graph Coloring a 4 f 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_199.png", "page_index": 621, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:41+07:00" } }, "CO1007-chapter-10-slide-622-0000": { "id": "CO1007-chapter-10-slide-622-0000", "text": "Example Graph connectivity Nguyen An Khuong. Example Tran Tuan Anh, Mai Xuan Toan. Tran Hong Step a b d f Tai c e O O 1 0 -2a BK 2 0 -2 -1b -1b TP.HCM 3 0 -2 -1 -6c -1 -4c 4 -1f -2 -1 -6 -3f -4 Contents 5 -1 -3a -1 -6 -3 -4 Connectivity 6 -1 -3 -2b -6 -3 -4 Paths and Circuits Euler and Hamilton 7 -1 -3 -2 -7c -3 -4 Paths Euler Paths and Circuits There exists a circle of negative length since Step 6 / Step 5 Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm b c Bellman-Ford Algorith m Floyd- Warshall Algorit hn -2 -5 Ford's algorith m Others 4 3 Graph Coloring a 1 e 10.50", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_200.png", "page_index": 622, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:49+07:00" } }, "CO1007-chapter-10-slide-623-0000": { "id": "CO1007-chapter-10-slide-623-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai 1 Xuan Toan. Tran Hong Tai 1 c . BK TP.HCM 5 a d Contents 4 3 Connectivity Paths and Circuits f 1 e Euler and Hamilton Paths Euler Paths and Circuits Example Hamilton Paths and Circuits Shortest Path Problem Step a b c d f e Dijkstra's Algorith m 0 0 8 8 Bellman-Ford Algorith m Floyd-Warshall Algorithm 1 0 5a 1a 4a Ford's algorith m Others 2 0 5a 6b 1a 4d 4a Graph Coloring 3 0 1e 6b 1a 4d 4a 4 0 1e 2b 1a 4d 4a 5 0 1e 2b 1a 4d 3c 6 0 1e 2b 1a 4d 3c 10.51", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_201.png", "page_index": 623, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:35:57+07:00" } }, "CO1007-chapter-10-slide-624-0000": { "id": "CO1007-chapter-10-slide-624-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK 2 TP.HCM (G1) 3 Contents B D Connectivity 4 Paths and Circuits 2 Euler and Hamilton Paths A 2 3 F Euler Paths and Circuits 6 5 Hamilton Paths and Circuits Shortest Path Problem 5 Dijkstra's Algorith m E Bellman-Ford Algorith m Floyd-Warshall Algorithm 12 Ford's algorith m Others Graph Coloring 10.52", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_202.png", "page_index": 624, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:01+07:00" } }, "CO1007-chapter-10-slide-625-0000": { "id": "CO1007-chapter-10-slide-625-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK TP.HCM 1 -5 c Contents 0 Connectivity Paths and Circuits Euler and Hamilton 5 Paths a d Euler Paths and Circuits X Hamilton Paths and Circuits 3 Shortest Path Problem Dijkstra's Algorith m 1 e Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.53", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_203.png", "page_index": 625, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:04+07:00" } }, "CO1007-chapter-10-slide-626-0000": { "id": "CO1007-chapter-10-slide-626-0000", "text": "Floyd-Warshall Algorithm [1962] Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM procedure FloydWarshall ( for k := 1 to n for i := 1 to n Contents for j := 1 to n Connectivity path[i,j] = min (path[i,jl, Paths and Circuits path[i,k]+path[k,j]); Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Property Shortest Path Problem any G, any weighted; all-to-all; this is an software algorithm; complexity Dijkstra's Algorithm O(V3). Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorithm O thers Graph Colring 10.54", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_204.png", "page_index": 626, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:07+07:00" } }, "CO1007-chapter-10-slide-627-0000": { "id": "CO1007-chapter-10-slide-627-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 000 40 20 00 -20 1 0 30 000 00 0 50 -10 BK a b 00 00 2 TP.HCM 3 5 Contents Connectivity Paths and Circuits d -1 Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_205.png", "page_index": 627, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:12+07:00" } }, "CO1007-chapter-10-slide-628-0000": { "id": "CO1007-chapter-10-slide-628-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 00 40 20 00 20 1 0 10 30 00 00 00 50 -10 BK a b 000 00 2 TP.HCM 00 10 000 40 20 L(1) = 3 .5 30 Contents 000 Connectivity Paths and Circuits * d -1 Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_206.png", "page_index": 628, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:16+07:00" } }, "CO1007-chapter-10-slide-629-0000": { "id": "CO1007-chapter-10-slide-629-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 X0 40 20 00 -20 1 0 10 30 00 00 0 -10 BK a 000 50 00 2 TP.HCM 00 10 000 40 20 00 - 20 L(1) = 61 3 .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity Paths and Circuits * d -1 Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_207.png", "page_index": 629, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:22+07:00" } }, "CO1007-chapter-10-slide-630-0000": { "id": "CO1007-chapter-10-slide-630-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 X0 40 20 00 -20 1 000 L(0) 30 00 00 0 BK a o b 000 50 -10 00 2 A TP.HCM 00 10 000 40 20 00 - 20 L(1) = 61 3 .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 1 10 Paths and Circuits * d 20 00 - 20 61 -1 L(2) Euler and Hamilton - 41 Paths 50 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_208.png", "page_index": 630, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:28+07:00" } }, "CO1007-chapter-10-slide-631-0000": { "id": "CO1007-chapter-10-slide-631-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 0 40 20 00 -20 1 00 L(0) 30 000 00 0 50 -10 BK a o b 000 00 2 A TP.HCM 00 10 000 40 20 00 20 L(1) = 61 3 .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 1 00 10 -12 40 Paths and Circuits * d 20 00 - 20 61 -1 L(2) Euler and Hamilton 30 41 00 71 Paths -32 50 -72 00 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorithm Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_209.png", "page_index": 631, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:35+07:00" } }, "CO1007-chapter-10-slide-632-0000": { "id": "CO1007-chapter-10-slide-632-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 0 40 20 00 -20 1 000 L(0) 30 000 00 0 50 -10 BK a o b 000 00 2 A TP.HCM 00 10 000 40 20 00 20 L(1) = 61 3 .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 1 00 10 -12 40 Paths and Circuits * d 20 00 20 61 -1 L(2) = Euler and Hamilton 30 41 00 71 Paths -32 50 -72 00 Euler Paths and Circuits Hamilton Paths and Circuits - 12 - 20 Shortest Path Problem L(3) = Dijkstra's Algorith m 3 41 00 71 Bellman-Ford Algorith m - 72 Floyd-Warshall Algorit hm Ford's algorithm Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_210.png", "page_index": 632, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:42+07:00" } }, "CO1007-chapter-10-slide-633-0000": { "id": "CO1007-chapter-10-slide-633-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 0 40 1 20 00 -20 000 L(0) 30 000 00 0 50 -10 BK a o b 000 00 2 A TP.HCM 00 10 000 40 20 00 20 61 L(1) = .5 30 41 00 71 Contents 000 50 10 00 Connectivity 00 10 -12 40 Paths and Circuits * d 20 00 20 L(2) = 61 -1 Euler and Hamilton 30 41 00 71 Paths -32 50 -72 00 Euler Paths and Circuits Hamilton Paths and Circuits 00 10 - 12 40 - 20 Shortest Path Problem 13 00 53 L(3) = Dijkstra's Algorith m 3 41 00 71 Bellman-Ford Algorithm -43 50 72 00 Floyd-Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_211.png", "page_index": 633, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:50+07:00" } }, "CO1007-chapter-10-slide-634-0000": { "id": "CO1007-chapter-10-slide-634-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 0 40 20 00 -20 1 000 L(0) 30 000 00 0 -10 BK a o b 000 50 00 2 A TP.HCM 00 10 000 40 20 00 - 20 L(1) = 61 3 .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 1 00 10 -12 40 Paths and Circuits d 20 00 20 61 -1 L(2) = Euler and Hamilton 30 41 00 71 Paths -32 50 -72 00 Euler Paths and Circuits Hamilton Paths and Circuits 00 10 - 12 40 Shortest Path Problem 13 00 20 53 L(3) = Dijkstra's Algorith m 3 41 00 71 Bellman-Ford Algorith m -43 50 72 00 Floyd- Warshall Algorit hm 40 Ford's algorith m Others 53 L(4) = Graph Coloring 71 50 - 72 00 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_212.png", "page_index": 634, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:36:59+07:00" } }, "CO1007-chapter-10-slide-635-0000": { "id": "CO1007-chapter-10-slide-635-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai 00 10 0 40 20 00 -20 1 000 L(0) 30 000 00 0 o b 50 -10 BK a 000 00 2 A TP.HCM 00 10 000 40 20 00 - 20 61 L(1) = .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 1 00 10 -12 40 Paths and Circuits d 20 00 20 L(2) = 61 -1 Euler and Hamilton 30 41 00 71 Paths -32 50 -72 00 Euler Paths and Circuits Hamilton Paths and Circuits 00 10 - 12 40 20 Shortest Path Problem 13 00 53 L(3) = Dijkstra's Algorith m 3 41 00 71 Bellman-Ford Algorith m -43 50 72 00 Floyd-Warshall Algorit hm 00 -14 -34 40 Ford's algorith m Others 13 00 - 20 53 L(4) = Graph Coloring 30 24 00 71 43 50 72 00 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_213.png", "page_index": 635, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:07+07:00" } }, "CO1007-chapter-10-slide-636-0000": { "id": "CO1007-chapter-10-slide-636-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 000 40 20 00 1 -20 L(0) X0 30 0 00 0 50 BK o b 000 -10 00 2 A TP.HCM 00 10 000 40 20 00 - 20 61 L(1) = .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 00 10 -12 40 Paths and Circuits d 20 00 20 61 -1 L(2) = Euler and Hamilton 30 41 00 71 Paths Shortest path from b to d -32 50 -72 00 Euler Paths and Circuits (53 from L(4)): Hamilton Paths and Circuits 00 10 - 12 40 Shortest Path Problem 13 00 20 53 L(3) = Dijkstra's Algorithm 3 41 00 71 Bellman-Ford Algorith m -43 50 72 00 Floyd- Warshall Algorit hm 00 -14 -34 40 Ford's algorithm Others 13 00 20 53 L(4) = Graph Coloring 30 24 00 71 43 50 - 72 00 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_214.png", "page_index": 636, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:17+07:00" } }, "CO1007-chapter-10-slide-637-0000": { "id": "CO1007-chapter-10-slide-637-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 00 40 20 00 -20 1 00 1(0) 30 0 00 0 o b 50 BK a 000 -10 00 2 TP.HCM 00 10 000 40 20 00 - 20 61 L(1) = 3 .5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 00 10 -12 40 Paths and Circuits d 20 00 20 61 L(2) = Euler and Hamilton -1 30 41 00 71 Paths Shortest path from b to d -32 50 -72 00 Euler Paths and Circuits (53 from L(4)): Hamilton Paths and Circuits 00 10 - 12 40 Shortest Path Problem bd = bc + cd 13 00 20 53 L(3) = Dijkstra's Algorithm (53 = -2o + 71 from L(3)) 3 41 00 71 Bellman-Ford Algorith m -43 50 72 00 Floyd- Warshall Algorit hm 00 -14 -34 40 Ford's algorith m Others 13 00 - 20 53 L(4) = Graph Coloring 30 24 00 71 43 50 72 00 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_215.png", "page_index": 637, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:27+07:00" } }, "CO1007-chapter-10-slide-638-0000": { "id": "CO1007-chapter-10-slide-638-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 000 40 20 00 -20 1 0 1(0) 30 000 00 0 -10 BK a o b 000 -50 00 2 TP.HCM 00 10 000 40 20 00 - 20 L(1) = 61 3 5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 00 10 -12 40 Paths and Circuits C d 20 00 20 61 L(2) = Euler and Hamilton -1 30 41 00 71 Paths Shortest path from b to d -32 50 -72 00 Euler Paths and Circuits (53 from L(4)): Hamilton Paths and Circuits 00 10 - 12 40 Shortest Path Problem bd = bc + cd 13 00 20 L(3) = 53 Dijkstra's Algorith m (53 = -2o + 71 from L(3)) 3 41 00 71 Bellman-Ford Algorith m cd = ca + ad -43 50 72 00 Floyd- Warshall Algorit hm (71 = 3o + 4o from L(1)) 00 -14 -34 40 Ford's algorith m Others 13 00 20 53 L(4) = Graph Coloring 30 24 00 71 43 50 72 00 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_216.png", "page_index": 638, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:37+07:00" } }, "CO1007-chapter-10-slide-639-0000": { "id": "CO1007-chapter-10-slide-639-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 00 10 000 40 20 00 -20 1 0 L(0) 30 000 00 0 -10 BK a o b 000 -50 00 2 TP.HCM 00 10 000 40 20 00 - 20 L(1) = 61 3 5 30 41 00 71 Contents 000 50 - 10 00 Connectivity 00 10 -12 40 Paths and Circuits C d 20 00 20 61 L(2) = Euler and Hamilton -1 30 41 00 71 Paths Shortest path from b to d -32 50 -72 00 Euler Paths and Circuits (53 from L(4)): Hamilton Paths and Circuits 00 10 - 12 40 Shortest Path Problem bd = bc + cd 13 00 20 L(3) = 53 Dijkstra's Algorith m (53 = -2o + 71 from L(3)) 3 41 00 71 Bellman-Ford Algorith m cd = ca + ad -43 50 72 00 Floyd- Warshall Algorit hm (71 = 3o + 4o from L(1)) 00 -14 34 40 Ford's algorith m Others = bd= bc + ca + ad 13 00 20 53 L(4) = Graph Coloring 30 24 00 71 43 50 72 00 10.55", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_217.png", "page_index": 639, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:47+07:00" } }, "CO1007-chapter-10-slide-640-0000": { "id": "CO1007-chapter-10-slide-640-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai 