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Algebra 2 lessons online free
Algebra 2 lessons online free
Discount applies only to current purchase. Discount cannot be combined with other offers.close window. Try Thinkwell Free. Algebra 2 is the third math course in high school and algebra 2 lessons online free guide you through among other things linear equations, inequalities, graphs, matrices, polynomials and radical expressions, onlinf equations, functions, exponential and logarithmic expressions, sequences and series, probability and trigonometry.This Algebra algeebra math course elssons divided into 13 chapters and each chapter is divided into alvebra lessons.
Then you found the right place to get help. We have more than forty free, text-based algebra lessons listed on the left. If not, try the site search at the top of every page. You will learn about functions, polynomials, graphing, complex numbers, exponential onlibe logarithmic equations, and much more, all through exploring real-world scenarios. Play TrailerPlay Trailer. The following topics will be covered. There will be two midterms and one final exam. To do it in the right way and get an accurate answer you will have to apply the order of operations.
As these are referred in multiple problems within the narrative that explains what you have to do to solve the problem. A great cook knows how to take algrbra ingredients and prepare a delicious meal. In this topic, you will become function-chefs. You will learn how to combine functions algebra 2 lessons online free arithmetic operations and how to compose functions. You will also learn how to transform functions in ways that shift, reflect, or stretch their graphs. Learn how to manipulate polynomials in order to prove identities and find the zeros of t. | 677.169 | 1 |
Course description:
The study of number theory originated in several ancient civilizations including China and India. Many famous results in number theory were proved in Europe in the 17th and 18th centuries. It is a core subject in mathematics in its own right and has important connections to algebra, geometry, combinatorics, and analysis. One motivation for studying number theory is that it provides many applications to coding theory and cryptography.
Number theory is known for having problems that are easy to state yet which can only be solved using complicated structures. For example, it took 300 years to find a complete proof of Fermat's Last Theorem.
Number theory is a vast subject; we will focus on the topics in the course description including Diophantine equations; distribution of primes; multiplicative functions; finite fields; quadratic reciprocity; quadratic number fields.
In addition, we will study elliptic curves and some applications to cryptography. We will use the computer program
SAGE to solve many problems in number theory.
Grading:
The course grades will be computed as follows:
20% homework; 20% midterm; 30% projects and presentations; 30% Final.
Borderline grades will be decided on the basis of class participation.
Homework: Due every week.
Doing homework problems is crucial for doing well in this class.
The process of doing homework will help you solve unfamiliar problems on the tests.
The homework problems will help you develop skills in computation and logical reasoning.
Homework must be neat, legible, and stapled.
I encourage you to brainstorm the problems in groups and write up your
solutions independently.
Project:
In this class, we will use the computer program SAGE, a free on-line math program which is helpful to solve complicated numerical problems.
It can also be used to collect data and develop greater understanding of topics in number theory.
Don't worry if you haven't used math software before - we will go over all the basics together.
There will be two group projects using SAGE and one group presentation.
This gives us a chance to have an overview of many fantastic topics in number theory that we wouldn't see otherwise.
Important Dates:
The midterm is in class on Friday March 2.
Group project 1 will be done in class during the week of March 5-9.
Group presentations will be the week of March 19-23.
Project 2 will be done in class during the week of March 26-30.
The final exam is Monday May 7 Mon 2-3 pm and Thurs 1-2 pm in Weber 118. | 677.169 | 1 |
Algebra I: Powerpoint Q and A Game - Exponents
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This powerpoint Q and A game ( a jeopardy type game) is designed to review the basic concepts of exponents and exponential models in algebra I using the categories: recursive sequences, exponential equations, multiplying and exponents, zero and negative exponents, and miscellaneous | 677.169 | 1 |
Revising/Exam Worries
Physics Factsheet: 03. Algebraic Manipulation I
Physics Factsheet: 03. Algebraic Manipulation I
Price £3.00
Code: PHA-FSPI-003
Quantity
What Is Algebra Relationships between various quantities like mass, Word Relationships between various quantities (like mass, volume and density) are best expressed in terms of some formula or equation (an equation is anything with an = sign in it) | 677.169 | 1 |
Heya guys! Is someone here know about pre algebra chapter 2? I have this set of problems about it that I just can't understand. Our class was assigned to answer it and understand how we came up with the answer . Our Math professor will select random people to solve the problem as well as show solutions to class so I need comprehensive explanation about pre algebra chapter 2. I tried answering some of the questions but I guess I got it completely incorrect. Please help me because it's a bit urgent and the due date is near already and I haven't yet understood how to answer this.
Well I do have a suggestion for you. There used to be time when even I was stuck on questions relating to pre algebra chapter 2, that's when my younger sister suggested that I should try Algebrator. It didn't just solve all my questions , but it also explained those answers in a very nice step-by-step manner. It's hard to believe but one night I was actually crying due the fact that I would miss yet another assignment deadline, and a couple of days from that I was actually helping my friends with their assignments as well. I know how weird it might sound, but really Algebrator helped me a lot.
I remember I faced similar problems with complex fractions, gcf and graphing circles. This Algebrator is rightly a great piece of math software program. This would simply give step by step solution to any math problem that I copied from workbook on clicking on Solve. I have been able to use the program through several Algebra 2, Intermediate algebra and College Algebra. I seriously recommend the program.
I am a regular user of Algebrator. It not only helps me get my homework faster, the detailed explanations given makes understanding the concepts easier. I strongly recommend using it to help improve problem solving skills. | 677.169 | 1 |
Text: one is to teach the basics of
set theory, logic, combinatorics and graph theory.
The other is to convey concepts essential to mathematics:
absolute clarity and precision in
definitions and statements of fact, and rigorous methods
for establishing that a statement is true.
Grading
We will have weekly assignments, three midterms and a final exam.
The midterms will be scheduled for Wednesdays, if not announced otherwise.
For the weekly assignments, there will be a small number of problems
(10 or so) which you should write up carefully.
They will be graded and returned to you promptly (to the best of my
ability).
I recommend that you do or at least attempt most of the exercises
in the book. That is the best way to learn!
Point value for the work will be as follows (plus or minus 50 points)
Weekly work
200
Test 1
150
Test 2
150
Test 3
150
Final
350
Total
1000
First Midterm
Wednesday, February 27th on Section 1 through
8, 10, 14.
Final Exam
Friday, May 17 8:00-10:00 am. in BA 260.
Grading Scale
A: 100-85%, B: 84-70%,
C: 69-55%, D: 55-40%, F: below 40%
Note
There will be no makeup exams. A missing midterm will amount
to F for the respective exam. For each exam a calculator and one arbitrarily
densely written letter-sized sheet of paper (cheat sheet) will be allowed. | 677.169 | 1 |
"This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically with minimal prerequisites and plenty of geometric intuition." — From the back cover
$39.95
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VERY GOOD: This book is in very good condition, showing only slight signs of use and wear. It was printed on acid-free paper.
Description
This book is Number 202 in Springer's Graduate Texts in Mathematics series.
Continued from the back cover: "A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the world 'manifold.' Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields.
"In his beautifully conceived Introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout."
"This book is based on lectures given to advance undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics." — From the back cover
"This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems." — From the back cover
"Despite its success as a mathematical tool in the general theory of relativity and its adaptability to a wide range of mathematical and physical problems, tensor analysis has always had a rather restricted level of use, with an emphasis on notation and the manipulation of indices. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject." — From the back cover
"The book contains an amazing wealth of material relating to the algebra, geometry, and analysis of the nineteenth century.... Written with accurate historical perspective and clear exposition, this book is truly hard to put down." — Zentralblatt für Mathematik (review of the first edition, taken from the back cover)Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course." — From the back cover
"This excellent text, long considered one of the best-written, most skillful expositions of group theory and its physical applications, is directed primarily to advanced undergraduate and graduate students in physics, especially quantum physics." — From the back cover
"The theory of groups is simultaneously a branch of abstract algebra and the study of symmetry. Designed to support a reader engaged in a first serious group theory course, or a mathematically mature reader approaching the subject for the first time, this book reviews the essentials." — From the back cover
Answers and hints to newly-added exercises in this "Study Edition" make this an appealing text to use by students and professionals alike when being introduced to the field of group theory as it applies to physics.
"As prerequisites I assume a first course in linear algebra (including matrix multiplication and the representation of linear maps between Euclidean spaces by matrices, though not the abstract theory of vector spaces) plus familiarity with the basic properties of the real and complex numbers. It would be a pity to teach group theory without matrix groups available as a rich source of examples, especially since matrices are so heavily used in application." — Mark Anthony Armstrong, Author, in the Preface
"In the 1990s it was realized that quantum physics has some spectacular applications in computer science. This book is a concise introduction to quantum computation, developing the basic elements of this new branch of computational theory without assuming any background in physics." — From the back cover
"[The late] Michio Kuga's lectures on Group Theory and Differential Equations are a realization of two dreams—one to see Galois groups used to attack the problems of differential equations—the other to do so in such a manner as to take students from a very basic level to an understanding of the heart of this fascinating mathematical problemElliptic curves have played an increasingly important role in number theory and relate fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, an few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory an Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications | 677.169 | 1 |
Description of the book "Framework Maths: Year 7 Core Students' Book":
Framework Maths is a brand new course designed to match the pitch, pace and progression of the Framework for Teaching Mathematics at Key Stage 3. This students Book is written for the Core tier in Year 7, and is firmly based on the Framework objectives. The book comprises units organised clearly into inspiring full-colour spreads. Each unit offers: *Prior learning points identified at the start so that revision is a continual process *Learning objectives identified so it is clear what students need to know *Clear explanations covered with examples showing the key techniques *Plenty of practice with questions pitched at the level suggested PDF in the Framework *Summaries and review questions to help students gain responsibility for their learning Framework Maths comprises a Students' Book and a Teacher's Book for each year of Key Stage 3, at three tiers of ability: Support, Core and Extension. There is also a CD-ROM for each year, containing assessment and further resources.
Reviews of the Framework Maths: Year 7 Core Students' Book
To date with regards to the book we have Framework Maths: Year 7 Core Students' Book comments end users are yet to nevertheless still left their particular review of the action, or not read it however. But, when you have presently read this guide and you're ready to create their discoveries well expect you to take your time to exit an evaluation on our site (we are able to release the two negative and positive critiques). Quite simply, "freedom regarding speech" All of us wholeheartedly helped. Your current responses to lease Framework Maths: Year 7 Core Students' Book -- different viewers is able to determine in regards to guide. This sort of guidance can certainly make you far more Usa!
David Capewell
However, at this time do not have got information regarding your musician David Capewell. Nonetheless, we will get pleasure from if you have just about any information regarding the idea, and they are able to offer the idea. Post the item to us! We have each of the check, in case all the info are usually genuine, we're going to publish on the web page. It's very important for individuals that accurate with regards to David Capewell. We thanks a lot upfront to be prepared to go to satisfy people! | 677.169 | 1 |
Review of the topics in a second-year high school algebra course taught at the college level. Includes: real numbers, 1st and 2nd degree equations and inequalities, linear systems, polynomials and rational expressions, exponents and radicals. Heavy emphasis on problem solving strategies and techniques.
Prereqs: One year high school geometry and either two years high school algebra, one semester high school precalculus, and a qualifying score on the Math Placement Exam; or a grade of C, P, or better in MATH 101.
Credit toward the degree may be earned in only one of MATH 102 or 103.
Applications of quantitative reasoning and methods to problems and decision making in the areas of management, statistics, and social choice. Includes networks, critical paths, linear programming, sampling, central tendency, inference, voting methods, power index, game theory, and fair division problems.
Not open to students with credit or concurrent enrollment in MATH 106 or MATH 203.
Applications of quantitative reasoning and methods to problems and decisions making in areas of particular relevance to College of Journalism and Mass Communication, such as governance, finance, statistics, social choice, and graphical presentation of data. Financial mathematics, statistics and probability (sampling, central tendency, and inference), voting methods, power index, and fair division problems.
MATH 300M is open only to a middle grades teaching endorsement program student. Credit towards degree may be earned in only one of: MATH 300, or MATH 300M. MATH 300M is designed to strengthen the mathematics knowledge of the middle-level mathematics teacher.
Develop a deeper understanding of "number and operations". The importance of careful reasoning, problem solving, and communicating mathematics, both orally and in writing. Connections with other areas of mathematics and the need for developing the "habits of mind of a mathematical thinker".
Using mathematics to model solutions or relationships for realistic problems taken from the middle school curriculum. The mathematics for these models are a mix of algebra, geometry, sequences (dynamical systems, queuing theory), functions (linear, exponential, logarithmic), and logic. Mathematical terminology, concepts and principles. Calculator based lab devices, graphing calculators, and computers as tools to collect data, to focus on concepts and ideas, and to made the mathematics more accessible.
Open only to middle grades teaching endorsement majors with a mathematics emphasis and/or to elementary education majors who want a mathematics concentration.
How to express mathematical solutions and ideas logically and coherently in both written and oral forms in the context of problem solving. Inductive and deductive logical reasoning skills through problem solving. Present and critique logical arguments in verbal and written forms. Problem topics taken from topics nationally recommended for middle school mathematics.
Elementary number theory, including induction, the Fundamental Theorem of Arithmetic, and modular arithmetic. Introduction to rings and fields as natural extension of the integers. Particular emphasis on the study of polynomials with coefficients in the rational, real, or complex numbers.
Uniform convergence of sequences and series of functions, Green's theorem, Stoke's theorem, divergence theorem, line integrals, implicit and inverse function theorems, and general coordinate transformations.
An introduction to mathematical reasoning, construction of proofs, and careful mathematical writing in the context of continuous mathematics and calculus. Topics may include the real number system, limits and continuity, the derivative, integration, and compactness in terms of the real number system.
Prereqs: Sophomore standing and removal of all entrance deficiencies in mathematics.
MATH 394 is not intended for students who are required to take calculus. MATH 394 may be repeated if the subtitles differ. See the Schedule of Classes each term for the specific sections and subtitles offered.
Topics course for students in academic fields not requiring calculus. Emphasis on understanding and mathematical thinking rather than mechanical skills. Topic varies.
Not open to MA or MS students in Mathematics. This course is for students seeking a mathematics major under the Education Option and for students in CEHS who are seeking their secondary mathematics teaching certificate.
This course is designed around a series of projects in which students create mathematical models to examine the mathematics underlying several socially-relevant questions.
Elementary group theory, including cyclic, dihedral, and permutation groups; subgroups, cosets, normality, and quotient groups; fundamental isomorphism theorems; the theorems of Cayley, Lagrange, and Cauchy; and if time allows, Sylow's theorems.
Derivation of the heat, wave, and potential equations; separation of variables method of solution; solutions of boundary value problems by use of Fourier series, Fourier transforms, eigenfunction expansions with emphasis on the Bessel and Legendre functions; interpretations of solutions in various physical settings.
Discrete and continuous models in ecology: population models, predation, food webs, the spread of infectious diseases, and life histories. Elementary biochemical reaction kinetics; random processes in nature. Use of software for computation and graphics.