2 BK a 6 TP.HCM 3 Contents Connectivity c Paths and Circuits Euler and Hamilton 00 20 0 Paths L(O) 30 00 -40 Euler Paths and Circuits 10 Hamilton Paths and Circuits 000 00 Shortest Path P roblem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.56", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_218.png", "page_index": 640, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:50+07:00" } }, "CO1007-chapter-10-slide-641-0000": { "id": "CO1007-chapter-10-slide-641-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 2 BK a 3 TP.HCM Contents 1 Connectivity c Paths and Circuits Euler and Hamilton 00 20 00 00 20 000 Paths L(0) 30 00 -40 30 00 -40 Euler Paths and Circuits 10 Hamilton Paths and Circuits 10 000 00 31 00 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorithm Others Graph Coloring 10.56", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_219.png", "page_index": 641, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:54+07:00" } }, "CO1007-chapter-10-slide-642-0000": { "id": "CO1007-chapter-10-slide-642-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 2 BK a 3 TP.HCM Contents 4 Connectivity c Paths and Circuits Euler and Hamilton 00 20 00 00 20 000 Paths L(0) 30 00 -40 L(1 30 00 -40 Euler Paths and Circuits 10 Hamilton Paths and Circuits 10 000 00 31 00 -20 Shortest Path Problem 00 20 Dijkstra's Algorith m L(2) = 30 00 -40 Bellman-Ford Algorith m 10 31 -12 Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.56", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_220.png", "page_index": 642, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:37:59+07:00" } }, "CO1007-chapter-10-slide-643-0000": { "id": "CO1007-chapter-10-slide-643-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 2 BK a 6 TP.HCM 3 Contents 1 Connectivity C Paths and Circuits Euler and Hamilton 00 20 00 00 20 0 Paths L(0) 30 00 -40 L(1 30 00 -40 Euler Paths and Circuits Hamilton Paths and Circuits 10 000 00 10 31 00 -20 Shortest Path Problem 00 20 Dijkstra's Algorith m L(2) 30 00 -40 Bellman-Ford Algorith m 10 31 - 12 Floyd-Warshall Algorithm Ford's algorith m Others STOP, there exists a circuit of negative length Graph Coloring 10.56", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_221.png", "page_index": 643, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:05+07:00" } }, "CO1007-chapter-10-slide-644-0000": { "id": "CO1007-chapter-10-slide-644-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK. TP.HCM 6 5 Contents Connectivity Paths and Circuits a Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem d Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others Graph Coloring 10.57", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_222.png", "page_index": 644, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:08+07:00" } }, "CO1007-chapter-10-slide-645-0000": { "id": "CO1007-chapter-10-slide-645-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Contents Connectivity 3) Paths and Circuits Euler and Hamilton (G2) 3 -4 Paths Euler Paths and Circuits -2 Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m O thers Graph Coloring 10.58", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_223.png", "page_index": 645, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:12+07:00" } }, "CO1007-chapter-10-slide-646-0000": { "id": "CO1007-chapter-10-slide-646-0000", "text": "Ford's algorithm Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM T(1) = 0 For each j E V do Contents Connectivity Paths and Circuits End Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Property Shortest Path Problem G without circle, positive length; one-to-all; rank table definition; complexity Dijkstra's Algorithm O(vl). Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.59", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_224.png", "page_index": 646, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:14+07:00" } }, "CO1007-chapter-10-slide-647-0000": { "id": "CO1007-chapter-10-slide-647-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK. TP.HCM 1 B 3 4 Contents 3 Connectivity 4 Paths and Circuits E Euler and Hamilton Paths 5 Euler Paths and Circuits 2 Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_225.png", "page_index": 647, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:17+07:00" } }, "CO1007-chapter-10-slide-648-0000": { "id": "CO1007-chapter-10-slide-648-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai 1 B BK 3 4 TP.HCM 3 4 E Contents 5 Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem r: rank(i) Dijkstra's Algorith m Bellman-Ford Algorith m A Floyd- Warshall Algorit hm B Ford's algorithm c Others D Graph Coloring E 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_226.png", "page_index": 648, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:22+07:00" } }, "CO1007-chapter-10-slide-649-0000": { "id": "CO1007-chapter-10-slide-649-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 1 B BK 4 TP.HCM 3 4 E Contents 5 Connectivity 2 Paths and Circuits Euler and Hamilton D Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem rank(i) Dijkstra's Algorith m Bellman-Ford Algorith m A Floyd-Warshall Algorit hm B A Ford's algorithm c A,B Others D B,C,E Graph Coloring E A 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_227.png", "page_index": 649, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:25+07:00" } }, "CO1007-chapter-10-slide-650-0000": { "id": "CO1007-chapter-10-slide-650-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 1 B BK 4 TP.HCM 3 4 E Contents 5 Connectivity 2 Paths and Circuits Euler and Hamilton D Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem rank(i) Dijkstra's Algorith m Bellman-Ford Algorith m A 0 Floyd-Warshall Algorit hm B A Ford's algorithm c A,B Others D B,C,E Graph Coloring E A 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_228.png", "page_index": 650, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:29+07:00" } }, "CO1007-chapter-10-slide-651-0000": { "id": "CO1007-chapter-10-slide-651-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 1 B BK 3 4 TP.HCM 3 4 E Contents 5 Connectivity 2 Paths and Circuits Euler and Hamilton D Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem rank(i) Dijkstra's Algorith m Bellman-Ford Algorith m A 0 Floyd-Warshall Algorit hm B 1 Ford's algorithm c B Others D B,C,E Graph Coloring E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_229.png", "page_index": 651, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:33+07:00" } }, "CO1007-chapter-10-slide-652-0000": { "id": "CO1007-chapter-10-slide-652-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai 1 B BK 3 4 TP.HCM 3 4 E Contents 5 Connectivity 2 Paths and Circuits Euler and Hamilton D Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem rank(i) Dijkstra's Algorithm Bellman-Ford Algorith m A 0 Floyd- Warshall Algorit hm B 1 Ford's algorithm c 2 Others D c Graph Coloring E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_230.png", "page_index": 652, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:37+07:00" } }, "CO1007-chapter-10-slide-653-0000": { "id": "CO1007-chapter-10-slide-653-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai 1 B BK 3 4 TP.HCM 3 4 E Contents 5 Connectivity 2 Paths and Circuits Euler and Hamilton D Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem T rank(i) Dijkstra's Algorithm Bellman-Ford Algorith m A 0 Floyd- Warshall Algorit hm B 1 Ford's algorithm c 2 Others D 3 Graph Coloring E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_231.png", "page_index": 653, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:42+07:00" } }, "CO1007-chapter-10-slide-654-0000": { "id": "CO1007-chapter-10-slide-654-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai B BK 4 TP.HCM 3 4 Contents 5 Connectivity 4 2 Paths and Circuits 2 Euler and Hamilton Paths D Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m rank(i) Bellman-Ford Algorith m i Floyd-Warshall Algorit hm A 0 Ford's algorithm B 1 Others c 2 Graph Coloring D 3 E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_232.png", "page_index": 654, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:46+07:00" } }, "CO1007-chapter-10-slide-655-0000": { "id": "CO1007-chapter-10-slide-655-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai B BK 4 TP.HCM 3 0 4 Contents 5 Connectivity 4 2 Paths and Circuits 2 Euler and Hamilton Paths D Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m rank(i) Bellman-Ford Algorith m i Floyd-Warshall Algorit hm A 0 Ford's algorithm B 1 Others c 2 Graph Coloring D 3 E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_233.png", "page_index": 655, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:50+07:00" } }, "CO1007-chapter-10-slide-656-0000": { "id": "CO1007-chapter-10-slide-656-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai B B BK TP.HCM 3 0 4 3 Contents 5 Connectivity 4 2 Paths and Circuits 2 4 Euler and Hamilton Paths E Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m rank(i) Bellman-Ford Algorith m i Floyd-Warshall Algorit hm A 0 Ford's algorith m B 1 Others c 2 Graph Coloring D 3 E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_234.png", "page_index": 656, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:54+07:00" } }, "CO1007-chapter-10-slide-657-0000": { "id": "CO1007-chapter-10-slide-657-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 7 B B BK TP.HCM 3 0 4 Contents 5 Connectivity 4 2 Paths and Circuits 2 4 Euler and Hamilton Paths E Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m rank(i) Bellman-Ford Algorith m i Floyd-Warshall Algorit hm A 0 Ford's algorith m B 1 Others c 2 Graph Coloring D 3 E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_235.png", "page_index": 657, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:38:59+07:00" } }, "CO1007-chapter-10-slide-658-0000": { "id": "CO1007-chapter-10-slide-658-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai 7 B B BK TP.HCM 3 0 T 5 4 C Contents 5 Connectivity 4 2 Paths and Circuits 2 4 Euler and Hamilton Paths E Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m rank(i) Bellman-Ford Algorith m i Floyd-Warshall Algorit hm A 0 Ford's algorith m B 1 Others c 2 Graph Coloring D 3 E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_236.png", "page_index": 658, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:03+07:00" } }, "CO1007-chapter-10-slide-659-0000": { "id": "CO1007-chapter-10-slide-659-0000", "text": "Example Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai - B B BK TP.HCM 4 3 5 4 3 5 Contents 5 Connectivity 4 2 Paths and Circuits 2 4 Euler and Hamilton Paths E Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m rank(i) Bellman-Ford Algorith m i Floyd-Warshall Algorit hm A 0 Ford's algorith m B 1 Others c 2 Graph Coloring D 3 E 1 10.60", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_237.png", "page_index": 659, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:09+07:00" } }, "CO1007-chapter-10-slide-660-0000": { "id": "CO1007-chapter-10-slide-660-0000", "text": "Exercise Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK 5 TP.HCM B 7 2 Contents Connectivity 1 Paths and Circuits 6 3 E Euler and Hamilton Paths 4 5 Euler Paths and Circuits 3 Hamilton Paths and Circuits 4 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m 2 Floyd- Warshall Algorit hm Ford's algorithm Others Graph Coloring 10.61", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_238.png", "page_index": 660, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:13+07:00" } }, "CO1007-chapter-10-slide-661-0000": { "id": "CO1007-chapter-10-slide-661-0000", "text": "Application Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Problem A young professor in Hue is invited to teach some years in Ho Chi Minh university of BK technology. He decides to represent the diverse operations of his transfer by a graph and. TP.HCM in this purpose, establishes the list of following operations: A: Find a house in Ho Chi Minh city. B: Choose a removal man and sign a contract of move Contents C: Make pack his furniture by the removal man Connectivity D: Make transport his furniture towards Ho Chi Minh city Paths and Circuits Euler and Hamilton E: Find an accommodation to HCM (from Hue) Paths F: Transport his family to HCM Euler Paths and Circuits Hamilton Paths and Circuits G: Move into his new accommodation Shortest Path Problem H: Register the children to their new school Dijkstra's Algorith m I: Look for a temporary work for his wife Bellman-Ford Algorith m Floyd- Warshall Algorit hm J: Fit out the new accommodation and pay this arrangement with the first treatment Ford's algorithm of his wife Others K: Find a small bar to celebrate in family the success of the move and express the Graph Coloring enjoyment to live in a good accommodation arrangement 10.62", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_239.png", "page_index": 661, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:17+07:00" } }, "CO1007-chapter-10-slide-662-0000": { "id": "CO1007-chapter-10-slide-662-0000", "text": "Application Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Considering