Derivation, analysis, and interpretation of mathematical models for problems in the physical and applied sciences. Scaling and dimensional analysis. Asymptotics, including regular and singular perturbation methods and asymptotic expansion of integrals. Calculus of variations.
Properties of stochastic processes and solutions of stochastic differential equations as a means of understanding modern financial instruments. Derivation and modeling of financial instruments, advanced financial models, advanced stochastic processes, partial differential equations, and numerical methods from a probabilistic point of view. | 677.169 | 1 |
Pages
3 Sep 2014
Best Recommended Books For IIT JEE Preparation
A not too bad book is one that
incorporates everything that streamlines the learning and cognizance process.
The top victories in Physics, Chemistry and Mathematics subjects contains
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area, example papers and fake test/movement questions at the end of the book.
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JEE Preparation By Subjects And Topics.
Mathematics:
1. First you must be able to solve
problems from NCERT class XI and class XII books.
2. For best concepts you can purchase
'IIT Mathematics' by M.L.Khanna.
3. For the most abundant supply of
difficult question topic-wise and their solutions, you may buy 'New Pattern
IITJEE Mathematics' of Arihant Publications by Dr. S.K.Goyal.
4. For trigonometry and coordinate
geometry the books by S.L.Loney are useful.
5. Tata Mcgraw Hills (TMH) books
7. Vector Algebra 19th Revised
Edition ( latest ) by Shanti Narayan
8. Problem in calculus of one
Variable For JEE Main & Advanced 2015 by I.E Maron
These Books Helps Students To Guide
What Books They Should Follow For Subjects And Major Topics For IIT JEE
Preparation.All These Books Are Authorized For IIT JEE Preparation Covering Almost Topics And Questions. | 677.169 | 1 |
Mathematics Department
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Student Colloquia
Most Tuesday afternoons during the academic year, the Mathematics students host a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels. Refreshments are served before the talks. | 677.169 | 1 |
A First direction in Numerical Methods is designed for college students and researchers who search sensible wisdom of contemporary suggestions in medical computing. keeping off encyclopedic and seriously theoretical exposition, the e-book presents an in-depth remedy of primary concerns and techniques, the explanations in the back of the luck and failure of numerical software program, and clean and easy-to-follow ways and methods.
The authors specialize in present equipment, concerns and software program whereas supplying a accomplished theoretical beginning, allowing those that have to observe the ideas to effectively layout options to nonstandard difficulties. The publication additionally illustrates algorithms utilizing the programming atmosphere of MATLAB(r), with the expectancy that the reader will progressively turn into knowledgeable in it whereas studying the fabric lined within the e-book. a number of workouts are supplied inside of every one bankruptcy besides evaluate questions geared toward self-testing.
The e-book takes an algorithmic method, concentrating on suggestions that experience a excessive point of applicability to engineering, laptop technological know-how, and business mathematics.
Audience:A First direction in Numerical Methods is aimed toward undergraduate and starting graduate scholars. it will possibly even be applicable for researchers whose major distinctiveness isn't medical computing and who're attracted to studying the elemental suggestions of the field some time past type of genuine research inequalities that are additionally mentioned in complete information.
Written through specialists in either arithmetic and biology, Algebraic and Discrete Mathematical tools for contemporary Biology deals a bridge among math and biology, delivering a framework for simulating, reading, predicting, and modulating the habit of advanced organic structures. each one bankruptcy starts off with a query from glossy biology, by way of the outline of yes mathematical equipment and conception acceptable within the seek of solutions.
Additional info for A First Course in Numerical Methods (Computational Science and Engineering)
Sample text
9. This is borne out by the value of the relative error. Let us denote the floating point representation mapping by x → fl(x), and suppose rounding is used. Then the quantity η in the formula on the current page is fundamental as it expresses a bound on the relative error when we represent a number in our prototype floating point system. 1 the rounding unit η. Furthermore, the negative of its exponent, t − 1 (for the rounding case), is often referred to as the number of significant digits. Rounding unit.
T − 1, which eliminates any ambiguity. 2. Floating point systems 23 The number 0 cannot be represented in a normalized fashion. It and the limits ±∞ are represented as special combinations of bits, according to an agreed upon convention for the given floating point system. 5. Consider a (toy) decimal floating point system with t = 4, U = 1, and L = −2. 666 is precisely representable because it has four digits in its mantissa and L < e < U. 01. How many different numbers do we have? The first digit can take on 9 different values, the other three digits 10 values each (because they may be zero, too).
63 × 10−3 > η. 100337 Thus, guard digits must be used to produce exact rounding. 7. Generally, proper rounding yields fl(1 + α) = 1 for any number α that satisfies |α| ≤ η. 1102e-16, beta = 0. 3, we can now explain why the curve of the error is flat for the very, very small values of h. For such values, fl( f (x 0 + h)) = fl( f (x 0 )), so the approximation is precisely zero and the recorded values are those of fl( f (x 0 )), which is independent of h. 2. Floating point systems 25 Spacing of floating point numbers If you think of how a given floating point system represents the real line you'll find that it has a somewhat uneven nature. | 677.169 | 1 |
Saturday, November 24, 2007
What connections can you make between what you are learning in class and the real world? That is, can you think of any applications besides those that we may have done in class or you have seen in the textbook? Explain the connections.
What do you foresee yourself studying in college or when you graduate high school? How do you think math is related? Find information online to back up your thoughts.
How is the math that we are doing in class related to your interests outside of class? Find information online to back up your thoughts. | 677.169 | 1 |
$235.00
List price $285 Jones and Bartlett International Series in Advanced Mathematics Completely revised and update, the second edition of An Introduction to Analysis presents a concise and sharply focused introdution to the basic concepts of analysis from the development of the real numbers through uniform convergences of a sequence of functions, and includes supplementary material on the calculus of functions of several variables and differential equations. This student-friendly text maintains a cautious and deliberate pace, and examples and figures are used extensively to assist the reader in understanding the concepts and then applying them. Students will become actively engaged in learning process with a broad and comprehensive collection of problems found at the end of each section. | 677.169 | 1 |
Core-Plus Mathematics
Core-Plus Mathematics is a four-year integrated curriculum
that replaces the traditional Algebra-Geometry-Algebra 2-Precalculus sequence.
Each course features interwoven strands of algebra and functions, statistics
and probability, geometry and trigonometry, and discrete mathematics. The
curriculum emphasizes mathematical modeling and is designed to make mathematics
accessible to more students, while challenging the most able students. The
curriculum materials also include a suite of computer software called
CPMP-Tools that provide powerful aids to learning mathematics and solving
mathematical problems. This use of technology permits the curriculum and
instruction to emphasize multiple representations and to focus on goals in
which mathematical thinking and problem solving are central. The materials
promote active learning and teaching centered around collaborative
investigations of problem situations.
Core-Plus Mathematics was developed by the Core-Plus Mathematics Project (CPMP)
at Western Michigan University. You can learn more here: | 677.169 | 1 |
Reviews of COMAP's Mathematics: Modeling Our World (MMOW /
ARISE)
Basic Information and Introduction
Mathematics: Modeling Our World is a Grades 9-12 curriculum developed
by the Consortium for Mathematics and its Applications (COMAP) in an
NSF-funded project Applications Reform in Secondary Education (ARISE).
The project was led by Solomon Garfunkel, Landy Godbold, and Henry
Pollak. The main COMAP web site is and within that site
the Mathematics: Modeling Our World curriculum is introduced here.
According to the description in the MMOW web pages, the curriculum
presents mathematics the way we use it in real-life. As in real-life,
MMOW's problems do not necessarily have perfect solutions, and MMOW
strengthens the students' ability to solve problems by strategic
thinking, trial and error and/or process of elimination, using
calculators and computers, and working cooperatively. MMOW is
designed to cover all the 1989 NCTM Standards at each level of the
curriculum.
The MMOW web pages further descibe the curriculum through these key
features: It is a Core curriculum, suitable for all students; it is an
integrated curriculum, in which any unit may involve a mix of
geometry, algebra, trigonometry, probability, or precalculus; and it
is a context-driven curriculum, in which units are arranged by context
rather than by mathematical topic. | 677.169 | 1 |
Differential Geometry of Curves and Surfaces (Hardback)
"Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one-and-two-dimensional objects in Euclidean spaces." — From the back cover
$35.99
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Condition
LIKE NEW: This book is in excellent, like-new condition.
Description
From the back cover: "The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems explore relationships between local and global properties.
"A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena."If you take five squares of the same size and join them in every possible way, you get 12 different shapes. These shapes are called 'Pentominoes' and they form the basis of an enormous range of interesting puzzles and investigations." — Inside FrontNot only does this textbook introduce your student to Geometry, but it also delves into using the LOGO and BASIC computer programming languages, graphing calculators, and spreadsheets. Examples of SAT and ACT problems are also included, as well as connections to other sciences, history, and art.In this remarkable volume, the author indeed fulfills the promise he makes in his preface: '—that each topic has been extricated from the mass of material in which it is usually found and given as elementary and full a treatment as reasonably possible.'" — From the back cover | 677.169 | 1 |
Calculus Test Prep
Introduction to calculus test preparation :
The common definition of Calculus is study of limits, derivatives, integrals, and infinite
series. Limits are simple mathematical tool in calculus. Calculus can be used to solve differentiation of any arbitrary equation and the corresponding output result. In calculus, Let y= f(x) be a
given continuous function. Then, y depends upon the x value and it changes with a change in the value of x. We can use the word increment to denote a small change of x and y. | 677.169 | 1 |
3.6.1 What do you need to know before you enroll in a math class?
Be sure you have memorized the addition and multiplication facts through 12 and be able to recall them quickly. Study them before taking the placement test or enrolling in a class. These are basic skills you will be expected to know without the use of a calculator.
For example, you should be able to find two numbers that multiply to be 36 and add up to be 15, in just a few seconds. If you need to brush up on these skills, try the link below.
Instructions: On the opening page choose the operation (addition, multiplication, . . .) from the list. In the second window enter your number in the white box and press the enter key to move on to the next problem.
3.6.2 Why is studying math different from other subjects?
Studying math is different than studying for other classes because math is skill-based, sequential, and cumulative. Studying math is more like becoming proficient at a sport or learning to play a musical instrument. It requires dedicated practice to be successful!
Skill-based means you can be shown the various techniques to work the problems but you will not know how to solve the problems until you practice and solve them on your own. So, understanding how your teacher works a problem in class is only the first step in learning. You must go home and practice until you can work the problems on your own without any assistance. Using examples to practice course material is great. However, you haven't learned the material if you still need them. It isn't so much that math is difficult, as it is that it requires dedicated practice throughout the course. Math problems always look just a bit different, so you need experience to be able to recognize which points are significant. Set time aside for studying math every day and use it.
Math is sequential, starting with basic skills and continuously building on to it. The new skill for today is a basic tool for the lesson tomorrow. So you must keep up with the work and don't skip sections. When you fail to grasp a concept, it hampers your ability to solve the problems in future lessons. If life gets complicated and you get behind, start where you left off and get caught up ASAP! Falling behind is frequently the death blow for success.
Cumulative means: just as the lessons build on themselves, so do the classes. In college algebra you will use the skills you learned in all the previous math classes you have taken. So, it is important this knowledge is in long term memory.
3.6.3 Why is math required for a college degree?
You may not see the purpose for learning mathematics for your particular degree. Students frequently make comments such as, "I am never going to need this for my job." So, why is math required for all college degrees? The answer in one word is: conditioning. Studying mathematics conditions your mind to think critically.
Everyone, no matter their job title, will have to problem solve at some point. The skills of organizing data, deciphering important facts from non-important ones, and developing a plan of action are all skills used in life.
You may still argue the point that specific math skills, such as factoring, are not going to be used in your particular job. So, why should you have to learn those skills? Consider the following example: You are on a basketball team. You go to practice. Your coach tells you to run the stairs. Why would the coach tell you to do that? You are not going to be running stairs during the basketball games, right? The answer is: conditioning.
Something else you may have not considered is that you may not keep the same job/career forever. So, the skills you need now may not satisfy the skills you need later.
3.6.4 Math Tutoring on TCC Campuses
Math Tutoring is available at all TCC campuses
Metro Campus - Math & Science Lab
Location: MC Academic Building, room 529 - Math & Science Lab
Phone: (918) 595-7000 ext. 6011. The Math Lab cannot be called directly from off-campus phone numbers. Call (918) 595-7000 and ask to be connected to the lab.
Hours: Monday - Thursday 8 a.m. to 9 p.m. Friday, 8 a.m. to 5 p.m.
Northeast Campus - Math Tutoring
Location: The FACET Center is located in A-1 of the Enterprise building. Math tutors provide one-on-one tutoring for all mathematics courses from Basic Math through Differential Equations. Students may also seek help in math-related work for chemistry and physics as well as other science and engineering courses.
3.6.5 How to be successful in math.
When you go to class:
- The brain can only absorb so much new information at a time. So, before class, read, or at least skim, the sections in the textbook to be covered during class, in order to extend your learning time.
- Turn off your phone during class.
- Stay focused, take notes, and anticipate the next step.
- Contribute to a learning environment: ask questions.
- Don't get distracted, or be a distraction for others during class.
- Use all the class time to cut back on your study time; don't leave early.
Attitude:
- Keep a positive attitude. It will help you concentrate and retain information.
- If you have had a bad experience with math in the past, let it go. Each semester is a new experience.
- Many students do better in math as they get older.
- Remember your IQ is flexible, just like a muscle. Exercise it to make it stronger.
- Your ability to do well in a math class is greatly dependent on your study habits!
- Do not merely try to pass the class; try to excel! You might just surprise yourself!
When to study:
Research has shown:
- Study as soon after class as possible.
- Four one-hour study periods on different days will be more productive than one 6 hour study period.
- If you study for more than hour, take a break every 50 minutes.
- Study every day and if possible at the same time of day.
- Study your hardest subject first.
Textbook strategies:
- Bring your textbook with you to class! Pay attention to any textbook references that your instructor makes.
- Read the examples provided in the textbook, along with examples your instructor showed you in class. These will be great resources when doing your homework!
- Many times, the textbook will refer you to a specific example before working a set of problems. Go to that example and read it!
- If your instructor assigns homework outside of the textbook, use your textbook as an extra study guide. Many times there are cumulative reviews that will help you when studying for an exam.
- The back of the book may contain answers to certain problems in order for you to check your work.
- Pay close attention to highlighted/boxed information. These areas usually cover important rules or tips for completing the problems.
Notebooks:
- Get a spiral notebook just for math.
- Keep the syllabus and handouts in the notebook.
- Take notes during class and when reviewing the textbook.
- Review your class notes as soon after class as possible and add more information when necessary.
- If you don't understand something as you review the notes, ask your instructor to clarify it for you. Do this before the next class session!!!
- Put notes in the margin over any material you want to ask about in class.
- Keep it neat, in order, and use proper notation.
- Work your homework problems in the notebook, too. Label the assignment, so you will know with which section it corresponds.
- Write down every step!