constraint of posteriority following: A < F; B < C; Cx->y-u-x-y-t,l=6 Bellman-Ford Algorith m Floyd-Warshall Algorithm ors-x-y-u-v-y-t,l=6 Ford's algorithm Othel Graph Coloring 10.66", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_246.png", "page_index": 668, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:42+07:00" } }, "CO1007-chapter-10-slide-669-0000": { "id": "CO1007-chapter-10-slide-669-0000", "text": "Top k shortest paths query Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai When the shortest path is not sufficient for application, top-k shortest paths are desired. BK TP.HCM Top 3 general shortest paths (allowing loops) Contents Connectivity Paths and Circuits 2nd. Euler and Hamilton Paths s->x->y-u-x-y-t,l=6 Euler Paths and Circuits ors-x-y-u-v-y->t,l=6 Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm 7 Top 2 elementary shortest paths (without Bellman-Ford Algorith m Floyd-Warshall Algorit hm loops) Ford's algorithm Othel Graph Coloring 10.66", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_247.png", "page_index": 669, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:46+07:00" } }, "CO1007-chapter-10-slide-670-0000": { "id": "CO1007-chapter-10-slide-670-0000", "text": "Top k shortest paths query Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai When the shortest path is not sufficient for application, top-k shortest paths are desired. BK TP.HCM Top 3 general shortest paths (allowing loops) Contents 1 Connectivity Paths and Circuits 2nd. Euler and Hamilton Paths s->x->y-u-x-y->t,l=6 Euler Paths and Circuits ors-x-y-u-v-y-t, l=6 Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm 1 Top 2 elementary shortest paths (without Bellman-Ford Algorith m Floyd-Warshall Algorit hm loops) Ford's algorithm Othel Graph Coloring 10.66", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_248.png", "page_index": 670, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:50+07:00" } }, "CO1007-chapter-10-slide-671-0000": { "id": "CO1007-chapter-10-slide-671-0000", "text": "Traveling Salesman Problem (TSP) Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Problem BK TP.HCM Given a set of m customers located in m cities and distances for each pair of cities, the problem involves finding a round-trip with the minimum traveling cost. Contents The vehicle must visit each customer exactly once and return to its point Connectivity Paths and Circuits of origin also called depot. Euler and Hamilton The objective function is the total cost of the tour. Paths Euler Paths and Circuits WP-complete: all known techniques for obtaining an exact solution Hamilton Paths and Circuits require an exponentially increasing number of steps (computing Shortest Path Problem resources) as the problems become larger. Dijkstra's Algorith m TSP is one of the most intensely studied problems in computational Bellman-Ford Algorith m Floyd- Warshall Algorit hm mathematics, yet no effective solution method. Ford's algorith m Others Graph Coloring 10.67", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_249.png", "page_index": 671, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:54+07:00" } }, "CO1007-chapter-10-slide-672-0000": { "id": "CO1007-chapter-10-slide-672-0000", "text": "Traveling Salesman Problem Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai b 3 c Xuan Toan. Tran Hong Tai 2 2 BK TP.HCM 3) 2 4 a d Contents Connectivity 4 Paths and Circuits Y Euler and Hamilton Paths Euler Paths and Circuits e Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m The total number of possible Hamilton circuit is (n - 1)!/2 Floyd-Warshall Algorithm Ford's algorith m For example, if there are 25 customers to visit, the total Others number of solutions is 24!/2 = 3.1 x 1023 Graph Coloring 10.68", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_250.png", "page_index": 672, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:39:59+07:00" } }, "CO1007-chapter-10-slide-673-0000": { "id": "CO1007-chapter-10-slide-673-0000", "text": "Traveling Salesman Problem Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai b 3 c Xuan Toan. Tran Hong Tai 2 2 BK TP.HCM 3 2 4 a d Contents Connectivity 4 Paths and Circuits 3 Y Euler and Hamilton Paths Euler Paths and Circuits e Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m The total number of possible Hamilton circuit is (n - 1)!/2 Floyd-Warshall Algorithm Ford's algorith m For example, if there are 25 customers to visit, the total Others number of solutions is 24!/2 = 3.1 x 1023 Graph Coloring If the depot is located at node 1, then the optimal tour is 1 - 5 - 2 - 3 - 4 - 1 with total cost equal to 11. 10.68", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_251.png", "page_index": 673, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:03+07:00" } }, "CO1007-chapter-10-slide-674-0000": { "id": "CO1007-chapter-10-slide-674-0000", "text": "Vehicle Routing Problem (VRP) Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Problem BK TP.HCM The vehicle routing problem involves finding a set of trips, one for each vehicle, to deliver known quantities of goods to a Contents set of customers. Connectivity The objective is to minimize the travel costs of all trips Paths and Circuits Euler and Hamilton combined. Paths Euler Paths and Circuits There may be upper bounds on the total load of each vehicle Hamilton Paths and Circuits and the total duration of its trip Shortest Path P roblem Dijkstra's Algorith m The most basic Vehicle Routing Problem (VRP) is the Bellman-Ford Algorithm Floyd-Warshall Algorithm single-depot capacitate VRP. Ford's algorithm Othen Graph Coloring 10.69", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_252.png", "page_index": 674, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:07+07:00" } }, "CO1007-chapter-10-slide-675-0000": { "id": "CO1007-chapter-10-slide-675-0000", "text": "Maps and Graphs Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition BK Every map can be represented by a graph. We call it dual TP.HCM graph. Problem of coloring the regions of a map -? coloring the Contents vertices of the dual graph so that no two adjacent vertices Connectivity have the same color. Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.70", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_253.png", "page_index": 675, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:10+07:00" } }, "CO1007-chapter-10-slide-676-0000": { "id": "CO1007-chapter-10-slide-676-0000", "text": "Maps and Graphs Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK Every map can be represented by a graph. We call it dual TP.HCM graph. Problem of coloring the regions of a map -> coloring the Contents vertices of the dual graph so that no two adiacent vertices Connectivity have the same color Paths and Circuits Euler and Hamilton Paths B Euler Paths and Circuits Hamilton Paths and Circuits B Shortest Path Problem Dijkstra's Algorith m E Bellman-Ford Algorith m C Floyd-Warshall Algorithm E Ford's algorith m D Others Graph Coloring D 10.70", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_254.png", "page_index": 676, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:14+07:00" } }, "CO1007-chapter-10-slide-677-0000": { "id": "CO1007-chapter-10-slide-677-0000", "text": "Graph coloring Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai A coloring (t mau) of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent BK vertices are assigned the same color. TP.HCM Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem a Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others Graph Coloring c d 10.71", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_255.png", "page_index": 677, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:19+07:00" } }, "CO1007-chapter-10-slide-678-0000": { "id": "CO1007-chapter-10-slide-678-0000", "text": "Graph coloring Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai : A coloring (t mäu) of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent BK vertices are assigned the same color. TP.HCM The chromatic number (só mäu) of a graph, denoted by X(G), is the least number of colors needed for a coloring of Contents this graph. Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem a Dijkstra's Algorith m Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others g Graph Coloring C d 10.71", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_256.png", "page_index": 678, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:23+07:00" } }, "CO1007-chapter-10-slide-679-0000": { "id": "CO1007-chapter-10-slide-679-0000", "text": "Graph coloring Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai : A coloring (t mäu) of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent BK vertices are assigned the same color. TP.HCM The chromatic number (só mäu) of a graph, denoted by X(G), is the least number of colors needed for a coloring of Contents this graph. Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem a Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others 4 9 Graph Coloring C d c d 10.71", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_257.png", "page_index": 679, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:27+07:00" } }, "CO1007-chapter-10-slide-680-0000": { "id": "CO1007-chapter-10-slide-680-0000", "text": "Four color theorem Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Theorem (Four color theorem) The chromatic number of a planar graph is no greater than four Contents Connectivity Paths and Circuits Was a conjecture in the 1850s Euler and Hamilton Paths Was not proved completely until 1976 by Kenneth Appel and Euler Paths and Circuits Wolfgang Haken, using computer Hamilton Paths and Circuits Shortest Path Problem No proof not relying on a computer has yet been found Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorit hm Ford's algorith m Others Graph Coloring 10.72", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_258.png", "page_index": 680, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:31+07:00" } }, "CO1007-chapter-10-slide-681-0000": { "id": "CO1007-chapter-10-slide-681-0000", "text": "Applications of Graph coloring Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Scheduling Final Exam Xuan Toan. Tran Hong Tai How can the final exams at a university be scheduled so that no student has two exams at the same time? BK Suppose we have 7 finals, numbered 1 through 7. TP.HCM The pairs of courses have common students are depicted in the following graph Contents Connectivity Paths and Circuits 1 Euler and Hamilton Paths 7 2 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd- Warshall Algorithm Ford's algorith m Others Graph Coloring 5 4 10.73", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_259.png", "page_index": 681, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:34+07:00" } }, "CO1007-chapter-10-slide-682-0000": { "id": "CO1007-chapter-10-slide-682-0000", "text": "Applications of Graph coloring Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Scheduling Final Exam Xuan Toan. Tran Hong Tai How can the final exams at a university be scheduled so that no student has two exams at the same time? BK Suppose we have 7 finals, numbered 1 through 7. TP.HCM The pairs of courses have common students are depicted in the following graph Contents Connectivity Paths and Circuits 1 Euler and Hamilton Paths 7 2 Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m 6 3 Others Graph Coloring 5 4 10.73", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_260.png", "page_index": 682, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:39+07:00" } }, "CO1007-chapter-10-slide-683-0000": { "id": "CO1007-chapter-10-slide-683-0000", "text": "Applications of Graph Coloring Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM Other Applications Frequency Assignments: Television channels 2 through 12 are Contents assigned to stations in North America so that no two stations Connectivity within 150 miles can operate on the same channel. How can Paths and Circuits the assignment of channels be modeled by graph coloring? Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorithm Others Graph Coloring 10.74", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_261.png", "page_index": 683, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:42+07:00" } }, "CO1007-chapter-10-slide-684-0000": { "id": "CO1007-chapter-10-slide-684-0000", "text": "Applications of Graph Coloring Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Other Applications Frequency Assignments: Television channels 2 through 12 are Contents assigned to stations in North America so that no two stations Connectivity within 150 miles can operate on the same channel. How can Paths and Circuits the assignment of channels be modeled by graph coloring? Euler and Hamilton Paths Euler Paths and Circuits Index Registers: In an execution of loop, the frequently used Hamilton Paths and Circuits variables should be stored in index registers to speed up. How Shortest Path Problem many index registers are needed? Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.74", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_262.png", "page_index": 684, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:46+07:00" } }, "CO1007-chapter-10-slide-685-0000": { "id": "CO1007-chapter-10-slide-685-0000", "text": "Revision Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Given H = (V,E) is a simple undirected graph. Which of the Contents following assessment is/are true? Connectivity A H and Hc are connected graphs Paths and Circuits Euler and Hamilton B H and Hc exist Euler path. Paths Euler Paths and Circuits H or Hc is connected graphs Hamilton Paths and Circuits Shortest Path Problem D H or Hc exists Hamilton path Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithn Ford's algorith m Others Graph Coloring 10.75", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_263.png", "page_index": 685, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:50+07:00" } }, "CO1007-chapter-10-slide-686-0000": { "id": "CO1007-chapter-10-slide-686-0000", "text": "Revision Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai a d g BK TP.HCM Contents Connectivity 6 h Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Which of the following assessments is/are true about the cut Shortest Path Problem vertex and cut edge of the graph above? Dijkstra's Algorithm Bellman-Ford Algorith m A The graph has 2 cut vertex. Floyd-Warshall Algorit hm Ford's algorith m B) The graph has 4 cut vertex. Others Graph Coloring The graph has 2 cut edge The graph has 2 cut vertex and 1 bridge. 10.76", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_264.png", "page_index": 686, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:54+07:00" } }, "CO1007-chapter-10-slide-687-0000": { "id": "CO1007-chapter-10-slide-687-0000", "text": "Revision Graph connectivity Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai BK TP.HCM Choose the correct answer about the reliation of cut edge and cut Contents vertex Connectivity A Two endpoints of a cut edge must be cut vertices Paths and Circuits Euler and Hamilton Two endpoints of a cut edge maybe not the cut vertices. Paths Euler Paths and Circuits One of two endpoints of a cut edge must be a cut vertex Hamilton Paths and Circuits Shortest Path Problem two endpoints of a cut edge are not cut vertices Dijkstra's Algorithm Bellman-Ford Algorith m Floyd-Warshall Algorithn Ford's algorith m Others Graph Coloring 10.77", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_265.png", "page_index": 687, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:40:57+07:00" } }, "CO1007-chapter-10-slide-688-0000": { "id": "CO1007-chapter-10-slide-688-0000", "text": "Revision Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Determine a shortest path from a to other vertices in the following BK TP.HCM graph. 1 Contents Connectivity -5 C Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits 5 Hamilton Paths and Circuits a d Shortest Path Problem Dijkstra's Algorith m 3 Bellman-Ford Algorith m Floyd-Warshall Algorithm 1 e Ford's algorithm Others Graph Coloring 10.78", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_266.png", "page_index": 688, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:01+07:00" } }, "CO1007-chapter-10-slide-689-0000": { "id": "CO1007-chapter-10-slide-689-0000", "text": "Revision Graph connectivity Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Determine a shortest path from any vertex to other vertex in the following graph (using Floyd-Warshall algorithm) BK TP.HCM 1 Contents Connectivity Paths and Circuits Euler and Hamilton Y Paths Euler Paths and Circuits (G2) 3 -5 Hamilton Paths and Circuits 3 Shortest Path Problem Dijkstra's Algorith m Bellman-Ford Algorith m Floyd-Warshall Algorithm Ford's algorith m Others Graph Coloring 10.79", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_10/slide_267.png", "page_index": 689, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:04+07:00" } }, "CO1007-chapter-11-slide-690-0000": { "id": "CO1007-chapter-11-slide-690-0000", "text": "Trees Nguyen An Khuong. Tran Tuan Anh, Mai Chapter 11 Xuan Toan,Tran Hon Tai Trees BK TP.HCM Discrete Structures for Computing on June 16, 2024 Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Faculty of Computer Science and Engineering University of Technology - VNUHCM trtanh@hcmut.edu.vn 11.1", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_001.png", "page_index": 690, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:07+07:00" } }, "CO1007-chapter-11-slide-691-0000": { "id": "CO1007-chapter-11-slide-691-0000", "text": "Contents Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM 11.2", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_002.png", "page_index": 691, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:08+07:00" } }, "CO1007-chapter-11-slide-692-0000": { "id": "CO1007-chapter-11-slide-692-0000", "text": "Course outcomes Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Course learning outcomes L.0.1 Understanding of logic and discrete structures BK L.O.1.1 - Describe definition of propositional and predicate logic TP.HCM L.0.1.2 - Define basic discrete structures: set, mapping, graphs L.0.2 Represent and model practical problems with discrete structures L.O.2.1 - Logically describe some problems arising in Computing L.O.2.2 - Use proving methods: direct, contrapositive, induction L.O.2.3 - Explain problem modeling using discrete structures L.0.3 Understanding of basic probability and random variables L.0.3.1 - Define basic probability theory L.0.3.2 - Explain discrete random variables L.0.4 Compute quantities of discrete structures and probabilities L.0.4.1 - Operate (compute/ optimize) on discrete structures L.0.4.2 - Compute probabilities of various events, conditional ones, Bayes theorem 11.3", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_003.png", "page_index": 692, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:12+07:00" } }, "CO1007-chapter-11-slide-693-0000": { "id": "CO1007-chapter-11-slide-693-0000", "text": "Introduction Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Very useful in computer science: search algorithm, game winning strategy, decision making, sorting, BK TP.HCM Other disciplines: chemical compounds, family trees, organizational tree, home bin tmp tan mail Is junk music latex scala 11.4", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_004.png", "page_index": 693, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:14+07:00" } }, "CO1007-chapter-11-slide-694-0000": { "id": "CO1007-chapter-11-slide-694-0000", "text": "Tree Trees Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan,Tran Hong Tai A tree (cay) is a connected undirected graph with no simple circuits. Consequently, a tree must be a simple graph. BK z X TP.HCM G1 G2 G3 G4 circuit exists not connected 11.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_005.png", "page_index": 694, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:17+07:00" } }, "CO1007-chapter-11-slide-695-0000": { "id": "CO1007-chapter-11-slide-695-0000", "text": "Tree Trees Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai A tree (cay) is a connected undirected graph with no simple circuits. Consequently, a tree must be a simple graph. BK x X TP.HCM G1 G2 G3 G 4 circuit exists not connected Definition Graphs containing no simple circuits that are not necessarily connected is forest (rüng), in which each connected component is a tree. 11.5", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_006.png", "page_index": 695, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:20+07:00" } }, "CO1007-chapter-11-slide-696-0000": { "id": "CO1007-chapter-11-slide-696-0000", "text": "Trees Rooted Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Definition A rooted tree (cay c6 gc) is a tree in which: BK One vertex has been designated as the root and TP.HCM Every edge is directed away from the root f 9 d e C 11.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_007.png", "page_index": 696, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:23+07:00" } }, "CO1007-chapter-11-slide-697-0000": { "id": "CO1007-chapter-11-slide-697-0000", "text": "Rooted Trees Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Definition A rooted tree (cay c6 göc) is a tree in which: BK One vertex has been designated as the root and TP.HCM Every edge is directed away from the root f 9 a d 6 d g e 11.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_008.png", "page_index": 697, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:26+07:00" } }, "CO1007-chapter-11-slide-698-0000": { "id": "CO1007-chapter-11-slide-698-0000", "text": "Rooted Trees Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Definition A rooted tree (cay c6 göc) is a tree in which: BK One vertex has been designated as the root and TP.HCM Every edge is directed away from the root f 9 a d 6 d g e 11.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_009.png", "page_index": 698, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:29+07:00" } }, "CO1007-chapter-11-slide-699-0000": { "id": "CO1007-chapter-11-slide-699-0000", "text": "Rooted Trees Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hon Tai Definition A rooted tree (cay c6 göc) is a tree in which: BK One vertex has been designated as the root and TP.HCM Every edge is directed away from the root 9 d 6 e e d g 11.6", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_010.png", "page_index": 699, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:31+07:00" } }, "CO1007-chapter-11-slide-700-0000": { "id": "CO1007-chapter-11-slide-700-0000", "text": "Terminology Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition BK a vertex of a tree is called a leaf (/a) if it has no children TP.HCM vertices that have children are called internal vertices (dinh trong) a 9 11.7", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_011.png", "page_index": 700, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:33+07:00" } }, "CO1007-chapter-11-slide-701-0000": { "id": "CO1007-chapter-11-slide-701-0000", "text": "Terminology Trees Nguyen An Khuong. Tran Tuan Anh, Mai Definition Xuan Toan. Tran Hong Tai parent (cha) of v is the unique u such that there is a directed edge from u to v BK when u is the parent of v, is called a child (con) of u TP.HCM vertices with the same parent are called siblings (anh em) the ancestors (t tién) of a vertex are the vertices in the path from the root to this vertex (excluding the vertex itself) descendants (con chäu) of a vertex v are those vertices that have u as an ancestor a b d g e 11.8", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_012.png", "page_index": 701, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:36+07:00" } }, "CO1007-chapter-11-slide-702-0000": { "id": "CO1007-chapter-11-slide-702-0000", "text": "Terminology Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Definition If a is a vertex in a tree, the subtree (cay con) with a as its root is BK the subgraph of the tree consisting of a and its descendants and TP.HCM all edges incident to these descendants. a 6 g e 11.9", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_013.png", "page_index": 702, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:39+07:00" } }, "CO1007-chapter-11-slide-703-0000": { "id": "CO1007-chapter-11-slide-703-0000", "text": "Trees m-ary tree Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Definition Tai m-ary tree (cay m-phan): at most m children on each internal vertex of a rooted tree. BK full m-ary tree (cay m-phan day d&): every internal vertex has TP.HCM exactly m children. An m-ary tree with m = 2 is called a binary tree (c&y nhi phan) Full 3-ary tree Binary tree Full 5-ary tree 3-ary tree 11.10", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_014.png", "page_index": 703, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:42+07:00" } }, "CO1007-chapter-11-slide-704-0000": { "id": "CO1007-chapter-11-slide-704-0000", "text": "Ordered Rooted Trees Trees Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai An ordered rooted tree (cay c6 göc có thú tu) is a rooted tree where the children of each internal vertex are ordered (e.g. in order from left to right). BK TP.HCM In an ordered binary tree (cay nhi phan c6 thü tu), if an Internal vertex has two children, the first child is called the left child (con bén trai) and the second is called the right child (con ben phai). a d g Left subtree of d Right subtree of a 11.11", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_015.png", "page_index": 704, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:44+07:00" } }, "CO1007-chapter-11-slide-705-0000": { "id": "CO1007-chapter-11-slide-705-0000", "text": "Properties & Theorems Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Theorem BK A tree with n vertices has n - 1 edges TP.HCM Theorem A full m-ary tree n vertices has (n - 1)/m internal vertices and m - 1)n + 1]/m leaves @ i interna/ vertices has m = mi + 1 vertices and (m - 1i + 1 leaves @ l leaves has n = (ml - 1/(m - 1) vertices and (l - 1)/(m - 1) internal vertices 11.12", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_016.png", "page_index": 705, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:48+07:00" } }, "CO1007-chapter-11-slide-706-0000": { "id": "CO1007-chapter-11-slide-706-0000", "text": "Example Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Example (Chain Letter Game) Tai Each person who receives the letter is asked to send it on to four other peoples. BK TP.HCM Some peoples do this, but others do not send any letters. How many people have seen the letter, including the first person, if no one receives more than one letter and if the chain letter ends after there have been 100 people who read it but did not send it out ? How many people sent out the letter? 