Homework skills:
- Homework is where you practice and hone your skills for the various techniques.
- It is important to do all the homework and in a timely manner, even if it is not worth points.
- Start by working the examples in the textbook. Then, if you have problems, you can check all the steps.
- Work the problems on paper!!! Whether you are assigned homework from the textbook, or from the computer, work the problems on paper. Then, if necessary, type the answers into the computer.
- Label the assignment name at the top of the paper, and label each problem with its specific number.
- Write all the steps and use proper notation.
- Work all the problems in order and leave a space if you need help with it. If you do not know how to complete a particular problem, then refer back to your notes and textbook examples. If they don't help, go see your instructor….and bring your work with you!
- After completing the homework, if you do not feel confident applying the various techniques, work more problems!
- When finished with a homework assignment, you should be able to do any of the problems, again, without any help from examples or your instructor.
- Math is a skill which REQUIRES practice.
- Practice also helps you identify which differences in notation are significant.
- Allow time ON-CAMPUS to get help from your instructor and/or the math lab, if you need it.
- Memory is the lowest form of learning; use it to make the higher forms of learning easier.
- Be sure to use long term memory since you will need this information in all your math classes.
Math software:
Many of TCC's math classes use online software which is user friendly. This can initially be intimidating, but feel free to open the software and look around. Click on all the buttons. You can't hurt anything.
- Try the "Take a tour" button, if available.
- Math software frequently has videos, animations, and other aids that simulate having a teacher 24/7!
- An online textbook may include "You Try It Buttons" and other multimedia aids.
- Online assignments usually provide you with instant feedback and frequently have links for additional help.
- Also, some online assignments may have the advantage of multiple attempts, which is a good way to raise your grade. So, be sure to check the syllabus for this type of information.
- You may be able to print some assignments and work them on paper before entering the answers online. Ask your instructor if this feature is available, and pay attention to instructions, if available.
- There are MANY computer labs for you to access the necessary homework. Be sure you know the location of all computer labs on your campus.
Basically the software is a tool like any other to help you be successful. Even if the software is unfamiliar, don't be discouraged. Just jump in! You will adjust to it quickly, and you will find it very helpful! Remember, your instructor is available to answer questions or assist you in getting started.
Test taking skills:
- Studying for a test starts the first day of class as you master the techniques and build your knowledge base.
- Cramming for a math test will not give you a desirable outcome since this is short term memory.
- Tests are more difficult than homework, since you must decide which techniques to apply to solve each problem. This requires a greater knowledge base and analytical skills. So, after each class session, strive to understand how the new topics relate to ones previously discussed. How are the similar? How are the different?
- It is important to review the material and integrate the different sections covered by the test. Strive to see the "big picture". Why do certain rules apply to some instances, and other rules apply to different situations?
- The night before a test, get a good night's rest to keep careless mistakes to a minimum.
- Eat before the test; research has shown this helps performance.
- Go to the bathroom before going to the classroom, so you won't be distracted.
- Show up at least 5 minutes early, so you are not rushed.
- Bring pencils and other allowed materials.
- When you are first given the test, write down formulas and procedural steps before you start work on the problems.
- Work out the problems using all the steps and proper notation.
- Skip problems with which you are not comfortable and come back to them.
- When you are finished, if you have time, go back and look at your answers. Be sure you answered every question, have simplified the answers, wrote the answer in the appropriate format, and check your answers to be sure they are reasonable.
- When you get your graded test back, be sure to look at all the problems you missed. Learn from your mistakes! Ask for help, if needed.
Math anxiety:
A "cure" for math anxiety may not exist. However, there are many things you can do to manage/minimize it. Most of the time, math anxiety develops due to repeated failure on math assignments and/or exams. You need to take steps to stop failure from occurring in the first place.
- Make sure you completely and thoroughly understand every single topic covered in class. If you don't, go see your instructor. Work on anything confusing, until you reach the point where you understand it.
- Be positive! Just because you may have misunderstood math in the past, does not mean you will always do that. YOU CAN LEARN IT!!!
- Before taking an exam, make your own practice exam. Write out several problems on blank paper. Include the directions with each problem. Put away all books, notes, etc. Put yourself in the same situation that you will be in during the exam. Then, take your exam. If you do not know what the directions are asking, skip the problem. Work all the ones you can. Then, get your book and notes back out. Study the problems you were unable to complete. Make sure you understand what the directions are asking.
- If you are worried because you think you will run out of time on an exam, then discuss this with your instructor. Ask how many questions will be on the exam. Determine how much time you will be able to devote to each problem. Then, study and practice until you can complete the problems in that amount of time.
Talk to your instructor:
- Your instructor is invested in your success. Do not hesitate to ask them questions, email them for help, or visit them during office hours.
- You may also go to the TCC Math Lab areas on any of the four campuses for additional help.
- Form a study group.
Consecutive classes:
- Once you start taking math classes, continue until you have completed your whole sequence.
- It is easier to continue while the previous course is still fresh in your mind.
- Skipping semesters other than summer creates additional refresher work for you.
3.6.6 Helpful and Entertaining Links
Practicing Number Facts: Choose the number of problems and the time limit. Then drag the numbers in the empty spaces to make the equation true. | 677.169 | 1 |
Math Corner Why do we need a math page on our wiki?Unfortunately, many times this class will feel like a second math class. That's because the ways in which we represent a chemical change will often be expressed quantitatively. Understanding how to solve algebraic equations, learning how towork with numbers written in scientific notation, expressing logarithms, and the graphing/analysis of data, will be vital skills that you will be required to use this year.
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Forgot your calculator? Then link to Creative Chemistry's on-line calculator and calculator tutorial. Be careful, you will still need to be able to enter exponents using the TI-83/TI-84 for the Regents exam.
Calculator TutorialHosted by Macon State College, this site provides detailed directions on how to use the TI-83/TI-84 calculators.
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"Lovett (Wheaton College) takes readers through the variegated landscape of algebra, from elementary modular arithmetic through groups, semigroups, and monoids, past rings and fields and group actions, beyond modules and algebras, to Galois theory, multivariable polynomial rings, and Gröbner bases." Choice Reviewed: Recommended
This text seeks to generate interest in abstract algebra by introducing each new structure and topic via a real-world application. The down-to-earth presentation is accessible to a readership with no prior knowledge of abstract algebra. Students are led to algebraic concepts and questions in a natural way through their everyday experiences. Applications include: Identification numbers and modular arithmetic (linear) error-correcting codes, including cyclic codes ruler and compass constructions cryptography symmetry of patterns in the real plane Abstract Algebra: Structure and Application is suitable as a text for a first course on abstract algebra whose main purpose is to generate interest in the subject or as a supplementary text for more advanced courses. The material paves the way to subsequent courses that further develop the theory of abstract algebra and will appeal to students of mathematics, mathematics education, computer science, and engineering interested in applications of algebraic concepts.
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This session deals with introducing the topic 'trigonometry' to anyone who wishes to learn about it in a very simple way. It does so by touching upon various dimensions within trigonometry. So it is advised to view this session before exploring rest of the sub topics. | 677.169 | 1 |
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Welcome to MATH 370
Welcome to Math 370: Functions & Modeling! We look forward to working with you to deepen and broaden your mathematics content knowledge, along with helping you gain familiarity with concepts needed to teach secondary mathematics at various levels. This course is designed to promote methods of discovery, problem solving, collaboration, and presentation of results. Ultimately, the goal is to think critically and creatively with an eye on the interconnections between algebra, analytic geometry, statistics, trigonometry, matrices, and calculus.
Toward this end, the course consists of five instructional Modules. For ease of access and use, materials for each are organized both by Module and by the date we use the materials in class. Please take some time to familiarize yourself with the course organization so we'll be ready to hit the ground running once class begins!
Feel free to contact us if you have any questions and/or concerns. We're here to make this semester as productive and valuable as possible! | 677.169 | 1 |
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to be offered odd years .
PREREQUISITE: A grade of C or better in M 234 or M 242, and junior standing.
Develop algebraic knowledge necessary to teach standards-based middle school mathematics. Investigate the underlying conceptual structure of topics in algebra and number appropriate to middle school. Explore the use of manipulative materials and technologies, and discuss related pedagogical issues and national standards. | 677.169 | 1 |
A Collection of Web Sites with Mathematical Content
The following is a list of web sites with mathematical content. The list is not
intended to be exhaustive. Rather, it is designed to give a feel for the range
of materials available. You certainly won't have time to look at all the material
on all the sites, but it will give you an idea of what is out there -- and what
you might be able to use as part of your project.
,
the general collection of MathDL. This is just now starting to put up content.
Much more will come soon.
Duke Material
·The Duke Connected Curriculum
Project (CCP): Interactive materials for courses from precalculus to
linear algebra, differential equations, and engineering mathematics. Most modules
have discussion and instructions in web pages with downloadable computer algebra
system worksheets for student exploration and reports. The following modules
have filled-in Maple worksheets available temporarily. Click on the workshop
icon to download these. (This will enable you to see the intent of the module
without having to work through the Maple.)
·The
Post CALC Project: These materials are designed for high school students
who have finished a yearlong course in calculus, but still have time left in
their high school career. The format is similar to the CCP materials, but these
modules are considerably longer.
3.MERLOT: This extensive project,
part of the emerging National Science Digital Library (NSDL), contains materials
that range across many disciplines besides mathematics.
4.iLumina:
Another collection in the NSDL. This site has extensive metadata available on
their entries. The material available covers biology, chemistry, computer science,
mathematics, and physics. Use Internet Explorer to search this site.
5.NCTM Illuminations
site: These materials are designed to illuminate the NCTM Standards. Click
on Interactive Mathlets. Check out the Car and Vector applet.
6.Virtual Laboratories in Probability
and Statistics: This site was created by Kyle Siegrist at the University
of Alabama at Huntsville. For a quick look, scroll down to the bottom of the
homepage and click on Applets. Check out the Interactive Histogram applet
and the Dice Experiment Applet.
7.Demos with Positive
Impact: This site was created by Dave Hill at Temple and Lila Roberts
at Georgia Southern. It is a collection of demos that use various technologies
and can be used for a variety of courses.
8.Interactive Mathematics:This site
is at Utah State.In the 9-12 Geometry section, check out The Pythagorean Theorem
and the Platonic solids
9.Math Forum:One of the oldest
web sites featuring mathematics, this site focuses on materials and services
for K-12.
The following sites feature a variety of applets. While we will not discuss
the creation of applets in this workshop, these collections may give you some
ideas on materials that might be usable in your project.
13.Joe Yanik's applets:
Joe is at Emporia State University. [As of 7/16, this site is down temporarily.
Check back later.] | 677.169 | 1 |
Currently out of print
CORD Algebra 2, Learning in Context
Interest Level : 10-12
Copyright 2008
By developing CORD Algebra 2: Learning in Context, CORD has
expanded its ground-breaking series of contextual-based math textbooks. More
importantly, teachers are provided a tool to help the student who would traditionally struggle
with math succeed and reach their full potential.
By combining real-world applications and lab activities with the mathematics being taught,
students receive a more meaningful education and are better prepared for state testing…
and life.
Common Core Standard Supplements
Makes All CORD textbooks Common Core compliant
Available on-line
Lesson Plans Online and CD-ROM
Fully editable for Word and PDF formats for specific classroom need
Reproducible and FREE for first edition.
Unique Features and Benefits
Engaging lessons address different learning styles to help ALL students learn algebra 2
Uses hands-on lessons to improve math content comprehension
Students see math at work in five key employment areas: Business and Marketing, Health Occupations, Industrial Technology, Family and Consumer Science and Agriculture/Agribusiness
The ideal learning tool for STEM Academy courses
Lesson assessments allow students to communicate their math knowledge through writing and discussion
Takes math from abstract concepts to concrete applications to which students can better relate.
Workplace Applications, real-world examples, labs and activities fit perfectly with the Common Core Standards mission statement of: " The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers."
Correlates to current Common Core, state and NCTM standards (see link below)
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CORD Communications strives to produce error-free materials. However, mistakes do happen. If you find errors in the textbook, please click here to tell us which book, page number and problem number. Provide a brief description of the error. We will look into the error and post any corrections needed to the website. | 677.169 | 1 |
United States of America. No part of this pub lication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
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SH SH 7 5 4 3 2 1 0
Preface
Finite mathematics has in recent years become an integral part of the mathematical background necessary for such diverse fields as biology, chemistry, economics, psychology, sociology, education, political science, business and engineering. This book, in presenting the more essential material, is designed for use as a supplement to all current standard texts or as a textbook for a formal course in finite mathematics. The material has been divided into twenty-five chapters, since the logical arrangement is thereby not disturbed while the usefulness as a text and reference book on any of several levels is greatly increased. The basic areas covered are: logic; set theory; vectors and matrices; counting - permutations, combinations and partitions; probability and Markov chains; linear programming and game theory. The area on vectors and matrices includes a chapter on systems of linear equations; it is in this context that the important concept of linear dependence and independence is introduced. The area on linear programming and game theory includes a chapter on inequalities and one on points, lines and hyperplanes; this is done to make this section self-contained. Furthermore, the simplex method is given for solving linear programming problems with more than two unknowns and for solving relatively large games. In using the book it is possible to change the order of many later chapters or even to omit certain chapters without difficulty and without loss of continuity. Each chapter begins with a clear statement of pertinent definitions, principles and theorems together with illustrative and other descriptive material. This is followed by graded sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the theory, bring into sharp focus those fine points without which the student continually feels himself on unsafe ground, and provide the repetition of basic principles so vital to effective learning. Proofs of theorems and derivations of basic results are included among the solved problems. The supplementary problems serve as a complete review of the material in each chapter. More material has been included here than can be covered in most first courses. This has been done to make the book more flexible, to provide a more useful book of reference and to stimulate further interest in the topics. I wish to thank many of my friends and colleagues, especially P. Hagis, J. Landman, B. Lide and T. Slook, for invaluable suggestions and critical review of the manuscript. I also wish to express my gratitude to the staff of the Schaum Publishing Company, particularly to N. Monti, for their unfailing cooperation.
Linear equation in two unknowns. Two linear equations in two unknowns. General linear equation. General system of linear equations. Homogeneous systems of linear equations.
Chapter
12
DETERMINANTS
OF ORDER TWO AND THREE
,
127
Introduction. Determinants of order one. Determinants of order two. Linear equations in two unknowns and determinants. Determinants of order three. Linear equations in three unknowns and determinants. Invertible matrices. Invertible matrices and determinants.
(iii) Where are you going? (iv) Put the homework on the blackboard. implicitly.1:
of a statement
(i)
r. the conjunction statement is true.
p q
1\
Any two statements can be combined by the word "and" to form a compound statement called the conjunction of the original statements. Symbolically. q. that is. Such composite statements are called comExample 2.
Consider the following expressions: Paris is in England. otherwise.
of two statements
is true only in the case when each sub-
1
. read "p and q". but not both. Statements will usually be denoted by the letters The truth or falsity
Example 1..
and let q be "The sun is shining". a compound statement sub statements "He is intelligent" and "He studies every night".
pl\q
denotes the conjunction
Example 3.
is false. The expressions (iii) and (iv) are not statements since neither is either true or false.