11.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_017.png", "page_index": 706, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:50+07:00" } }, "CO1007-chapter-11-slide-707-0000": { "id": "CO1007-chapter-11-slide-707-0000", "text": "Example Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example (Chain Letter Game) Tai Each person who receives the letter is asked to send it on to four other peoples. BK TP.HCM Some peoples do this, but others do not send any letters. How many people have seen the letter, including the first person, if no one receives more than one letter and if the chain letter ends after there have been 100 people who read it but did not send it out ? How many people sent out the letter? Solution Using 4-ary tree with 100 leaves corresponding to 100 persons who did not send out the letter. -> n = (ml - 1)/(m - 1) = (4 x 100 - 1)/(4 - 1) = 133 vertices and i = n - l = 133 - 100 = 33 interna/ vertices. 11.13", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_018.png", "page_index": 707, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:53+07:00" } }, "CO1007-chapter-11-slide-708-0000": { "id": "CO1007-chapter-11-slide-708-0000", "text": "Level and Height Trees Nguyen An Khuong. Definition Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai The level (múc) of a vertex v in a rooted tree is the length of the unique path from the root to this vertex. BK The level of the root is defined to be zero. TP.HCM The height (d cao) of a rooted tree is the maximum of the levels of vertices (i.e. the length of the longest path from the root to any vertex). a Example 6 k o Level of root a = 0 .2 b,i,k= 1 and c,e,f,l=2.. C . Because the largest 9 level of any vertex is d m n 4, this tree has height 4. h 11.14", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_019.png", "page_index": 708, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:57+07:00" } }, "CO1007-chapter-11-slide-709-0000": { "id": "CO1007-chapter-11-slide-709-0000", "text": "Balanced m-ary Trees Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Definition A rooted m-ary tree of height h is balanced (cän dói) if all leaves are at levels h or h - 1. BK TP.HCM T1 T2 11.15", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_020.png", "page_index": 709, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:41:59+07:00" } }, "CO1007-chapter-11-slide-710-0000": { "id": "CO1007-chapter-11-slide-710-0000", "text": "Balanced m-ary Tree Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Theorem There are at most mh leaves in an m-ary tree of height h. lt can be proved by using mathematical induction on the height Corollary If an m-ary tree of height h has I leaves, then h [logm £]. If the m-ary tree is full and balanced, then h = logm £: 11.16", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_021.png", "page_index": 710, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:02+07:00" } }, "CO1007-chapter-11-slide-711-0000": { "id": "CO1007-chapter-11-slide-711-0000", "text": "Exercise Trees Nguyen An Khuong, Tran Tuan Anh, Mai Exercise (Chess tournament) Xuan Toan,Tran Hong Tai Suppose 1000 people enter a chess tournament. Use a rooted tree model of the tournament to determine how many games must be BK played to determine a champion. If a player is eliminated after one TP.HCM loss and games are played until only one entrant has not lost. (Assume there are no ties) 11.17", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_022.png", "page_index": 711, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:04+07:00" } }, "CO1007-chapter-11-slide-712-0000": { "id": "CO1007-chapter-11-slide-712-0000", "text": "Question Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM Exercise How many vertices and how many leaves does a complete m-ary tree of height h have? : Show that a full m-ary balanced tree (cay m-phan hoan häo of height h has more than mh-1 leaves. How many edges are there in a forest of t trees containing a total of n vertices? 11.18", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_023.png", "page_index": 712, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:07+07:00" } }, "CO1007-chapter-11-slide-713-0000": { "id": "CO1007-chapter-11-slide-713-0000", "text": "Labeling Ordered Rooted Trees Trees Nguyen An Khuong. Tran Tuan Anh, Mai Ordered rooted trees are often used to store information Xuan Toan. Tran Hong Tai Need a procedure for visiting each vertex of an ordered rooted tree to access data. Ordering and labeling the vertices is important to traverse them in any BK procedure TP.HCM Universal address system (h@ dia chi ph dung) 0<1 <1.1 <1.1.1<1.2<1.3<...<2<3<3.1 < 0 . 1.2 1.3 3.1 1.3.1 1.3.2 3.1.1 3.1.2 1.3.1.1 11.19", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_024.png", "page_index": 713, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:10+07:00" } }, "CO1007-chapter-11-slide-714-0000": { "id": "CO1007-chapter-11-slide-714-0000", "text": "Traversal Algorithms (Thuat toan duyét cay) Trees Nguyen An Khuong, Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d 9 e 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_025.png", "page_index": 714, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:12+07:00" } }, "CO1007-chapter-11-slide-715-0000": { "id": "CO1007-chapter-11-slide-715-0000", "text": "Traversal Algorithms (Thuat toan duyét cay) Trees Nguyen An Khuong, Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d 9 a 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_026.png", "page_index": 715, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:15+07:00" } }, "CO1007-chapter-11-slide-716-0000": { "id": "CO1007-chapter-11-slide-716-0000", "text": "Traversal Algorithms (Thuat toan duyét cay) Trees Nguyen An Khuong, Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d a 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_027.png", "page_index": 716, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:17+07:00" } }, "CO1007-chapter-11-slide-717-0000": { "id": "CO1007-chapter-11-slide-717-0000", "text": "Traversal Algorithms (Thuat toan duyét cay) Trees Nguyen An Khuong, Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d g a 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_028.png", "page_index": 717, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:19+07:00" } }, "CO1007-chapter-11-slide-718-0000": { "id": "CO1007-chapter-11-slide-718-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a 9 a d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_029.png", "page_index": 718, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:22+07:00" } }, "CO1007-chapter-11-slide-719-0000": { "id": "CO1007-chapter-11-slide-719-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hon Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d a d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_030.png", "page_index": 719, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:24+07:00" } }, "CO1007-chapter-11-slide-720-0000": { "id": "CO1007-chapter-11-slide-720-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hon Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d y a b d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_031.png", "page_index": 720, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:27+07:00" } }, "CO1007-chapter-11-slide-721-0000": { "id": "CO1007-chapter-11-slide-721-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d 9 a b f d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_032.png", "page_index": 721, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:31+07:00" } }, "CO1007-chapter-11-slide-722-0000": { "id": "CO1007-chapter-11-slide-722-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d a b f g d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_033.png", "page_index": 722, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:33+07:00" } }, "CO1007-chapter-11-slide-723-0000": { "id": "CO1007-chapter-11-slide-723-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d 4 a b f g d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_034.png", "page_index": 723, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:36+07:00" } }, "CO1007-chapter-11-slide-724-0000": { "id": "CO1007-chapter-11-slide-724-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d g X a b f g d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_035.png", "page_index": 724, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:39+07:00" } }, "CO1007-chapter-11-slide-725-0000": { "id": "CO1007-chapter-11-slide-725-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hong Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d g a b f g c d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_036.png", "page_index": 725, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:42+07:00" } }, "CO1007-chapter-11-slide-726-0000": { "id": "CO1007-chapter-11-slide-726-0000", "text": "Traversal Algorithms (Thuat toán duyét cay) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Preorder Traversal (duyét tién thü tu - NLR) Xuan Toan,Tran Hon Tai procedure preorder(T: ordered rooted tree) r := root of T BK print r TP.HCM for each child c of r from left to right T(c) := subtree with c as its root preorder(T(c)) a d g a b f g c e d 11.20", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_037.png", "page_index": 726, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:45+07:00" } }, "CO1007-chapter-11-slide-727-0000": { "id": "CO1007-chapter-11-slide-727-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM . , Tn from left to right, inorder traversal: T1-r-T2-...-Tn: a d 9 C 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_038.png", "page_index": 727, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:47+07:00" } }, "CO1007-chapter-11-slide-728-0000": { "id": "CO1007-chapter-11-slide-728-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM : , Tn from left to right, inorder traversal: T-r-T2-...-Tn. a d 9 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_039.png", "page_index": 728, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:50+07:00" } }, "CO1007-chapter-11-slide-729-0000": { "id": "CO1007-chapter-11-slide-729-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM . , Tn from left to right, inorder traversal: T1-r-T2...-Tn: a d 9 C a 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_040.png", "page_index": 729, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:52+07:00" } }, "CO1007-chapter-11-slide-730-0000": { "id": "CO1007-chapter-11-slide-730-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM . , Tn from left to right, inorder traversal: T1-r-T2-...-Tn: a d 9 a 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_041.png", "page_index": 730, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:55+07:00" } }, "CO1007-chapter-11-slide-731-0000": { "id": "CO1007-chapter-11-slide-731-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T-r-T2-...-Tn a 9 C a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_042.png", "page_index": 731, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:42:58+07:00" } }, "CO1007-chapter-11-slide-732-0000": { "id": "CO1007-chapter-11-slide-732-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM . , Tn from left to right, inorder traversal: T1-r-T2...-Tn: a d 9 a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_043.png", "page_index": 732, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:01+07:00" } }, "CO1007-chapter-11-slide-733-0000": { "id": "CO1007-chapter-11-slide-733-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T-r-T2-...-Tn a d 4 f a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_044.png", "page_index": 733, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:04+07:00" } }, "CO1007-chapter-11-slide-734-0000": { "id": "CO1007-chapter-11-slide-734-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T-r-T2-...-Tn a d f b a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_045.png", "page_index": 734, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:07+07:00" } }, "CO1007-chapter-11-slide-735-0000": { "id": "CO1007-chapter-11-slide-735-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T-r-T2-...