COMPOUND
STATEMENTS
pound statements.Chapter 1
Propositions and Truth Tables
STATEMENTS A statement (or verbal assertion) is any collection of symbols (or sounds) which is either true or false.
p
1\
The truth [TIl
value of the compound
1\
statement
q
q
satisfies
p
1\
the following
q
property:
If p is true and q is true. is called its truth value..
In other words. We begin with a study of some of these connectives.
Some statements are composite.
Let p be "It is raining" Then p
1\
q denotes the statement
"It is raining and the sun is shining".1:
of the statements
p and q.1: "Roses are red and violets are blue" is a compound statement "Roses are red" and "Violets are blue". then p
is true. with substatements
Example 2. composed of sub statements and various logical connectives which we discuss subsequently.
p.
with
The fundamental property of a compound statement is that its truth value is completely determined by the truth values of its substatements together with the way in which they are connected to form the compound statement. (ii) 2
+2 =
4
The expressions (i) and (11) are statements.2:
"He is intelligent or studies every night" is. CONJUNCTION. the first is false and the second is true. .
satisfies the following property:
If p is true or q is true or both p and q are true. pvq denotes the disjunction of the statements p and q and is read "p or q". which we regard as defining p v q:
p q pvq
T T F F Example 4. the disjunction of two statements is false only when both substatements are false. Each of the other statements is false since at least one of its substatements is false. 1
Consider the following four statements: (i) (ii) Paris is in France and Paris is in France and 2+2 = 4. otherwise p v q
Accordingly. 2 + 2 = 5. Each of the other statements least one of its sub statements is true. The property lTzJ can also be written in the form of the table below. By property [T2].1: Let p be "Marc studied French at the university". Symbolically. is true since at
. (iv) Paris is in England and
By property [Ttl.
pvq
Any two statements can be combined by the word "or" (in the sense of "and/or") to form a new statement which is called the disjunction of the original two statements.
(iii) Paris is in England and 2 +2 = 4.2:
PROPOSITIONS
AND TRUTH
TABLES
[CHAP.
Example 4. only the first statement is true. (iv) Paris is in England Or 2 + 2 = 5.
(ii)
(iii) Paris is in England or 2 + 2 = 4. DISJUNCTION. We regard this table as defining precisely the truth value of the compound statement p (\ q as a function of the truth values of p and of q. pvq
The truth value of the compound statement [T2] is false. then p v q is true. The other lines have analogous meaning. only (iv) is false. 2+ 2 = 5.2
Example 3.2:
T F T F
T T T F
Consider the following four statements:
(i)
Paris is in France or Paris is in France or
2 + 2 = 4. the first line is a short way of saying that if p is true and q is true then p (\ q is true. 2+ 2 = 5.
A convenient way to state property [T1J is by means of a table as follows:
p
q T F T F
P(\q T F F F
T T F F
Here. and let q be "Marc lived in France". Then p v q is the statement "Marc studied French at the university or (Marc) lived in France".
Symbolically.
PROPOSITIONS
AND TRUTH TABLES
By repetitive use of the logical connectives (1\.q) is constructed as follows:
. Now the truth value of a proposition depends exclusively upon the truth values of its variables. " before p or. .1.2: Consider the following statements: (i) (ii)
2
+2 =
5
It is false that
2+2
5. Observe that (i) is false and (ii) and (iii). For example. each its negation. . as above. each its negation. v. for example. denotes the negation of p (read "not p").
Example 5. we can construct compound statements that are more involved. Example 5. Then (ii) and (iii) are each the negation of (i). q..
Consider the statements in Example 5. Sometimes it is used in the sense of "p or q or both". A simple concise way to show this relationship is through a truth table. exactly one of the two alternatives occurs.. In the case where the substatements p. and sometimes it is used in the sense of "p or q but not both". q. called the negation of p. are false.. called the exclusive disjunction.
Given any statement p. at least one of the two alternates occurs._. ) are variables. .and others discussed subsequently). by inserting in p the word "not". "or" shall be used in the former sense. i. This discussion points out the precision we gain from our symbolic language: p v q is defined by its truth table and always means "p and/or q".
-p
NEGATION. . are true..2..
The truth value of the negation of a statement [T3]
satisfies the following property: is true:
If p is true.1: Consider the following three statements: (i) (ii) Paris is in France. can be formed by writing "It is false that. The defining property [T31 of the connective can also be written in the form of a table:
Example 5. . if p is false. then -p
Thus the truth value of the negation of any statement is always the opposite of the truth value of the original statement.pis false..4:
Consider the statements in Example 5. It is false that Paris is in France. the sentence "He will go to Harvard or to Yale" uses "or" in the latter sense. we call the compound statement a proposition. that is. then .CHAP.
(iii) 2 + 2 "'" 5 Then (ii) and (iii) are each the negation of (i).
11
PROPOSITIONS
AND TRUTH
TABLES
3
Remark:
The English word "or" is commonly used in two distinct ways.. the truth value of a proposition is known once the truth values of its variables are known.
(iii) Paris is not in France.e. another statement.e.
Observe that (i) is true and (ii) and (iii). of a compound statement Pip. i.. The truth table.3: Example 5. of the proposition -t» 1\ . if possible. Unless otherwise stated.
2" rows are required. in general. Remark: The truth table of the above proposition consists precisely of the columns under the variables and the column under the proposition:
p T T F F q T F T F -(p
A
-q)
T F T
T
The other columns were merely used in the construction Another way to construct the above truth table for construct the following table:
p
of the truth is as follows. as above. and. 8 rows are necessary. the truth value at each step being determined from the previous stages by the definitions of the connectives A. and that there is a column under each variable or connective in the proposition. v.. the last step. q. for n variables. r«. which appears in the last column. and that there are enough rows in the table to allow for all possible combinations of T and F for these variables. 4 rows are necessary.
..
table.4
PROPOSITIONS
AND
TRUTH
TABLES
[CHAP. Finally we obtain the truth value of the proposition.) There is then a column for each "elementary" stage of the construction of the proposition. for 3 variables. First
-(p
q)
A
-q)
q T F T F Step
-
(p
A
-
T T F F
Observe that the proposition is written on the top row to the right of its variables. (For 2 variables. . i. 1
p T T F F
q T F T F
~q F T F T
P
A
=«
~(p
A
-q)
F T F F
T F T T
Observe that the first columns of the table are for the variables p. Truth values are then entered into the truth table in various steps as follows:
p
q T F T F
-
(p
A
-
q) T F T F 1
p T T F F Step
q T F T F
-
(p T T F F 1 (b)
A
F T F T 2
q) T F T F 1
T T F F Step
T T F F 1
(a)
p
q T F T
-
(p T T F F 1
(c)
A
F T
q)
p T T F F Step
q T F T F
T F T T 4
(p T T
A
F T F T 2
q)
T T
F T
T F T F 1
F T F F 3
T F T F 1
F
F Step
F
F 3
F
T 2
F
F 1
F
(d)
The truth table of the proposition then consists of the original columns under the variables and the last column entered into the table.e.
(iii) False. (5) -p 1\ -q. Hence: (i) True.
1. (6) It is not true that he is short or not handsome. v and ~ to read "and".4.
1. (6) It is not true that it is not raining.3.
. (3) It is false that he is short or handsome." Give a simple verbal sentence which describes each of the following statements: (1) -p. (1) He is tall and handsome.
=
8. (5) He is tall. (ii) False. Paris is in France or 2 + 1 = 6.e. (iii) False. (4) He is neither tall nor handsome.
OF COMPOUND STATEMENTS
Determine the truth value of each of the following statements.. "or" and "It is false that" or "not".
(4) It is raining or it is not cold. or he is short and handsome. (3) It is cold or it is raining. (2) p 1\ q. (i) 3 + 2 = 7 and 4 + 4 1 + 1 = 3. respectively. and then simplify the English sentence. (ii) True. (6) ." statements in symbolic form using p and q. the compound statement Hence: (i) False. the truth value of the negation of p is the opposite of the truth value of p. (3) p v q.1.2. 1]
PROPOSITIONS
AND TRUTH
TABLES
5
Solved Problems
STATEMENTS 1.
In each case.
Determine the truth value of each of the following statements.
(iii) 6 + 4
=
10 and
By property [Td. Let p be "It is cold" and let q be "It is raining. (i) (ii) (iii)
= 7. (5) It is not cold and it is not raining. i.
"p and q" is true only when p and q are both true. (ii) It is not true that London is in England.
Paris is in England or 3 + 4
"p or q" is false only when p and q are both false. translate 1\.)
(1) (2) p p
1\ 1\
Write each of the following
q
(3) (4)
~(~p ~p
1\
v q)
(5) (6)
p v (-p
-(-p
1\
q)
=«
=«
v ~q)
TRUTH VALUES
1.CHAP.
1. (1) It is not cold.
(Assume that "He is short" means "He is not tall". (ii) True.
Determine the truth value of each of the following statements.
By property [T2]. (i) It is not true that London is in France. ~p.5. the compound statement Hence: (i) True.
By property [T3]. (2) It is cold and raining.q. (2) He is tall but not handsome. (4) q v -p.
(ii) 2 + 1
=
3 and 7 + 2
=
9.
Let p be "He is tall" and let q be "He is handsome. London is in France or 5 + 2 = 3.
It is a statement of the type: If monkeys are human..
Now consider the truth table of the proposition
p q -p
""'p v q:
-p v q
T
T F
T
F T
F
F T
T
F T
F
F
T
T
Observe that the above truth table is identical to the truth is logically equivalent to the proposition .
The truth table of the conditional statement follows:
T T T F T F
F
T
T
F
Example 1. then the earth is flat.. 18
. by definition. (iv) is a true statement even though its substatements "Paris is in England" and "2 + 2 = 5" are false...
p~ q
3
Such
Many statements...p v q:
p ~ q == . particularly in mathematics.. We may regard p ~ q as an abbreviation for an oft-recurring statement.
By the property [T4]. the conditional statement "If p then q" is logically equivalent to the statement "Not p or q" which only involves the connectives v and . 2 + 2 = 5. then If Paris is in France. are of the form "If p then q". then
=
5. We emphasize that. only (ii) is a false statement.Chapter
Conditional Statements
CONDITIONAL. statements are called conditional statements and are denoted by
p~q
The conditional p ~ q can also be read: (i) (ii) [T41 The conditional
p p
implies only if
q q
(iii) 1) is sufficient for q (iv)
q
is necessary for
p.1:
F
T
2 + 2 = 4.
(iii) If Paris is in England.p
v
table of p ~ q.and thus was already a part of our language. the others are true.. then
2 + 2 = 4. 2+2
Consider the following statements: (i) (ii) If Paris is in France.
The truth value of p ~ q satisfies:
p~ q
is true except in the case that
p
is true and
q
is false. then (iv) If Paris is in England.
Hence p ~ q
q
In other words.
. the statements (i) and (iv) are true. In other words. the above truth table establishes Theorem 3.
-p
-q
-p
called respectively the converse. but then..
.
p -'> q and its contrapositive -q
-'>
-p are logically
. then p ~ q is true. Theorem 3. if p and q have opposite truth values. then p ~ q is false. ) is always true. By property [Tol. the composite proposition P(p. .. q.. ... by property [T5].e. ) and Q(p.q. simply..
P(p. 2 + 2 = 5..
p~ q
Another common statement Such statements. ) are logically equivalent if and only if they have the same truth table.2: A conditional statement equivalent. ) == Q(p.. . On the other hand. ) ~ Q(p. ) ~
Q(p. satisfies the following property: [To]
The truth value of the biconditional statement
p~ q
If p and q have the same truth value. )
CONDITIONAL
STATEMENTS
AND VARIATIONS
p
-'>
-'>
Consider the conditional proposition tions which contain p and q:
q
-'>
q
and the other simple conditional proposiand
-q
-'>
p.
Recall that propositions Ptp. inverse.1:
Ptp. q. and (ii) and (iii) are false..CHAP. q.
The truth table of the biconditional follows:
p q p~ q
T T F F Example 2. . is a tautology. .
. of these four propositions follow:
Conditional
p q p->q
and contmpositive
Converse
q->p
propositions..
(ii)
(iii) Paris is in England if and only if 2 + 2 = 4..q. "p iff q".
are called biconditional statements. q.
Contra positive
-q
The truth tables
Inverse
~p
+r:«
T
+>»
T F T T
T T
T
T
T T F T
F
T F
F
T T
T
F F
F
T
Observe first that a conditional statement and its converse or inverse are not logically equivalent. q.. denoted by
is of the form
p~q
"v
if and only if q" or. (iv) Paris is in England if and only if 2 + 2 = 5.. q.... i.1:
T F T F
T F F T
Consider the following statements:
(i)
Paris is in France if and only if Paris is in France if and only if
2 + 2 = 4. 3]
CONDITIONAL
STATEMENTS
19
BICONDITIONAL. ) if and only if the proposition
is a tautology.
Let p denote "It is cold" and let in symbolic form. p.2. "p is sufficient for q" or
"a
is necessary
for p". q --. p is false.p "Whenever it rains it is cold" is equivalent to "If it rains then it is cold". but q --.
Note that p --. A sufficient condition for it to be cold is that it rain.
(iv) Now the statement That is.
to "If it is cold then it does not
3.
(i)
q--. q is true.
Recall that "If p then q" is equivalent to "Not p or q".q
(iii) q--. Since the contra positive statement -q --.1. It never rains when it is cold. 3
Consider the following statements
p --. then A is equilateral.
If A is isosceles.3. the original conditional statement p --> q is also true. "If x is even then x2 is even". (v)
The statement "It never rains when it is cold" is equivalent rain".
then A is isosceles.
(i) If it is cold. (ii) If productivity increases.2: Prove:
(p --> q)
If x2 is odd then x is odd. Whenever it rains it is cold.20
Example 3.
Rewrite the following statements
without using the conditional. he wears a hat. +q.p
(ii) p--.
(i)
It is not cold or he wears a hat. then x == 2n where n is an integer.
We show that the contrapositive -q --> =».1:
CONDITIONAL
STATEMENTS
[CHAP. Productivity does not increase or wages rise.
(ii)
3.
Determine the truth table of (p
p T T q T
-->
q)
-->
(p
pAq
A
q).
(p
-->
p-->q T
q)
-->
(p
A
q)
T
T T
F
T
F
T T
F F
F
F F F
F F
. q can be read "p only if q".
Recall that p --. p:
about a triangle
A:
If A is equilateral. (i) (ii) (iii) (iv) (v)
It rains only if it is cold. p --. That is. is true.
q
denote "It rains". Let x be even. Hence x2 == (2n)(2n) == 2(2n2) is also even. then wages rise.
Solved Problems
CONDITIONAL
3.
Write the following statements
A necessary condition for it to be cold is that it rain. q: q --.-p is true. Example 3.
then he will not pass the test.
(ii) He swims if and only if the water is warm.
.