-Tn a d f b g a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_046.png", "page_index": 735, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:09+07:00" } }, "CO1007-chapter-11-slide-736-0000": { "id": "CO1007-chapter-11-slide-736-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T-r-T2-...-Tn a d f 9 f b g a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_047.png", "page_index": 736, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:12+07:00" } }, "CO1007-chapter-11-slide-737-0000": { "id": "CO1007-chapter-11-slide-737-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T-r-T2-...-Tn a d 9 f b g a d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_048.png", "page_index": 737, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:15+07:00" } }, "CO1007-chapter-11-slide-738-0000": { "id": "CO1007-chapter-11-slide-738-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T1-r-T2-...-Tn a d g f b g a e d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_049.png", "page_index": 738, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:18+07:00" } }, "CO1007-chapter-11-slide-739-0000": { "id": "CO1007-chapter-11-slide-739-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Inorder Traversal (Duyét trung thü tu - LNR) Suppose a tree T with root r. If T consists only of r, then r is BK inorder traversal of T. Otherwise, suppose r has subtrees T1, T2 TP.HCM , Tn from left to right, inorder traversal: T1-r-T2-...-Tn a d 9 f b g a e c d 11.21", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_050.png", "page_index": 739, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:22+07:00" } }, "CO1007-chapter-11-slide-740-0000": { "id": "CO1007-chapter-11-slide-740-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong, Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d 9 C 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_051.png", "page_index": 740, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:24+07:00" } }, "CO1007-chapter-11-slide-741-0000": { "id": "CO1007-chapter-11-slide-741-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d y 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_052.png", "page_index": 741, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:27+07:00" } }, "CO1007-chapter-11-slide-742-0000": { "id": "CO1007-chapter-11-slide-742-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong, Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d 9 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_053.png", "page_index": 742, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:30+07:00" } }, "CO1007-chapter-11-slide-743-0000": { "id": "CO1007-chapter-11-slide-743-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a 9 d 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_054.png", "page_index": 743, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:33+07:00" } }, "CO1007-chapter-11-slide-744-0000": { "id": "CO1007-chapter-11-slide-744-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong, Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r d 9 d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_055.png", "page_index": 744, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:36+07:00" } }, "CO1007-chapter-11-slide-745-0000": { "id": "CO1007-chapter-11-slide-745-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_056.png", "page_index": 745, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:39+07:00" } }, "CO1007-chapter-11-slide-746-0000": { "id": "CO1007-chapter-11-slide-746-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hon Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d 4 + d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_057.png", "page_index": 746, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:41+07:00" } }, "CO1007-chapter-11-slide-747-0000": { "id": "CO1007-chapter-11-slide-747-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hon Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d f g d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_058.png", "page_index": 747, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:43+07:00" } }, "CO1007-chapter-11-slide-748-0000": { "id": "CO1007-chapter-11-slide-748-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d y f g b d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_059.png", "page_index": 748, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:46+07:00" } }, "CO1007-chapter-11-slide-749-0000": { "id": "CO1007-chapter-11-slide-749-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d f g b d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_060.png", "page_index": 749, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:49+07:00" } }, "CO1007-chapter-11-slide-750-0000": { "id": "CO1007-chapter-11-slide-750-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d 9 f g b d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_061.png", "page_index": 750, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:52+07:00" } }, "CO1007-chapter-11-slide-751-0000": { "id": "CO1007-chapter-11-slide-751-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong, Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d g f g b e d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_062.png", "page_index": 751, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:55+07:00" } }, "CO1007-chapter-11-slide-752-0000": { "id": "CO1007-chapter-11-slide-752-0000", "text": "Traversal Algorithms Trees Nguyen An Khuong. Tran Tuan Anh, Mai Postorder Traversal (Duyét hau thu tu - LRN) Xuan Toan,Tran Hong Tai procedure postorder(T: ordered rooted tree) r := root of T BK for each child c of r from left to right TP.HCM T(c) := subtree with c as its root postorder(T(c)) print r a d 9 f g b e c d a 11.22", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_063.png", "page_index": 752, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:43:58+07:00" } }, "CO1007-chapter-11-slide-753-0000": { "id": "CO1007-chapter-11-slide-753-0000", "text": "Infix, Prefix and Postfix Notations Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Infix (trung t6): BK TP.HCM ((x + y) 1 2) + ((x - 4)/3) Prefix (tién t6): + T +xy2/ - x43 1 2 3 x y x 4 Postfix (hau t6): xy + 2T x4 -3/+ 11.23", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_064.png", "page_index": 753, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:01+07:00" } }, "CO1007-chapter-11-slide-754-0000": { "id": "CO1007-chapter-11-slide-754-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Exercise Tai Find the ordered rooted tree representing BK (-(pAq) v(-qAr))->(-pV-r) TP.HCM Then use this rooted tree to find the prefix, postfix and infix forms of this expression 11.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_065.png", "page_index": 754, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:03+07:00" } }, "CO1007-chapter-11-slide-755-0000": { "id": "CO1007-chapter-11-slide-755-0000", "text": "Exercise Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan, Tran Hong Exercise Tai Find the ordered rooted tree representing BK (-(pA q) v(-q r))->(-p V-r) TP.HCM Then use this rooted tree to find the prefix, postfix and infix forms of this expression Solution Constructing the rooted tree from the bottom up : Preorder traversal creates prefix notation >V-Ap qA-q rV-p nr Postorder traversal creates postfix notation p q A - V qr p-r-V - Inorder traversal creates infix notation (with parentheses) p q V q-A r ->p V r- 11.24", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_066.png", "page_index": 755, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:06+07:00" } }, "CO1007-chapter-11-slide-756-0000": { "id": "CO1007-chapter-11-slide-756-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise Find postorder traversal of a binary tree with inorder D B H E I A BK F C J G K and preorder A B D E H l C F G J K TP.HCM 11.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_067.png", "page_index": 756, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:08+07:00" } }, "CO1007-chapter-11-slide-757-0000": { "id": "CO1007-chapter-11-slide-757-0000", "text": "Exercise Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise Find postorder traversal of a binary tree with inorder D B H E I A BK F C J G K and preorder A B D E H I C F G J K. TP.HCM Solution 4 B H K Post order: D H 1 E B F J K G C A 11.25", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_068.png", "page_index": 757, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:11+07:00" } }, "CO1007-chapter-11-slide-758-0000": { "id": "CO1007-chapter-11-slide-758-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise Find in-order traversal of a binary tree with pre-order A D E B J C BK F H l G and post-order E J B D H l F G C A. TP.HCM 11.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_069.png", "page_index": 758, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:13+07:00" } }, "CO1007-chapter-11-slide-759-0000": { "id": "CO1007-chapter-11-slide-759-0000", "text": "Exercise Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise Find in-order traversal of a binary tree with pre-order A D E B J C BK F H I G and post-order E J B D H I F G C A. TP.HCM Solution A B E H ln-order: E D J B A H F 1 C G. 11.26", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_070.png", "page_index": 759, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:15+07:00" } }, "CO1007-chapter-11-slide-760-0000": { "id": "CO1007-chapter-11-slide-760-0000", "text": "Exercise Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Exercise How many different trees are there with the in-order of K E B J C BK A H G I D F and father-child relations respecting to the alphabet TP.HCM order. 11.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_071.png", "page_index": 760, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:17+07:00" } }, "CO1007-chapter-11-slide-761-0000": { "id": "CO1007-chapter-11-slide-761-0000", "text": "Exercise Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise How many different trees are there with the in-order of K E B J C BK A H G I D F and father-child relations respecting to the alphabet TP.HCM order. Solution B 4 K Pre-order: E D J B A H F1 C G 11.27", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_072.png", "page_index": 761, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:20+07:00" } }, "CO1007-chapter-11-slide-762-0000": { "id": "CO1007-chapter-11-slide-762-0000", "text": "Binary Search Trees Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Definition Binary search tree (cay tim kiém nhi phan - BST) is a binary tree BK in which the assigned key of a vertex is: TP.HCM larger than the keys of all vertices in its left subtree, and smaller than the keys of all vertices in its right subtree 5 11.28", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_073.png", "page_index": 762, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:22+07:00" } }, "CO1007-chapter-11-slide-763-0000": { "id": "CO1007-chapter-11-slide-763-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_074.png", "page_index": 763, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:24+07:00" } }, "CO1007-chapter-11-slide-764-0000": { "id": "CO1007-chapter-11-slide-764-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Example Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_075.png", "page_index": 764, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:25+07:00" } }, "CO1007-chapter-11-slide-765-0000": { "id": "CO1007-chapter-11-slide-765-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_076.png", "page_index": 765, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:28+07:00" } }, "CO1007-chapter-11-slide-766-0000": { "id": "CO1007-chapter-11-slide-766-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics physics 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_077.png", "page_index": 766, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:30+07:00" } }, "CO1007-chapter-11-slide-767-0000": { "id": "CO1007-chapter-11-slide-767-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics geography physics 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_078.png", "page_index": 767, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:33+07:00" } }, "CO1007-chapter-11-slide-768-0000": { "id": "CO1007-chapter-11-slide-768-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan,Tran Hong Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order mathematics geography physics zoology 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_079.png", "page_index": 768, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:35+07:00" } }, "CO1007-chapter-11-slide-769-0000": { "id": "CO1007-chapter-11-slide-769-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai Example Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics geography physics meteorology zoology 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_080.png", "page_index": 769, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:37+07:00" } }, "CO1007-chapter-11-slide-770-0000": { "id": "CO1007-chapter-11-slide-770-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics geography physics geology meteorology zoology 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_081.png", "page_index": 770, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:40+07:00" } }, "CO1007-chapter-11-slide-771-0000": { "id": "CO1007-chapter-11-slide-771-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics geography physics geology meteorology zoology psychology 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_082.png", "page_index": 771, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:42+07:00" } }, "CO1007-chapter-11-slide-772-0000": { "id": "CO1007-chapter-11-slide-772-0000", "text": "Adding and Locating an Item in BST Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Example Tai Form a BST for the words mathematics, physics, geography zoology, meteorology, geology, psychology, chemistry using BK TP.HCM alphabetical order. mathematics geography physics chemistry geology meteorology zoology psychology Complexity in searching O(log(n)) vs. O(n) in linear list 11.29", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_083.png", "page_index": 772, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:45+07:00" } }, "CO1007-chapter-11-slide-773-0000": { "id": "CO1007-chapter-11-slide-773-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. 