Write the negation of each statement in as simple a sentence as possible. hence the negation of (ii) is either of the following: He swims if and only if the water is not warm.
CONDITIONAL
3.14. (i) If it is cold.
~q)
(ii)
~(~lJ
p~q p~q
~
q).13. statement can be written as p --> (q 1\ ~r"). 3]
CONDITIONAL
STATEMENTS
23
-->
3.
Write the negation of each statement as simply as possible.
(i) By Problem 3.
-t»
-->
q) =: P
1\
r-
q.
3. (iii) It is necessary to have snow in order for Eric to ski. then he is poor. (iii) Note that
~(p --> ~q)
== P
1\ ~
~q
== P
1\
q. He does not swim if and only if the water is warm.
STATEMENTS
AND VARIATIONS of each statement. Hence the
If Marc does not study. then x is not positive.
hence the negation of (i) is
He studies and he will not pass the exam. then he wears a coat but no sweater. (iii) If it snows. then he studied". then he will go to college or to art school.12.
(i) Let p be "It is cold". (i i) The given statement is equivalent contra positive of (ii) is to "If Marc passes the test. ~(p ~ q) =' p ~ =« =0 ~p ~ q.12.
(ii)
The given statement
is of the form
~[p
--> (q v ~')I
p
-->
(q v ~'). he will pass the exam.
Hence the negation of (iii) is
It snows and he drives the car. (i)
If he studies. then he does not drive the car. q be "He wears a coat" and r be "He wears a sweater".
(iii)
~(~lJ
~q).
Simplify:
(i) (ii)
~(p ~
(i) ~(p ~ ~q).15. p
1\
But
V
==
~(q
r)
P
1\
~q
1\
+r
Thus the negation of (ii) is He studies and he does not go to college or to art school.
~(~p
~
q)
3. (ii) By Problem 3.CHAP. Now
~[p
-->
Then the given
(q
1\
~r)]
==
p
1\
~(q
1\
~r")
==
P
1\
t-«
V
r)
Hence the negation of (i) is
It is cold and he wears a sweater
or no coat. (iv) If x is less than zero.16.
Determine the contrapositive (i)
If John is a poet.
(i) The contra positive of p
-->
q is ~q
-->
+p. (ii) If he studies. then he is not a poet.
Hence the contra positive of (i) is
If John is not poor.
(ii) Only if Marc studies will he pass the test.
If it did not snow.. then
=
4.. .. -po
Hence the contra positive of (iv) is
If x is positive..
1+1 2 iff 3 + 4 == 5.. 5 + 5 = 10..
It is not true that
(iii) A necessary condition that (iv) It is not true that (v) If 3 < 5. The contrapositive of p ......
The contra positive of -p .
then -3
< -5. He is poor or else he is both rich and happy. then 3 + 3 == 5 or 2 + 1 = 3 and 1 + 1 == 2...q. Being rich is a sufficient condition to being happy. q. Assume "He is poor" is equivalent to -po
Determine the truth value of each statement.
3.-q is
==
q ..18.
Problems
Write each statement in symbolic form
Let p denote "He is rich" and let q denote "He is happy". then -3
1 + 1 = 5 iff 3+3=1. of p . of -q .
(vii) One is never happy when one is rich.-rp... q. Determine (i) (ii) the truth
3 is that
4+4
4. which is the inverse . q. (ix) (x) To be rich means the same as to be happy.. (i) (ii) If 5
< 3.-q .q is -q . Being rich is a necessary condition to being happy. 0) (ii) (iii) (iv) (v) (vi) If he is rich then he is unhappy.-p is . then Eric will not ski.
< -5.20.
value of each statement..17. then it snowed".. and the conditional and contrapositive are contrapositives of each other!
Supplementary
STATEMENTS 3... then x is not less than zero....p is -p . which is the original conditional The contrapositive proposition.....
q... 3.
It is not true that if
If 2 + 2 = 4.-q.. He is neither rich nor happy.19.. -p
Hence the contrapositive
==
q ..
Find and simplify: (i) Contrapositive of the contrapositive of p . p ... then it is not true that 3 + 3 == 7 iff 1 + 1
(iii) If 2 + 2 = 4... (ii) Contrapositive of the converse of p .
1+2
(vi) A sufficient condition that 3.... using p and q..-q is .-q
The converse of p .p.-p
(iii) The inverse of p .-q ...p..q is -r-p which is the converse of p
q.-p . (viii) He is poor only if he is happy..q is q .. the inverse and converse are contra positives of each other.. It is necessary to be poor in order to be happy. 2 + 2 == 4.24
(iii) The given statement of (iii) is (iv) The contrapositive
CONDITIONAL
STATEMENTS
[CHAP. To be poor is to be unhappy. (iii) Contrapositive of the inverse of p q. 1+2 = 3 is that 4+4 4. 3
is equivalent to "If Eric skis.
-r
of q ... Note..
In other words..
(i) (ii) The contrapositive of p .
....
(ii)
[q ~
(r--> -p)]
V
[(-q--> p) ~rl. then unemployment rises.
3. Find the truth table of each proposition: (i) [p /\ (-q--> p)]/\ -[(p ~ -q) --> (q v -p)].
(i) FTFF.22. Betty smokes Kent or Salem. Prove: (i) (p /\ q)
--> r
3. (iii) Contra positive of the converse of p --> -q.
3. (i) (ii)
p --> -q -p /\ -q
Problems
(vii) p --> -q (viii) -p --> q (ix) p ~ q (x) -p v (p /\ q)
(iii) q --> -p (iv) -p ~-q
(v)
p --> q
--> p
(vi) q
3.27. then he reads neither Life nor Time. (iii) Only if he does not tire will he win. (ii) FFFT (ii) FTFT (ii) TTTFTTFT the appropriate truth tables. (ii) Contrapositive of -p --> q. If he tires. then he does not have courage.
(i) T. 3. (i) (ii) (iii) (iv)
Construct
3.
Write the negation of each statement in as simple a sentence as possible.25. Mary speaks Spanish or French if and only if she speaks Italian.
--> q.
3.
(ii) p
--> (q --> r)
==
(p /\ -r)
CONDITIONAL AND VARIATIONS 3. then he is not a sailor. (i) If stock prices fall. If Marc is rich. Find: (i) Contra positive of p --> +q. (ii) F. (ii) T.23.24.
3. then both Eric and Audrey are happy.21. If it is not a rectangle. (iv) F.19. John reads Newsweek. Find the truth
table of each proposition:
(i)
t-» v
q) --> p. and Life or Time. 3]
CONDITIONAL
STATEMENTS
25
3. Mary speaks Spanish or French. (ii) (-q v p) ~ (q --> -pl.
==
(p --> r) v (q --> r).CHAP. Marc is rich and Eric or Audrey is unhappy.
(i) TTFF.27. (iii) T (i) (ii) (iii) (iv) (v) (vi) Stock prices fall and unemployment does not rise.23. (vi) T (i) F.
Hint.
If he is not strong. 3.18. then he will not win.
(vi) If John reads Newsweek TRUTH TABLES 3.24.
Answers to Supplementary
3.22.26.26. (ii) (iii) (iv) (v) He has blond hair if and only if he has blue eyes.
(iii) -p
(iv) p
--> q
.
then it is not a square.
(ii) q ~ (-q /\ pl.
If he does not win. Determine the contra positive of each statement.21.
Find the truth table of each proposition: (i) (p ~ -q) --> (-p /\ q). (ii) It is necessary to be strong in order to be a sailor. (iv) Converse of the contra positive of -p --> -q. and Camels. (v) F. (i) If he has courage he will win.
--> -q.
(i) TFTT.25. 3.
(i) q --> -p. (iii) F.
(ii) -rq
--> p. Betty smokes Kent or Salem only if she doesn't smoke Camels. He has blond hair but does not have blue eyes. but not Italian. 3. 3. (iv) It is sufficient for it to be a square in order to be a rectangle.
3.20.
then p implies r"
that is. 3 and 4.P2. ••• .2: is valid. Pn are true simultaneously if and only if the proposition P1/\P21\'" I\Pn is true.P2.». yields (has as a consequence) another proposition Q. If an argument is true it is called a valid argument.Q
4
called Such
Logical Implication
. otherwise the argument is false.Q is true if Q is true whenever all the premises P l' P2. Thus the argument P1. and p --> q is true in Cases 1.P For p --> q and q are both true in Case (line) 3 in the above truth this Case p is false. P2.Chapter
Arguments.
Thus an argument is a statement.
p q p-->q
T T F F
T
F
T F
T F T T
For p is true in Cases (lines) 1 and 2.1: The argument P1.1: The following argument is valid:
I-
p. if the proposition (P1I\P21\'" I\Pn) -> Q is a tautology.e.
A fundamental principle of logical reasoning
"If p implies q and q implies
Example 1. the following argument
p --> q. p --> q
q
(Law of Detachment)
The proof of this rule follows from the following truth table. but in
The following argument
p --> q. has a truth value.
The truth value of an argument is determined as follows:
[Ts]
An argument P1. q I. P2. Pn I. ••• . ••• .
ARGUMENTS An argument is an assertion that a given set of propositions P l' P 2' ••. i.>
26
. Theorem 4. P n' premises. the argument Example 1. •• '.. called the conclusion. if an argument is false it is called a fallacy. q -->
is valid:
I-
r
p -->
r
(Law of Syllogism)
This fact is verified by the following truth table which shows that the proposition [(p --> q) 1\ (q --> r)] --> (p r) is a tautology:
--.3:
states:
r.Pn I.
Example 1. an argument is denoted by Pl'P2.
Now the propositions P1.Q is valid if and only if Q is true whenever P11\ P2 1\ ••• 1\ P n is true or.Q is valid if and only if the proposition (P1I\P21\ •• 'I\PJ -> Q is a tautology.·· I. hence p and p --> q are true simultaneously in Case 1. We state this result formally. is a fallacy: table. . equivalently. Since in this Case q is true. •• "Pn I. P n are true.
then 5 does not divide 15.-p where p is "5 is a prime number" and q is "5 divides 15".4 this argument is a fallacy. . g. ) if
q pvq
We claim that p logically implies p v q.
S2: 7 is not less than 4.
For the argument is of the form p -? -q. ) is true. -p f-.. the argument P f-. Observe that p is true in Cases (lines) 1 and 2. and by Example 1. a tautology. . then the argument
Pt». then the opposite angles are equal. ) is true whenever Pip. )
f-.3: We claim that the following argument is valid:
SI:
If 5 is a prime number.
q.
LOGICAL IMPLICATION
[CHAP..5.
-p
f--
q
T F T F
F T F T
F T T T
F F T T
The argument is a fallacy since in Case (line) 4 of the adjacent truth table. q. )
is valid. p -? -q and -p are true but q is false. Furthermore.
S2: 5 divides 15..Q is valid if and only if the conditional statement p-? Q is always true.Q(p. Then the argument is of the form
p
-?
T T F F
-q. We remark that although the conclusion here is obviously a false statement..
LOGICAL IMPLICATION
Q(p.. We state this result formally. S: The opposite angles are not equal.. ) is said to logically «. the above argument does not constitute such a proof since the argument is a fallacy. q.
For the argument is of the form p-? q. and in these Cases pv q is also true. Example 2. .
p q -q P -?-q -p
We translate the argument into symbolic form. and conversely. In other words. where p is "Two sides of a triangle are equal" and q is "The opposite angles are equal". the argument as given is still valid.
The fact that the conclusion of the argument happens to be a true statement is irrelevant to the fact that the argument is a fallacy. q.
Example 3.. . S: 7 is a prime number. Let p be "7 is less than 4" and q be "7 is a prime number". . g.. ) is true. ) is true whenever P(p.4: Determine the validity of the following argument:
SI:
If 7 is less than 4. . S: 5 is not a prime number. ..
. q f-. For consider the truth tables of p and p v q in the adjacent table. Example 2... p logically implies pv q..1:
imply
a proposition
p
Q(p.
T T F F
T F
T T
T
F
T
F
Now if Q(p.-q.2:
ARGUMENTS. 4
We claim that the following argument is not valid: S 1: If two sides of a triangle are equal. S2: Two sides of a triangle are not equal.. i. It is because of the false premise SI that we can logically arrive at the false conclusion.e.
A proposition P(p.28
Example 2...
a. . Although the conclusion S does follow from S2 and axioms of Euclidean geometry.. then 7 is not a prime number. and we proved this argument is valid in Example 1.
But 6 is even. p p q -p -q -p --> -q
T T F F
T F T F
F F T T
F T F T
T T F T of the
r-
q
Now in the adjacent truth table.
. -p --> -q and p are both true in Case (line) 2. Therefore.8. then I do not like mathematics.6.
If I fail. and r be "I fail". then I will not fail mathematics.
Either I study or I fail.
Translate the argument into symbolic form. hence the argument is a fallacy. then I will study. then 5 is not prime. q v r
r. But I failed mathematics.
does not affect the fact that the argument
4.
4.
First translate the argument into symbolic form.
Test the validity of the following argument:
If I study. Hence the argument is a fallacy.
LOGICAL IMPLICATION
31
4." Then the argument is of the form
-p --> -q.
then I will study.
Test the validity of the following argument:
If 6 is not even.
If I do not play basketball. I played basketball. Let p be "6 is even" and let q be "5 is prime.
Test the validity of the following argument:
If I like mathematics. Then the given argument is of the form p To test the validity p --> q. Therefore 5 is prime. but in this Case q is false.
r p-->q
construct
qvr
truth
tables
of
the
propositions
r --> -p
T T T T F F F F
T T F F T T F F
T F T F T F T F
T T F F T T T T
T T T F T T T F
F F F F T T T T
F T F T T T T T
Recall that an argument is valid if the conclusion is true whenever the premises are true.r
-->
-p the
-p
of
the
q
argument. Now in Case (line) 1 of the above truth table.7. q v rand r --> -p :
p
-->
q be "I study"
q. 4]
ARGUMENTS. The fact that the conclusion is a true statement is a fallacy.
The argument can also be shown to be a fallacy by constructing the truth table proposition [(-p--> -q) I\pJ --> q and observing that the proposition is not a tautology.CHAP. Let p be "I like mathematics". the premises p --> q and q v r are both true but the conclusion r --> -p is false.
32
ARGUMENTS,
LOGICAL IMPLICATION
[CHAP. 4
and
First translate the argument into symbolic form. Let p be "I study", q be "I fail mathematics" j- be "I play basketball". Then the given argument is as follows:
p -->
=«. ~r
--> p,
q
I-
r
tables
<
r:
To test the validity -rq, =r:" », q and r':
of the argument,
p q
construct
p --> ~q
the truth ~r F T F T F T F T
of the given
propositions
r
T F T F T F T F
~q
r ..... p T T T T T F T F
T T T T F F F F
T T F F T T F F
F F T T F F T T
F F T T T T T T
Now the premises p --> ~q, ~r --> p and q are true simultaneously only in Case (line) 5, and in that case the conclusion r is also true; hence the argument is valid.