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Give an algorithm for finding this counterfeit coin. lig lig 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_087.png", "page_index": 776, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:55+07:00" } }, "CO1007-chapter-11-slide-777-0000": { "id": "CO1007-chapter-11-slide-777-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_088.png", "page_index": 777, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:44:57+07:00" } }, "CO1007-chapter-11-slide-778-0000": { "id": "CO1007-chapter-11-slide-778-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_089.png", "page_index": 778, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:00+07:00" } }, "CO1007-chapter-11-slide-779-0000": { "id": "CO1007-chapter-11-slide-779-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig 1 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_090.png", "page_index": 779, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:02+07:00" } }, "CO1007-chapter-11-slide-780-0000": { "id": "CO1007-chapter-11-slide-780-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig lig 1 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_091.png", "page_index": 780, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:05+07:00" } }, "CO1007-chapter-11-slide-781-0000": { "id": "CO1007-chapter-11-slide-781-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig bal lig 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_092.png", "page_index": 781, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:08+07:00" } }, "CO1007-chapter-11-slide-782-0000": { "id": "CO1007-chapter-11-slide-782-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig bal lig 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_093.png", "page_index": 782, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:12+07:00" } }, "CO1007-chapter-11-slide-783-0000": { "id": "CO1007-chapter-11-slide-783-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig bal lig lig bal lig Null Y 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_094.png", "page_index": 783, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:15+07:00" } }, "CO1007-chapter-11-slide-784-0000": { "id": "CO1007-chapter-11-slide-784-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig bal lig lig bal lig Null Y 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_095.png", "page_index": 784, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:18+07:00" } }, "CO1007-chapter-11-slide-785-0000": { "id": "CO1007-chapter-11-slide-785-0000", "text": "Decision Trees (Cay quyét dinh) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Example Xuan Toan. Tran Hong Tai There are seven coins, all with the same weight, and a counterfeit coin that weighs less than the others. How many weighings are BK necessary using a balance scale to determine which of the eight TP.HCM coins is the counterfeit one? Give an algorithm for finding this counterfeit coin. lig bal lig lig bal lig lig bal lig lig bal lig Null Y 8 C 11.30", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_096.png", "page_index": 785, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:22+07:00" } }, "CO1007-chapter-11-slide-786-0000": { "id": "CO1007-chapter-11-slide-786-0000", "text": "Yet Another Application Trees Nguyen An Khuong, Tran Tuan Anh, Mai Xuan Toan,Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? Start! 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_097.png", "page_index": 786, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:24+07:00" } }, "CO1007-chapter-11-slide-787-0000": { "id": "CO1007-chapter-11-slide-787-0000", "text": "Yet Another Application Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan,Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? TB 0.0005 Start! 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_098.png", "page_index": 787, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:26+07:00" } }, "CO1007-chapter-11-slide-788-0000": { "id": "CO1007-chapter-11-slide-788-0000", "text": "Yet Another Application Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? TB 0.0005 0.9995 Start! TB 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_099.png", "page_index": 788, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:29+07:00" } }, "CO1007-chapter-11-slide-789-0000": { "id": "CO1007-chapter-11-slide-789-0000", "text": "Yet Another Application Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? 0.999 TB 0.0005 Start! 0.9995 TB 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_100.png", "page_index": 789, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:31+07:00" } }, "CO1007-chapter-11-slide-790-0000": { "id": "CO1007-chapter-11-slide-790-0000", "text": "Yet Another Application Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? 0.999 TB 0.001 0.0005 Start! 0.9995 TB 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_101.png", "page_index": 790, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:33+07:00" } }, "CO1007-chapter-11-slide-791-0000": { "id": "CO1007-chapter-11-slide-791-0000", "text": "Yet Another Application Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai If we know that the probability that a person has tuberculosis TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? 0.999 TB 0.001 0.0005 Start! 0.9995 0.01 TB 0.99 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_102.png", "page_index": 791, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:36+07:00" } }, "CO1007-chapter-11-slide-792-0000": { "id": "CO1007-chapter-11-slide-792-0000", "text": "Yet Another Application Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? 0.999 TB & + = 0.0004995 TB 0.001 TB & - = 0.0000005 0.0005 0.9995 Start! 0.01 TB& + = 0.009995 TB 0.99 TB & - = 0.989505 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_103.png", "page_index": 792, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:40+07:00" } }, "CO1007-chapter-11-slide-793-0000": { "id": "CO1007-chapter-11-slide-793-0000", "text": "Yet Another Application Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Example Tai If we know that the probability that a person has tuberculosis (TB) is p(TB) = 0.0005. BK We also know p(+TB) = 0.999 and p(-TB) = 0.99. TP.HCM What is p(TB+) and p(TB-)? 0.999 TB & + = 0.0004995 TB 0.001 TB & - = 0.0000005 0.0005 0.9995 Start! 0.01 TB& + = 0.009995 TB 0.99 TB & - = 0.989505 p(TB+) = p(TBn+) 0.0004995 0.0476 p(+) 0.0004995+0.009995 11.31", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_104.png", "page_index": 793, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:44+07:00" } }, "CO1007-chapter-11-slide-794-0000": { "id": "CO1007-chapter-11-slide-794-0000", "text": "Problem Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK Definition TP.HCM A spanning tree (cay khung) in a graph G is a subgraph of G that is a tree which contains all vertices of G. 11.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_105.png", "page_index": 794, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:46+07:00" } }, "CO1007-chapter-11-slide-795-0000": { "id": "CO1007-chapter-11-slide-795-0000", "text": "Problem Trees Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. 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Tran Hong Tai BK Definition TP.HCM A spanning tree (cay khung) in a graph G is a subgraph of G that is a tree which contains all vertices of G. a b C d g 11.32", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_108.png", "page_index": 797, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:45:53+07:00" } }, "CO1007-chapter-11-slide-798-0000": { "id": "CO1007-chapter-11-slide-798-0000", "text": "Problem Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. 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Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai BK TP.HCM 3 Property Go deeper as you can Backtrack (quay /ui) to possible branch when you are stuck O(e) or O(n2) 11.33", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_127.png", "page_index": 816, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:46:27+07:00" } }, "CO1007-chapter-11-slide-817-0000": { "id": "CO1007-chapter-11-slide-817-0000", "text": "Trees Depth-First Search Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. 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Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai vertex L 0 BK 1 2,3,4 TP.HCM 2 3,4,5,6 3 4,5,6 4 5.6,7,8 5 6,7,8 6 8 11.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_137.png", "page_index": 826, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:46:50+07:00" } }, "CO1007-chapter-11-slide-827-0000": { "id": "CO1007-chapter-11-slide-827-0000", "text": "Breadth-First Search (Tim kiém uu tién chiéu röng) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai vertex L 0 BK 1 2,3,4 TP.HCM 2 3,4,5,6 3 4,5,6 4 5,6,7,8 5 6,7,8 6 7,8 8 11.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_138.png", "page_index": 827, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:46:53+07:00" } }, "CO1007-chapter-11-slide-828-0000": { "id": "CO1007-chapter-11-slide-828-0000", "text": "Breadth-First Search (Tim kiém uu tién chiéu röng) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai vertex L 0 BK 1 2,3,4 TP.HCM 2 3,4,5,6 3 4,5,6 4 5,6,7,8 5 6,7,8 6 7,8 7 8 6 11.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_139.png", "page_index": 828, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:46:56+07:00" } }, "CO1007-chapter-11-slide-829-0000": { "id": "CO1007-chapter-11-slide-829-0000", "text": "Breadth-First Search (Tim kiém uu tién chiéu röng) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai vertex L 0 BK 1 2,3,4 TP.HCM 2 3,4,5,6 3 4,5,6 4 5.6,7,8 5 6,7,8 6 7,8 7 8 End! 8 0 6 11.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_140.png", "page_index": 829, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:46:59+07:00" } }, "CO1007-chapter-11-slide-830-0000": { "id": "CO1007-chapter-11-slide-830-0000", "text": "Breadth-First Search (Tim kiém uu tién chiéu röng) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai vertex L 0 BK 1 2,3,4 TP.HCM 2 3,4,5,6 3 4,5,6 4 5,6,7,8 5 6,7,8 6 7,8 7 8 8 0 h 6 8 Property : 0(e) or 0(n2) 11.35", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_141.png", "page_index": 830, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:03+07:00" } }, "CO1007-chapter-11-slide-831-0000": { "id": "CO1007-chapter-11-slide-831-0000", "text": "Breadth-First Search Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Algorithm BK TP.HCM procedure BFS (G) T := tree consisting only vertex v1 L := empty list put 1 in the list L of unprocessed vertices while L is not empty remove the first vertex, v, from [ for each neighbor w of v if w is not in L and not in T then add w to the end of the list T add w and edge{v,w} to T 11.36", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_142.png", "page_index": 831, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:05+07:00" } }, "CO1007-chapter-11-slide-832-0000": { "id": "CO1007-chapter-11-slide-832-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Exercise Xuan Toan, Tran Hong Tai Find spanning tree in the following graphs BK a b TP.HCM a h 2 11.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_143.png", "page_index": 832, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:08+07:00" } }, "CO1007-chapter-11-slide-833-0000": { "id": "CO1007-chapter-11-slide-833-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Exercise Xuan Toan, Tran Hong Tai Find spanning tree in the following graphs BK a b TP.HCM h 2 a b c h 2 11.37", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_144.png", "page_index": 833, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:10+07:00" } }, "CO1007-chapter-11-slide-834-0000": { "id": "CO1007-chapter-11-slide-834-0000", "text": "Minimum Spanning Trees Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A minimum spanning tree (cay khung nhö nhat) in a BK connected weighted graph is a spanning tree that has the TP.HCM smallest possible sum of weights of its edges. 