LOGICAL IMPLICATION 4.9. Show that
p
A
q
logically implies
table for
p
p ~ q.
-->
Construct
the truth
(p A q)
(p ~
q) :
q
T
pAq T
p~q
(p
A
q) --> (p ~ T
q)
T T F F Since (p
A
T F F T
A
F T F
F F F
p
T T T implies p ~ q.
q) _, (p ~
q) is a tautology,
q logically
4.10.
Show that 1)~
q
logically implies
p
p
-->
q.
Consider the truth
tables of p ~ q and p _, q :
q p~q p-->q
T T F F
T F T F
T F F T
T F T T
-e
Now p ~ q is true in lines 1 and 4, and in these cases p implies 11..... . q
Determine the number of nonequivalent the proposition 1)B q.
Consider the adjacent truth table of p B q. plies p B q if p B q is true whenever Pi p, q) is only in Cases (lines) 1 and 4; hence P(p, q) cannot There are four such propositions which are listed
l' I
propositions
P(p, q) which logically imply
p
q
pBq
Now P(p, q) logically imtrue. But p B q is true be true in Cases 2 and 3. below:
Supplementary Problems
ARGUMENTS 4.14. 4.15. 4.16. Test the validity Test the validity Test the validity of each argument: of each argument: of each argument: (i) -p ~ q, P
I+
q; (ii) -p ~ -q, q
I1'..c,
I-
p.
-1' ~
(i) p ~ q, l' --> -q (i) p~
-q,
1'~
-p;
(ii) p ~ -q, (ii) p..c, q,
TV
-q
I-
I-
P~
-1'.
p, q
I-
-1';
-q,
-1'
-po
ARGUMENTS AND STATEMENTS 4.17. Test the validity But Paris Therefore, 4.18. Test the validity of the argument: then Paris is not in France. If London is not in Denmark, is in France. London is in Denmark. of the argument:
If I study, then I will not fail mathematics. I did not study. I failed mathematics.
34
4.19. Translate
(a)
ARGUMENTS,
LOGICAL IMPLICATION
[CHAP. 4
into symbolic form and test the validity
of the argument:
If 6 is even, then 2 does not divide 7. Either 5 is not prime or 2 divides 7. But 5 is prime. Therefore, 6 is odd (not even).
(b)
On my wife's birthday, I bring her flowers. Either it's my wife's birthday or I work late.
r
did not bring my wife flowers today. today I worked late.
Therefore,
(e)
If I work, I cannot study. Either r work, or I pass mathematics. r passed mathematics. Therefore, I studied.
(el)
If r work, r cannot study. Either I study, or I pass mathematics. r worked. Therefore, I passed mathematics.
Chapter 5
Set Theory
SETS AND ELEMENTS The concept of a set appears in all branches of mathematics. Intuitively, a set is any well-defined list or collection of objects, and will be denoted by capital letters A, B, X, Y, .... The objects comprising the set are called its elements or members and will be denoted by lower case letters a, b, x, y, . . . The statement tip is an element of A" or, equivalently, tip belongs to A" is written The negation of pEA is written
p
e: A.
A
pEA
There are essentially two ways to specify a particular is to list its members. For example,
set.
One way, if it is possible,
=
{a,e,i,o,u}
denotes the set A whose elements are the letters a, e, i, 0, u. Note that the elements are separated by commas and enclosed in braces { }. The second way is to state those properties which characterize the elements in the set. For example, B
=
{x:
x is an integer,
x> O}
which reads "B is the set of x such that x is an integer and x is greater than zero," denotes the set B whose elements are the positive integers. A letter, usually x, is used to denote a typical member of the set; the colon is read as "such that" and the comma as "and".
Example 1.1: The set B above can also be written as B Observe that Example 1.2: -6
e: B,
3 E Band
7T
e: B.
=
{l, 2, 3, ... }.
The set A above can also be written as A
=
{x: b
x is a letter in the English alphabet, x is a vowel}
Observe that Example 1.3:
e: A,
eEA
and p
e: A.
Let E = {x: x2 - 3x + 2 = O}. In other words, E consists of those numbers which are solutions of the equation x2 - 3x + 2 = 0, sometimes called the solution set of the given equation. Since the solutions of the equation are 1 and 2, we could also write E {1,2}.
=
Two sets A and B are equal, written A = B, if they consist of the same elements, i.e, if each member of A belongs to B and each member of B belongs to A. The negation of A =B is written A #B.
Example 1.4: Let E
=
{x:
Then E F G. Observe that a set does not depend on the way in which its elements are displayed. A set remains the same if its elements are repeated or rearranged.
==
x2
-
3x
+2 =
O},
F
=
{2,1}
and
G
=
{1, 2, 2, 1, 6/3}.
35
36 FINITE AND INFINITE SETS
SET THEORY
[CHAP. 5
Sets can be finite or infinite. A set is finite if it consists of exactly n different elements, where n is some positive integer; otherwise it is infinite.
Example 2.1: Let M be the set of the days of the week. M In other words, Sunday}
Let P {x : x is a river on the earth}. Although it may be difficult to count the number of rivers on the earth, P is a finite set.
=
SUBSETS
A set A is a subset of a set B or, equivalently, B is a superset A of A, written
cB
or
B => A
iff each element in A also belongs to B; that is, x E A implies x E B. We also say that A is contained in B or B contains A. The negation of A c B is written A ¢ B or B fJ A and states that there is an x E A such that x tl B.
Example 3.1: Consider the sets A
=
{1, 3, 5, 7, ... }, xisprime,
B x>2}
=
{5, 10, 15, 20, = {3,5,7,11, than 2 is odd.
} } On the other hand,
C = {x:
Then C c A since every prime number greater B ¢. A since 10 E B but 10 El A. Example 3.2:
Let N denote the set of positive integers, Z denote the set of integers, Q denote the set of rational numbers and R denote the set of real numbers. Then
NeZ
Example 3.3:
cQcR
The set E {2, 4, 6} is a subset of the set F {6, 2, 4}, since each number 2,4 and 6 belonging to E also belongs to F. In fact, E = F. In a similar manner it can be shown that every set is a subset of itself.
=
=
I Definition: I
As noted in the preceding example, A c B does not exclude the possibility that A In fact, we may restate the definition of equality of sets as follows: Two sets A and B are equal if A
= B.
c Band Be A.
In the case that A c B but A =1= B, we say that A is a proper subset of B or B contains A properly. The reader should be warned that some authors use the symbol C for a subset and the symbol c only for a proper subset. The following theorem is a consequence of the preceding definitions:
Theorem 5.1:
Let A, Band C be sets. Then: A = B; and (iii) if A c Band
(i) A c A; (ii) if A c Band B c C, then A c C.
Be A, then
UNIVERSAL AND NULL SETS
In any application of the theory of sets, all sets under investigation are regarded as subsets of a fixed set. We call this set the universal set or universe of discourse and denote it (in this chapter) by U.
Example 4.1: Example 4.2: In plane geometry, the universal In human population world. set consists of all the points in the plane. set consists of all the people in the
studies, the universal
CHAP. 5]
SET THEORY
37
It is also convenient to introduce the concept of the empty or null set, that is, a set which contains no elements. This set, denoted by 0, is considered finite and a subset of every other set. Thus, for any set A, 0 cAe U.
Example 4.3: Example 4.4: Let A
=
{x:
x2
= 4,
x is odd}.
Then A is empty, i.e. A
= 0.
According
Let B be the set of people in the world who are older than 200 years. to known statistics, B is the null set.
CLASS, COLLECTION, F AMIL Y
Frequently, the members of a set are sets themselves. For example, each line in a set of lines is a set of points. To help clarify these situations, other words, such as "class", "collection" and "family" are used. Usually we use class or collection for a set of sets, and family for a set of classes. The words subclass, subcollection and subfamily have meanings analogous to subset.
Example 5.1: Example 5.2: The members of the class {{2, 3}, {2}, {5,6}} are the sets {2,3}, {2} and {5,6}.
Consider any set A. The power set of A, denoted by P(A) or 2A, is the class of all subsets of A. In particular, if A {a, b, c}, then
=
P(A)
=
{A, {a, b}, {a, c}, {b, c}, {a}, {b}, {e}, 0}
In general, if A is finite and has n elements, then P(A) will have 2n elements.
SET OPERATIONS The union of two sets A and B, denoted by A u B, is the set of all elements which belong to A or to B: AuB {x: x E A or x E B} Here "or" is used in the sense of and/or. The intersection of two sets A and B, denoted by A n B, is the set of elements which belong to both A and B: AnB {x: x E A and x E B} If A n B = 0, that is, if A and B do not have any elements in common, then A and B are said to be disjoint or non-intersecting. The relative complement of a set B with respect to a set A or, simply, the difference of A and B, denoted by A "'- B, is the set of elements which belong to A but which do not belong to B: A "'-B = {x: x E A, x tl B} Observe that A "'-B and B are disjoint, i.e. (A "'-B)
n
B=
0.
The absolute complement or, simply, complement of a set A, denoted by Ac, is the set of elements which do not belong to A: Ac
=
{x:
x E U, x tl A}
That is, Ac is the difference of the universal set U and A.
Example 6.1: The following diagrams, called Venn diagrams, illustrate the above set operations. Here sets are represented by simple plane areas and U, the universal set, by the area in the entire rectangle.
38
SET THEORY
[CHAP. 5
A u B is shaded.
A n B is shaded.
A "-.B is shaded.
Example 6.2: Let A
A c is shaded.
=
{I, 2, 3, 4} and A
U
B
=
{3, 4, 5, 6} where
U
=
{I, 2, 3, ... }.
Then:
B
{I, 2, 3, 4, 5, 6}
A n B = {3, 4} Ac
A"-.B
=
{1,2}
=
{5,6,7,
... }
Sets under the above operations satisfy various laws or identities which are listed in Table 5.1 below. In fact we state: Theorem 5.2: Sets satisfy the laws in Table 5.1.
LAWS OF THE ALGEBRA OF SETS Idempotent A Laws lb. AnA
la.
AuA
=
=
A
2a.
(AUB)UC
=
BuA
Associative AU(BUC) Commutative
Laws 2b. Laws 3b. AnB (AnB)nC
=
BnA
An(BnC)
3a.
AuB
=
=
Distributive 4a. Au(BnC)
Laws 4b. An(BuC) = (AnB)u(AnC)
=
A U
(AuB)n(AuC) Identity
Laws 5b. 6b. AnU An0
5a. 6a.
Au0 AuU
=
=
=
=
A
0 0
U
7a. Sa.
AuAc
=
=
Complement U
Laws 7b. Sb. AnAc U»
= =
(Ac)c = A De Morgan's
= 0, 0c =
Laws 9b. (A n B)c
9a.
(A uB)c
AcnBc Table 5.1
A c u Be
Remark:
Each of the above laws follows from the analogous logical law in Table 2.1, Page 11. For example, A
nB
=
{x:
x E A and x E B}
{x:
x E B and x E A}
B
nA
the set of babies is disjoint from the set of people who can manage crocodiles.3:
q is logically
between set inclusion and the above set operations:
Each of the following conditions is equivalent to A (i) A
c B:
(v) B
U
n
B
A B
(iii) Be cAe (iv) A n Be
Ae
U
(ii) A u B
=0
ARGUMENTS
AND VENN DIAGRAMS
Many verbal statements can be translated into equivalent statements about sets which can be described by Venn diagrams. or "Babies cannot manage crocodiles" is a consequence of Sl' S2 and S3.
(The above argument is adapted from Lewis Carroll.) Now by S1> the set of babies is a subset of the set of illogical people:
illogical people
By S3' the set of illogical people is contained in the set of despised people:
Furthermore. the set of despised people and the set of people who can manage a crocodile are disjoint:
But by the above Venn diagram. by S2.
s. is valid.CHAP. Symbolic Logic. he is also the author of Alice in Wonderland.1: Consider the following argument: Sl: Babies are illogical. Lastly we state the relationship Theorem 5. Thus the above argument. then p equivalent to q A p: P A q == q A p. Hence Venn diagrams are very often used to determine the validity of an argument. 5]
SET THEORY
39
A
Here we use the fact that if p is x E A and q is x E B.
S:
Babies cannot manage crocodiles.
Example 7. S2'
S3
f-
S
. S2: Nobody is despised who can manage a crocodile. S3: Illogical people are despised.
I(iii):
If A
c Band
B
C
C.4.5. 2.
Z is not the empty set
SUBSETS
5. Hence P(S) Note that there are 23
=
{S. 2. . {l.
Determine whether or not each set is the null set: (i) X
(i) (ii)
=
{x:
X2
=
9.2}. i. We have shown that x E A implies x E C. . X = ~. ~}
=
8 subsets of S.
{r.3. Z"#~. 4.
(iii) Z
=
{x: x
+8 =
8}. some texts define the null
~
== {x: x"# x}
Z
(iii) The number zero satisfies since it contains O.t.
{s.
.1.r.
{t. That
x +8 8..s. these are {l}.s}. 3} . The set ~ contains no elements. {2}. Now 3 E A and. hence X is empty.3}. } (iii) The number of people living on the earth. that is.s. The number 2 belongs to A.
{s. Let A
=
{X:
3x
=
6}.
Each is different from the other.
5. since B consists of even numbers. 2. The set {O} contains one element. {O}.
5. set as follows: In fact. 99. 2x
=
4}. Now A c B implies x E B. so Y is also empty.6.r. {l. 3}. the number zero. then A
C
C.r}. We assume that any object is itself. {l. {3}. {l}.
5.
The power set P(S) of S is the class of all subsets of S.
The first three sets are finite. hence A is not a subset of B. 3 ~ B. that is.
Which of the following sets are finite? (iv) {x: x is an even number} (i) The months of the year. A = {2}. (ii) {I. 100} (v) {I. hence x E C.s}.40
SET THEORY
[CHAP.t. 5
Solved Problems
SETS.
It is necessary to show that at least one element in A does not belong to B..t.
5. the last two sets are infinite.e.2}. {2. ELEMENTS
5.
Determine which of the following sets are equal:
0. {l.
{l. that A c C. {2}. it is the empty set.3.
(ii) Y
=
{x: x # x}. it does not equal A.
Does A
= 2?
A is the set which consists of the single element 2. 3. 5} is not a subset of B
=
{x: x is even} .3}.2.
5.
Prove that
A
=
{2. 3}.
There is no number which satisfies both x2 = 9 and 2x = 4.8.
Prove Theorem 5. 3.. {2.. But Be C. hence is.
Which of these sets are equal:
They are all equal. 2. Let x E A.t}?
Order and repetition
do not change a set.
We must show that each element in A also belongs to C.
Find the power set P(S) of the set S
=
{l. the null set.
Accordingly. {OJ.
5. The set {~} also contains one element.
=
=
{O}. 3}. . There is a basic difference between an element p and the singleton set {p}.7. {3} and the empty set~.
Let V={d}.
Y c X is true.11.4}. A
=
n
{I. 8. hence We Y is false. (iv) V is not a subset of X since d E V but d <l X. by hypothesis.CHAP.
(i)
A u B consists of those elements which belong to A or B (or both).d}. An C = {3. (iii) Z::J V.9.8. hence
=
{2.
Since each element in Y is a member
Now a E Z but a <l W.
=
are not in An C. hence V C X is false. Determine
5. 8} and
Ac
C
=
{3. A
The null set hence A 0. (v) B"" C.
(i) (ii)
X={a.
0c
A. 9}.
(iii) The only element in V is d and it also belongs to Z. W={c. 8. 6. 2.. 2. hence
=
{5. hence
An C = {3.
(ii) W =/. hence X = W is false. 4}.
5. (ii) A
n
C. 5]
SET THEORY
41 and Z={a. 6. whether each statement (i) Y eX. then A = 0.
SET OPERATIONS 5. 3. (ii) A n B.8}. (iii) (A
B.c}. 6. hence W # Z is true.
=
0
is a subset
of every set.
5.
Find:
(i) A c. But. (iv) A
U
=
{2. B". 4. 9}. Let
(i) (ii)
U
=
{1. 7.b} is true or false:
of X.b. 4. 7.
Ac consists of the elements
in U that
are not in A. (vi) We Y. hence Z:) V is true.b.d}. shade:
(i) A
U
B. 6}. hence shade the area in A and in B as follows:
•
(ii)
A u B is shaded. 8}.10. In each Venn diagram
below.. (v) Now a E X but a <l W. 6.Z. Y={a. 4. .
(vi) W is not a subset of Y since c E W but c <l Y. as follows:
Then A n B consists of the cross-hatched
••
area which is shaded
•
. A with strokes slanting upward to the right (!///) and then downward to the right (\ \ \ \).4} and
(iii) (A n C)c consists of the elements in U that so (A n C)c {l. hence
(iv) A uB consists of the elements in A or B (or both): (v) B '" C consists of the elements
A uB
=
{l. 9}.12.2. (iv) V c X.
•
To compute A n B. 5. Now by (ii). (v) X = W. Prove:
If A is a subset of the empty
set
0. in particular. 5. C
in B which are not in C. first shade shade B with strokes slanting below:
A n B consists of the area that is common to both A and B.
c 0. B C)C.
An C consists of the elements in both A and C. . 2. 3.
(ii) (AUB)C.
5.
5
A n B is shaded.
(iv) AenBc.AY.
AnB AnB
= B if BcA.
(B" A)c is shaded. (iii) First B"A: shade B"
(A u BY is shaded.
(i)
Be consists of the elements which do not belong to B. then (A
U B)c
is the area outside Au B:
Au B is shaded. then (B"
A
B "A is shaded.
In the Venn diagram
below. hence shade the area outside B as follows:
B
Be is shaded. and then shade Be with strokes slanting downward to the right (\\\\). shade:
(i) Be. the area in B which does not lie in A. then AcnBc is the cross-hatched area:
. with strokes slanting upward to the right (/111). the area outside of A.13. =A if AcB.42
SET
THEORY
[CHAP. A)e is the area outside
A. (ii) First shade Au B. (iii) (B". Observe the following: (a) A nB is empty
(b) (e)
if A and B are disjoint.
(iv) First shade Ac.
CHAP.
A en Bc is shaded.
=
(AnB)u(AnC).x(lA}
=
{x:
xEB.
as expected by De Morgan's law. Notice that An(BuC)
(A n B) u (A n C) is shaded.
5.
A n (B u C) is shaded.xEAc}
BnAc
. and then shade An C with downward slanted strokes.
B"A
in
=
{x:
xEB.
5.
First shade An B with upward slanted strokes. as expected by the distributive law.
(ii) (AnB)U(AnC). shade
(i) An(BUC). Prove: B '" A = B n A c.15. and then shade B u C with downward strokes.5J
SET
THEORY
43
A C and Bs are shaded. now An (B u C) is the cross-hatched area:
slanted
A and B (ii)
UC
are shaded. now (A n B) u (A n C) is the total area shaded:
A n B and A n C are shaded.
Observe that (A uB)c = AcnBc.14. Thus the set operation of difference can be written terms of the operations of intersection and complementation.
(i)
First shade A with upward slanted strokes. In the Venn diagram below.
S3: No temperamental person is wealthy. Since the diagram represents a case in which the conclusion is false. even though the premises are true.
S2: Poets are temperamental. the argument is false. For an argument to be valid.
Consider the following Venn diagram:
wealthy people
Now the premises hold in the above diagram. find a conclusion such that the argument is valid:
Sl: All lawyers are wealthy. hence the argument
5. Nobody who is wealthy is a student.CHAP. Lazy people are not wealthy. the set of wealthy people and the set of temperamental people are disjoint. Thus
.
S: ___
By 81. but the conclusion does not hold.
but the conclusion does not hold. the set of lawyers is a subset of the set of wealthy people. Show that the following argument is not valid: All students are lazy. It is possible to construct a Venn diagram in which the premises and conclusion hold. 5]
SET THEORY
45
Notice that both premises hold. the conclusion must always be true whenever the premises are true. and by 83. For the following set of premises.22.23. such as
5. is not valid.
6. 4 and the set {4. 3. Which statements (i) 5 E A. (iii) {5} c A.
==
'P(8)
==
{8. {4. {2.
subset of an infinite set is infinite.28.e.3.5} C A.
The elements of A are 2.
(iii) X
cA
but X
1.46
SET THEORY
[CHAP.24. 2. Therefore (ii) is correct. {3}.5}} cA.9}. Furthermore. . Find the power set 'P(S) of the set
22
S
=
{3.5} and E
Let A {1.4}. Let A = {2. (ii) {4.4}}. hence (i) and (ii) are incorrect. (ii) XeD but X
=
=
=
{3.5}. 4}.5.
are also valid
MISCELLANEOUS
PROBLEMS
5. The elements of A are 2.
5. the set of poets is a subset of the set of temperamental
people. 5}. .2. so (iii) is also incorrect. A. Let A = {2.
State whether
each statement
is true or false:
(U) Every
(i) Every subset of a finite set is finite. Which statements are incorrect and why? (i) {4. 4}}. (ii) {5} E A.4.5} which belongs to A is a subset of A.
5.
1.
5. 4}.
Find the power set 'P(A) of
A
=
{I.
are incorrect and why?
Each statement is incorrect. but (i) is an incorrect statement. 4} and the power set P(B) of
B
=
{l.
Note first that 8 contains two elements. and the two singleton sets which contain the elements 3 and {1. hence
Thus the statement conclusion. 0}
Supplementary
SETS. {1. D Which sets can equal X if we are given the following information? (i) X and B are disjoint. 5}.4}}.25.
Problems
= =
{3.9}. B {2.
5. i. C.
5
By 82. C {1. (iii) {{4.
"No poet is a lawyer"
or equivalently
"No lawyer is a poet" is temperamental"
is a valid
The statements "No poet is wealthy" and "No lawyer conclusions which do not make use of all the premises.5} E A.27.4} respectively. {4..8}. In other words. {{1. 4}.
(iv) X
cC
but X
1. There are eight subsets of A and {5} is not one of them.. (iii) is also a correct statement since the set consisting of the single element {4.26. 5}. Therefore 'P(8) contains 4 elements: 8 itself.4. 4 and the set {4.29.7. the empty set 0.8. SUBSETS 5. 5}.
. B. 3 and the set {1. {3} and {{l. 3}.
(A "'-Be)e
Find:
(iii) G"'-B (iv) BeuG below. 1}
0c
{{4}}
=
{a. 3.e.
In the Venn diagrams
w"'-
V
(ii) Veu W
(a)
5.31. (iv) The set of animals living on the earth.40.35.d./. 1}
(iv) {4} E {{4}} (v) (vi) {4} c {{4}}
(ii)
{1.3(ii): Prove: If AnB=0. 3}
(iii) {1.39.
X27
+ 26x18
-
17xll
+ 7x3
-
10
o..0).We. Band G so that A.
Prove: (i)
(ii)
AcB
implies Au(B"'-A) An (B" G)
=
B.(A u G). in terms of the operations of intersection and complement. (v) The set of numbers which are solutions of the equation the origin (0.41.CHAP. shade
(i)
(vi) (A -. 3} = {3.
(b)
Draw a Venn diagram of three non-empty sets A. G¢B.B = A nBc defines the difference operation in terms of the operations of intersection and complement. Au (B" G) # (A u B) "'. Let U
=
{2.
State (i) (ii)
whether
each set is finite or infinite: to the x axis. Determine the validity of each argument for each proposed conclusion. A cB then if and only if A uB AcBe.c. 1.y}. Prove Theorem 5.(A n G).33.. 1.
(i)
Some poets are college professors. 4.
BcG.37. Some poets are wealthy. 2.2} SET OPERATIONS 5. 2} c {1.34.b. 1.
(vi) The set of circles through 5. AnG #
G have the following
0
(iv) Ac(BnG). Band properties: (iii) AcG. 5. 2.
Give an example to show that
ARGUMENTS AND VENN DIAGRAMS 5. Au B.
cv
(vii)
(viii) (A nAe)e (iii) Vn We (iv) Ve".
(ii) Some poets are not college professors. 3.32. 2.
The formula A "'.
.
B
=
(v)
{a. in the English
The set of lines parallel The set of letters
(iii) The set of numbers which are multiples of 5. State whether each statement
(i)
is true or false:
{1.36.e. 4.
5.c.30./.
5. No college professor is wealthy.d.
A#G
5. GcB. AnG = 0 (ii) AcB.c.
= B.e. 5. 5]
SET THEORY
47
5.BnG=0 (i) AcB.y}.A#G. alphabet. Find a formula that defines the union of two sets.y} GenA
and
G
=
{b.
5.
Prove:
=
(A n B) .e}.
G#B. (i) AuG (ii) BnA
A
=
{a.b.
{l.
Some interesting
(iii) Some teachers are not interesting (iv) Some mathematicians (v)
Some teachers are not mathematicians.3}}. e.. 5
of each argument
for each proposed conclusion. (i) C and E.4}. from college.
Answers to Supplementary
5. people. then he is not a poet.g}
(A".
(vi) If Eric is a mathematician. {2.44. 5.43. one must graduate No college graduate
(i)
is poor.3}.
(i) T. t. (a)
= {a.48
5.e} = {b.
All mathematicians Only uninteresting (i) (ii) Insurance
are interesting
Some teachers sell insurance. g} (A ".e. are teachers.31. 0} (ii) finite. (v) F.c.C)c = {b. people become insurance salesmen.f} = {b.
Teachers are not poor. 4}.42. (iv) T. (iv) finite. (vi) T (v) (vi) (vii) CenA (vi) infinite
5. {{2... (iii) T. {l}.33./. person.29. people.
All poets are interesting Audrey is an interesting
(i)
Audrey is a poet.
(i) infinite.-Be)e = {b.28.27. {4}.
salesmen are not mathematicians.
All poets are poor. Determine the validity of the argument
for each proposed conclusion. c. Determine the validity of the argument for each proposed conclusion. Determine the validity
SET THEORY
[CHAP. (iv) None
(i) T. (v) finite.32. In order to be a teacher. {1. 5.g}
=
{a. Band D.. (iii) infinite.d. people.
5.
Problems
(ii) D and E./. 5..-V
V
VcuW
VnWc
.d. (ii) F
'P(B)
=
{B.
(ii) Audrey is not a poet.-B (iv) BcuC 5. (iii) A. Poets are not teachers. 5.30.
(ii)
(iii) If Marc is a college graduate. people are not teachers. {{2. then he does not sell insurance. d}
=
C»
(viii) (A nAe)c
=
U
(]I
W".3}}. 5. (ii) T. (i) (ii) AuC= BnA
U
(iii) C".
say A
Example 2. d) are equal if and only if a
Example 1.3: Example 1. such as (1. b) where a E A and bE B: AxB The product of a set with itself. 6-1 below represent
The set {2. b) Two ordered pairs (a. the vertical line through P meets the x axis at a. the fundamental property of ordered proven: (a. b) and (c. (4. b): a E A. say a.1:
{(a. d) if and only if a = c and b = d PRODUCT SETS
pairs can be
Let A and B be two sets. Such an ordered pair is written (a.
The reader is familiar with the Cartesian plane R2 ::::R X R (Fig. and the horizontal line through P meets the y axis at b. b}
From this definition.1: Example 1. say a and b. 6-1 below). 2) are different. bE B}
X
A. Ordered pairs can have the same first and second elements. is sometimes denoted by A2. b) = (c.3) and (3. b) can be defined rigorously (a. 6-1
Fig. 6-2
50
. consists of all ordered pairs (a.
Remark:
An ordered pair (a. in which one of them. plane in Fig. {a.Chapter 6
Product Sets
ORDERED PAIRS An ordered pair consists of two elements.4) and (5.2: Example 1.3} is not an ordered pair since the elements 2 and 3 are not distinguished.
{a}
=
as follows:
a as the
{{a}. b) of real numbers and vice versa. Here each point P represents an ordered pair (a.4:
=c
and b
= d. b) the key here being that first element. b}} which we use to distinguish
c
{a.1).5). written A x B. The points in the Cartesian real numbers. is designated as the first element and the other as the second element. The product set (or Cartesian product) of A and B.
h
-1 0
P
2 2 3
B
hr---~----~P----~ab-----+-----+-----~-
-8 -2
a
-1
-2
-8
~--~----~----~A
2 3
Fig.
ordered pairs of
The ordered pairs (2.
. TFF. (2. the Cartesian product of n sets AI' A2. it is possible to represent A X B by a coordinate diagram as shown in Fig. y). 1. y). its y-component and its z-component. denoted by A x B x C. c) where a E A. (3. 2. The Cartesian product of sets A. Here the vertical lines through the points of A and the horizontal lines through the points of B meet in 6 points which represent A X B in the obvious way. x). TTT for (T. then A x B is also infinite.
c
E C}
X . 3.·
Example 3. 1. B
=
{I. (2. say.2:
Let A = {I. TFT. The point P is the ordered pair (2. x). 3. etc. bE Band c E C: A
X
B
X
C
{(a. then A x B has s times t elements. 1. . 2. consists of all ordered triplets (a. an
E
An}
Here an ordered n-tuple has the obvious intuitive meaning.
PRODUCT
SETS
IN GENERAL
The concept of a product set can be extended to more than two sets in a natural way. 3} and B = {a. (a.··
. x).
AXB
Then
=
{(I. y). FTF. b). Let A
Example 3.1:
••
. consists of all ordered n-tuples (aI' a2. Furthermore. if a finite set A has s elements and a finite set B has t elements.
b. y). (b. b). in which one of them is designated as the first element. Lastly. that is. (b. 3. 6-2 above. x). a). value to each of the eight cases below:
p
p. (1. c):
a
E A. y}. 3. q and r.
bn)
iff
al = bI.
•••
. 2. (a.
•••
. (b. . 3} and C =
=
{x.an)
= (bl.. 1. If either A or B is empty. it consists of n elements. an) where Al
X
al al
An' denoted by Al X A2 E AI' . an E An:
E
An'
A2
X '"
X
An
{(aI.