11.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_145.png", "page_index": 834, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:12+07:00" } }, "CO1007-chapter-11-slide-835-0000": { "id": "CO1007-chapter-11-slide-835-0000", "text": "Minimum Spanning Trees Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A minimum spanning tree (cay khung nhö nhät) in a BK connected weighted graph is a spanning tree that has the TP.HCM smallest possible sum of weights of its edges. $2000 Chicago $1200 $1000 New York San Francisco $900 $1300 $700 Denver $1600 $800 $1400 $2200 Atlanta 11.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_146.png", "page_index": 835, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:15+07:00" } }, "CO1007-chapter-11-slide-836-0000": { "id": "CO1007-chapter-11-slide-836-0000", "text": "Minimum Spanning Trees Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Definition A minimum spanning tree (cay khung nhö nhät) in a BK connected weighted graph is a spanning tree that has the TP.HCM smallest possible sum of weights of its edges. $2000 Chicago $1200 $1000 New York San Francisco $900 $1300 Denver $700 $1600 $800 $1400 $2200 Atlanta 11.38", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_147.png", "page_index": 836, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:18+07:00" } }, "CO1007-chapter-11-slide-837-0000": { "id": "CO1007-chapter-11-slide-837-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Prim's Algorithm (1957) procedure Prim(G) T := a minimum-weight edge for i := 1 to n - 2 e := an edge of minimum weight incident to a vertex in T and not forming a simple circuit in T if added to T T := T with e added return T 11.39", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_148.png", "page_index": 837, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:21+07:00" } }, "CO1007-chapter-11-slide-838-0000": { "id": "CO1007-chapter-11-slide-838-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hong Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 4 1 1 10 d e 5 10 f 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_149.png", "page_index": 838, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:24+07:00" } }, "CO1007-chapter-11-slide-839-0000": { "id": "CO1007-chapter-11-slide-839-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hong Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 4 1 1 10 d e 5 10 f 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_150.png", "page_index": 839, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:26+07:00" } }, "CO1007-chapter-11-slide-840-0000": { "id": "CO1007-chapter-11-slide-840-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hong Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 1 10 d e 5 10 f 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_151.png", "page_index": 840, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:30+07:00" } }, "CO1007-chapter-11-slide-841-0000": { "id": "CO1007-chapter-11-slide-841-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hong Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 1 7 10 d e 5 10 f 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_152.png", "page_index": 841, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:33+07:00" } }, "CO1007-chapter-11-slide-842-0000": { "id": "CO1007-chapter-11-slide-842-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hon Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c A 1 7 10 d e 5 10 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_153.png", "page_index": 842, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:36+07:00" } }, "CO1007-chapter-11-slide-843-0000": { "id": "CO1007-chapter-11-slide-843-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hon Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 1 1 1 10 d e 5 10 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_154.png", "page_index": 843, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:39+07:00" } }, "CO1007-chapter-11-slide-844-0000": { "id": "CO1007-chapter-11-slide-844-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Pick a vertex to start from Xuan Toan, Tran Hong Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 1 1 1 10 a e 5 10 3 f 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_155.png", "page_index": 844, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:42+07:00" } }, "CO1007-chapter-11-slide-845-0000": { "id": "CO1007-chapter-11-slide-845-0000", "text": "Prim's Algorithm (Nearest-Neighbor) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Pick a vertex to start from Tai Iteratively absorb smallest edge possible BK TP.HCM a 4 b 5 c 1 1 1 10 a e 5 10 3 f 10 g 11.40", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_156.png", "page_index": 845, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:44+07:00" } }, "CO1007-chapter-11-slide-846-0000": { "id": "CO1007-chapter-11-slide-846-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM Kruskal's Algorithm (1958 procedure Kruska/(G) T := empty graph for i := 1 to m - 1 e := any edge in G with smallest weight that does not form a simple circuit when added to T T := T with e added return T 11.41", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_157.png", "page_index": 846, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:46+07:00" } }, "CO1007-chapter-11-slide-847-0000": { "id": "CO1007-chapter-11-slide-847-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 10 a e 5 10 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_158.png", "page_index": 847, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:49+07:00" } }, "CO1007-chapter-11-slide-848-0000": { "id": "CO1007-chapter-11-slide-848-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 10 a e 5 10 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_159.png", "page_index": 848, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:52+07:00" } }, "CO1007-chapter-11-slide-849-0000": { "id": "CO1007-chapter-11-slide-849-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 10 a e 5 10 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_160.png", "page_index": 849, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:55+07:00" } }, "CO1007-chapter-11-slide-850-0000": { "id": "CO1007-chapter-11-slide-850-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 10 d e 5 10 3 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_161.png", "page_index": 850, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:47:58+07:00" } }, "CO1007-chapter-11-slide-851-0000": { "id": "CO1007-chapter-11-slide-851-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 10 d e 5 10 3 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_162.png", "page_index": 851, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:01+07:00" } }, "CO1007-chapter-11-slide-852-0000": { "id": "CO1007-chapter-11-slide-852-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 10 d e 5 10 3 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_163.png", "page_index": 852, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:04+07:00" } }, "CO1007-chapter-11-slide-853-0000": { "id": "CO1007-chapter-11-slide-853-0000", "text": "Kruskal's Algorithm (Lightest-Edge) Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hong Tai Iteratively add smallest edge possible BK TP.HCM a 4 b 5 c 4 1 1 10 d e 5 10 3 f 10 g 11.42", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_164.png", "page_index": 853, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:07+07:00" } }, "CO1007-chapter-11-slide-854-0000": { "id": "CO1007-chapter-11-slide-854-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan,Tran Hong Tai BK TP.HCM Exercise By using Prim's and Kruskal's algorithm, determine minimum spanning tree in the following graphs. 11.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_165.png", "page_index": 854, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:09+07:00" } }, "CO1007-chapter-11-slide-855-0000": { "id": "CO1007-chapter-11-slide-855-0000", "text": "Exercise Trees Nguyen An K huong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise BK By using Prim's and Kruskal's algorithm, determine minimum TP.HCM spanning tree in the following graphs a 4 C 1 2 5 2 1 d 2 g 1 2 1 h 2 11.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_166.png", "page_index": 855, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:13+07:00" } }, "CO1007-chapter-11-slide-856-0000": { "id": "CO1007-chapter-11-slide-856-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan, Tran Hon Tai Exercise BK By using Prim's and Kruskal's algorithm, determine minimum TP.HCM spanning tree in the following graphs a 1 b 8 c 4 2 4 6 2 d e 9 5 1 3 2 7 h 2 11.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_167.png", "page_index": 856, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:16+07:00" } }, "CO1007-chapter-11-slide-857-0000": { "id": "CO1007-chapter-11-slide-857-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai Exercise By using Prim's and Kruskal's algorithm, determine minimum BK spanning tree in the following graphs TP.HCM a 2 b 3 c 1 d 3 1 2 5 3 g 3 e h 4 2 3 3 3 1 2 j k /. 11.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_168.png", "page_index": 857, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:20+07:00" } }, "CO1007-chapter-11-slide-858-0000": { "id": "CO1007-chapter-11-slide-858-0000", "text": "Exercise Trees Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan, Tran Hong Tai Exercise By using Prim's and Kruskal's algorithm, determine minimum BK spanning tree in the following graphs. (and maximum spanning TP.HCM tree (cay khung cuc dai) a 2 b 3 c 1 d 3 1 2 5 4 3 9 3 e h 4 2 3 3 3 2 j k 11.43", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_169.png", "page_index": 858, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:25+07:00" } }, "CO1007-chapter-11-slide-859-0000": { "id": "CO1007-chapter-11-slide-859-0000", "text": "Revision Trees Nguyen An Khuong. Given two graphs G and H as follow: Tran Tuan Anh, Mai Xuan Toan, Tran Hong 2 3 B Tai BK TP.HCM 6 h F E G H) Which of the following assessments is/are correct? G is a tree. G and H are isomorphic If we delete an edge of G then we can get a tree. We can get a tree if we delete an edge of H. 11.44", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_170.png", "page_index": 859, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:28+07:00" } }, "CO1007-chapter-11-slide-860-0000": { "id": "CO1007-chapter-11-slide-860-0000", "text": "Trees Revision Nguyen An Khuong. Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai BK TP.HCM What is the value of each of these prefix expressions? @ - * 2 / 8 4 3 b * -* 33* 425 c +- * 3 2 + 2 3/ 6-42 d *+3 + 3* 3 + 3 33 11.45", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_171.png", "page_index": 860, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:30+07:00" } }, "CO1007-chapter-11-slide-861-0000": { "id": "CO1007-chapter-11-slide-861-0000", "text": "Revision Trees Nguyen An Khuong. Determine the minimum spanning tree of the following graph (by Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai two methods) 11 5 c E BK 10 TP.HCM 6 2 2 4 10 8 7 B D F 8 H 3 14 4 6 11.46", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_172.png", "page_index": 861, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:33+07:00" } }, "CO1007-chapter-11-slide-862-0000": { "id": "CO1007-chapter-11-slide-862-0000", "text": "Revision Trees Nguyen An Khuong. Determine the minimum spanning tree of the following graph (by Tran Tuan Anh, Mai Xuan Toan. Tran Hong Tai two methods) 11 5 E BK 10 TP.HCM 6 2 2 4 10 8 7 B D F 8 H 3 14 4 6 By using Prim's or Kruskal's algorithm, could we determine a minimum spanning tree in a directed graph? Explain it. 11.46", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_173.png", "page_index": 862, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:37+07:00" } }, "CO1007-chapter-11-slide-863-0000": { "id": "CO1007-chapter-11-slide-863-0000", "text": "Revision Trees Nguyen An Khuong. Tran Tuan Anh. Mai Xuan Toan. Tran Hong Tai BK Given G = (V, E) is an undirected graph that has m vertices. The TP.HCM complement graph of G is Gc = (V,F) such that: G U Gc = Kn va E n F = 0. Let T be the spanning tree of K6, What is the number of edge of Tc. A 5 B 10 15 D 20 11.47", "metadata": { "doc_type": "slide", "course_id": "CO1007", "source_file": "/workspace/data/converted/CO1007_Discrete_Structures_for _Computing/Chapter_11/slide_174.png", "page_index": 863, "language": "en", "ocr_engine": "PaddleOCR 3.2", "extractor_version": "1.0.0", "timestamp": "2025-10-30T16:48:40+07:00" } } }