In general. B. T. (a. a).)
convenience
. FFT. b E B. C.
Then:
A XBXC
{(a. x). (3.
••• . T). (b. b). x). FFF} we have written.
(b. 2. TTF. another as the second element. FTT.. y)}
TRUTH
SETS
OF PROPOSITIONS say. 2. (a. 2. b}.6J
PRODUCT SETS
51
Example 2. b)}
Since A and B do not contain many elements. if either A or B is infinite. assigns
a truth
T T T
T
T T F
F
T F T
F
F F F F Let U denote the set consisting U (For notational
T T F F
T F T F in the table above:
of the eight 3-tuples appearing
=
{TTT. (a. b}. b.
(aI. a). (b. y). and the other is not empty.2:
=
{a.an):
AI'
.. then A x B is empty.CHAP. three variables
q
r
Recall that any proposition P containing. not necessarily distinct.
an = bn
an ordered triplet:
In three dimensional Euclidean geometry each point represents its x-component.• X
Analogously.
For. written aRb (ii) "a is not related to b". we redefine a relation by Definition:
I
A relation R from A to B is a subset of Ax
Let R be the following relation from B={a. c). symbolized by "< ". 3}
Then 1 R a. either m is married to w or m is not married to w.2: Set inclusion is a relation in any class of sets. c). the sentence "x is less than y". given any pair of sets A and B. b) in A x B exactly one of the following statements:
(i) "a is related to b". simply. b). is a relation in any set of real numbers. given any ordered pair (a. any subset R* of Ax B defines a relation R from A to B as follows: aRb iff (a. b. 3 R a.2: Let R be the following relation R Then a in W
*
=
-*
an
1 2 3
=
{a. a). relation R from a set A to a set B assigns to each pair
(a. either AcB or A¢B.Chapter
Relations
RELATIONS
7
A binary relation or.
Example 1.
eRe
*
58
.1: Example 1.
Marriage is a relation from the set M of men to the set W of women. either a is perpendicular to b or a is not perpendicular to b. (3. b). (a. For. For. Order.
=
{(a. c}:
-* a. (c. either a<b or a<t:b Perpendicularity is a relation in the set of lines in the plane.b}: R {(1. b)
E
R*
In view of the correspondence between relations R from A to B and subsets of A x B. 2 b. or. The relation R is displayed on the coordinate diagram of AX B on the right. Example 2. b) : aRb} On the other hand.
B. b): a is related to b} = {(a.
to
Example 2. b) of real numbers.1:
A = {I. (c. b)} and c a. and 3 b.4:
RELATIONS
AS SETS OF ORDERED
PAIRS
Now any relation R from a set A to a set B uniquely defines a subset R* of Ax B as follows: R* = {(a. given any pair of lines a and b. 2. written a$b A relation from a set A to the same set A is called a relation in A. (1.
Example 1. For.
aRb. equivalently.3:
Example 1. given any man mE M and any woman w E W.
e.
R is an equivalence
relation
if R is
(i) reflexive.
<
b.2: (i)
if whenever
aRb
then
bRa.e. c) E R implies (a. b) E R implies (b. b) E R}
Consider the relation in A
=
= =
b
{(I.3).1: (i)
R. denoted by ~ or pairs in A X A with equal coordinates:
~A
is the set of all
=
{(a.3). (3. 7]
RELATIONS
59
~A'
Example 2.
(ii) The relation R of perpendicularity of lines in the plane is not transitive. b) E R
<
and
(a.1). then f3 is similar to a. 3}.
A relation (iii) transitive.
A relation R in a set A is called symmetric (a. then a is parallel and not perpendicular to e. 3)} {(2.
Then
(ii) Let R be the relation < in any set of real numbers.
For if triangle
(ii) The relation R = {(1. 2).
for every
a
E
A.
Example 4.
iff a
<
and
>
(a.3). a) E R. c) E R. (2. (a.
=
{1. The identity relation in A. i. symmetric and transitive.
INVERSE
RELATION
Let R be a relation from A to B. (1. (3. if (a.
a is
The relation R of similarity of triangles is transitive since if triangle similar to f3 and f3 is similar to y. (3. (2. a)
E
A relation R in a set A is called reflexive
Example 4.2)} to the relations iff a> b
{I. i. respectively. Since if line a is perpendicular to line b and line b is perpendicular to line c.1:
=
Then
{(b.
Example 4. I)} in A since (2. Rand
Observe that in A.
and
is an
Example 4.e. i. 2) ~ R.. 2. b) E R iff a Then R is not reflexive since a <t: a for any real number a.3) E R but (3.
Let R be the relation of similarity in the set of triangles in the plane.3:
Let A be any set. similar to triangle f3.1). 3} is not symmetric
A relation R in a set A is called transitive if whenever i. The inverse of R.2). denoted by R::'. a): R
R-l
(a.
.2:
The inverses of the relations are respectively
defined by and and
"x is the husband of y" "x is the wife of y"
"x is taller than y"
"x is shorter than y"
EQUIVALENCE
RELATIONS if aRa. a): a
E
A}
The identity relation is also called the diagonal by virtue of its position in a coordinate diagram of Ax A. b) E Rand (b.
i. if
a is
The relation R of similarity of triangles is symmetric.e. R is reflexive since every triangle is similar to itself. (2.e. then a is similar to y.4:
By the three preceding examples. the relation R of similarity of triangles equivalence relation since it is reflexive.
(ii) symmetric.CHAP. (a.3: (i)
aRb
and b R c then aRc. is the relation from B to A which consists of those ordered pairs which when reversed belong to R:
R:!
Example 3. b) E R-l
Example 3. 2.
R=! are identical.
{2. 3. 5}. 5}.
x ==
y
(mod 5)
which reads "x is congruent to y modulo 5" and which means that the difference x . (i) (ii) Then the quotient set A I R
a E raj. i. is called the quotient
{raj : a
E
A}
property of a quotient set is contained in the following theorem. (ii) is not a partition of X since 5 E X and 5 belongs to both {I. {7. 14. such that each a E X belongs to one and only one of the subsets..1:
{x: (a.1:
Consider the following classes of subsets of X (i) (ii) (iii)
=
{I.. (iii) is a partition of X since each element of X belongs to exactly one cell. .
= =
{
-5..60
PARTITIONS
RELATIONS
[CHAP. {5... 9}] [{I. 5}.
-4. .2. 4. (i) X
••• . 9.. 4. -7.. -3. . denoted by A I R.
Example 5.. 8. for every a E A. 9}:
[{I. -9. 3.. Furthermore. called the equivalence class of A. 6}.e. } · . . 7
A partition of a set X is a subdivision of X into subsets which are disjoint and whose union is X. 7.
· . be the set of elements to which a is related:
[aj AIR
The fundamental Theorem 7..}
2. 5. Thus the collection {Al' A2. 4. [aj = [b] if and only if (a. let raj. . = {. Then R5 is an equivalence relation in Z. 3.
(iii) if [aJ
Example 6. 11.} =
Observe that each integer x which is uniquely expressible in the form x 5q + r where 0"" r < 5 is a member of the equivalence AT where r is the remainder. 5} and {5.1: Let R5 be the relation
[b]. 12. is a partition of A.8. 8. 9}. .
· .
Am} of subsets of X is a partition (ii) for any Ai. On the other hand. 7. the set of integers.
{ . {2. for each a E A. 13.} · ... Let R be an equivalence relation in a set A.y is divisible by 5. {2. 9}]
Then (i) is not a partition of X since 7 E X but 7 does not belong to any of the cells. -10. The subsets in a partition are called cells.Aj' either Ai=Aj
of X iff: or AinAj=~
=
A1UA2U···UAm. {4. 8}. 3.. -1. = { . 0.. 6.
.. 3. } · .
-6.. 6. 7. 9}] [{I.
then [aJ and [bJ are disjoint. 8}. Specifically. -8. b)
=1= E
R. . -2.
defined by
in Z. There are exactly five distinct equivalence classes in Z / R5: Ao Al A2 As A4
= {. 10. 6. Note that the equivalence classes are pairwise disjoint and that Z
=
Ao
U Al U A2 U
Ag
U A4
.
EQUIVALENCE
RELATIONS
AND PARTITIONS
Let R be an equivalence relation in a set A and. 1. x) E R}
of A by R:
The collection of equivalence classes of A..
{x: (d. c). (2.5). 7. (3. (a.2). write the elements of R but in reverse order: R-l
3r--r~f--+--+--+2 3 4 5
=
{(6. d).2). (3. (d. 1). b). d} and let R be the relation in M consisting of those points which are displayed on the coordinate diagram of M X M on the right. b)}
n-!
= {(b. then R = {(2. (c. (b. b). (3.. i.. (5. d).5).1. a). (i) Find all the elements in M which are related to b. 5)} (ii)
R is sketched on the coordinate
shown in the figure.
Let R be the relation from E = {2. (5. (6.10)} (ii) R is sketched on the coordinate shown in the figure.--+a t---+-+--I---+--
(ii) Find all those elements in M to which d is related.
diagram
of Ax B as
3r--'--~--+---~
lr-~---+---r--'_2
(iii) The inverse of R consists of the same pairs as are in R but in the reverse order.f--.4}
to
B
f1. b. hence
pairs
(a. (2. b).3.5). (10. (3.3. (b.6. (5. b). (10.
< from A
=
{1. b). (d. (b. (b.2. 5} to F = {3. diagram of Ex F as
71--+--+--+-f-+6r-
----. 5}. a). Hence the desired set is {a. b). (5. defined by
(i) Write R as a set of ordered pairs. (ii) Plot R on a coordinate diagram of Ax B. 3). (ii) Plot R on a coordinate diagram of Ex F. b). that is.
Let M = {a. b. (a. c). that is. b} is the desired set. b).3).e. (iii) Find the inverse relation R: '.10).e.
as the second
The vertical line through d contains all the points of R in which d appears as the first element: (d. b) and (d.3). 5)}
7. Hence {a.. (b. 4)} is the relation
3
4
Observe that R-l greater than y". (3. Let R be the relation "x is less than y".1).3).3). (iii) Find the inverse relation R:». hence
u-!
= {(3. (c. (2.
(i) Choose from the sixteen ordered pairs in Ex F those in which the first element divides the second. (iii) Find the inverse relation R: '. (4. b) E R}. (b. and then write the pairs
R
(iii) First
in reverse order:
{(a. a) and (d.2.
(i)
R
consists of those ordered which a < b.--+-~--
CI--+---r--.2). d)}
. d}. 4. c.CHAP.>. (1. 3. (5. 10} defined by "x divides y". defined by "x is
7. write R as a set of ordered pairs. {x: (x.3). i.2). (i) Write R as a set of ordered pairs._--+---r-
(iii) To find the inverse of R. a).
dt---+-.. b) E A X B
for
R = {(1. 7]
RELATIONS
61
Solved Problems
RELATIONS 7.
(i) (ii)
a
b
C
d
The horizontal line through b contains all points of R in which b appears element: (a. x) E R}.3.6).
. (d. 6).
e. Thus R-l c R. Hence (a. Thus R is reflexive iff (a. 2).3). 4}. (a.62
EQUIVALENCE 7.
Now a relation R is not symmetric if there exists an ordered pair (a. R3 is transitive.t R
since (4. (2. b) E R:». (3. . then (b. and so R C R:».
=
Let
(a. b) E R.
then
(b. 3 "" 5 but 5/= 3. (iii) transitive. since if a since if a
(ii)
II f3
then and f3
II a f311 a. b) E Rand
but
(a.)
(i)
R is reflexive since. 7
Let R be the relation ~ in N = {I. (ii) symmetric.3) E Rl and (3. if the line a is parallel to the line f3 then f3 is
(iii) R is transitive
II f3
II y
then a
II y.2) E R2 and (2.3) f. Determine whether R is (i) reflexive. then (b. b) E R. a):
Suppose R is symmetric. not necessarily
distinct. b) E R iff a ~ b. and so R Rr». R
= R-l. (i) R is reflexive iff ~
(i) (ii) Recall that
A A
Show that: (ii) R is symmetric iff R = R::'.
such
(a. (2.4. c) f. i. (4. Let (a. (3. a) e: R.
=
{(a.
Let R be a relation in A.
Let
W
{I. On the other hand.. symmetric
7.3) E R3 but (3.6. (iv) an equivalence relation.
(ii)
A relation R is not transitive that Hence Rj is not transitive
if there exist elements a. c) E R
c.
such that
whether each relation is (i) symmetric. but (4. (iv) an equivalence relation. R3 is not symmetric since On the other hand. for example. 2. (iii) transitive. band
(b. (3.
R is not symmetric since since..
(i)
R is reflexive since a"" a for every a E N. (ii) symmetric.1) f. b) E Rr ).2). i.2)} {(1.4) f.3) f.t R2
(3.
c R.1) f. R2 is symmetric. Now suppose symmetric.
Furthermore. Determine whether R is (i) reflexive.t R3.
R2 is not transitive
since but (3.
and transitive.
a E A}. b) E R (b.e. by assumption. (2. b) E R.t Rl.3)} (ii) transitive.
Consider the following relations
in W:
s. RELATIONS
RELATIONS
[CHAP. let (a. a) E R for every a E A iff
c R.
(iv) R is not an equivalence relation
since it is not symmetric.t
n.
(ii)
(iii) R is transitive
a""
band
b "" c implies
a""
c.
7.5) E R but (5. 3. I)} {(2.7.
R is symmetric parallel to a. (Assume that every line is parallel to itself. Hence: Rl is not symmetric since (4.
Accordingly
R is
.
7.3) E R2 On the other hand.5. 2. by symmetry.
Let R be the relation II (parallel) in the set of lines in the plane.3).
R2 R3 Determine
(i)
{(1.1) E ti. (a.2).3. i. a) E R and.e.1).
(iv) R is an equivalence relation
since it is reflexive.3) E Rl but (3. }. (1. a) E R-l
= R. a) E R by symmetry.
a
for every line a.t R.
of such assignments is called a function (or mapping) from (or on) A into B and is written f:A ~ B or A_!_"B The unique element in B assigned to a E A by I is denoted by I(a). and to each irrational number the number -1. that is. Here I(a) = b. d} A -> E. c
c and d
->
b
define a function 1 from A into E. the co-domain B. y. The domain of I is A. c}..
The following diagram defines a function
Here I(a) = y. I(b) = c. Thus I(x) I if x is rational { -1 if x is irrational
The range of 1 consists of 1 and -1: I[R] = {I. c. for every real number x let 2• Then the image of -3 is 9 and so we may write 1(-3) 9 or I: -3 -> 9.
I:
=
and
B
=
{x. b. Here the domain of 1 is the set of countries in the world. that is.1: Let I(x)
=
{f(a):
a E A}
I
=x
assign to each real number its square. and called the image of a under I or the value of f at a. c. that is. that is. t. the range
66
. d} and B a
Example 1. I(France) = Paris. Let A = {a.
I[A]
Example 1. b
->
c.c}. b.5: Let A {a. the collection. and the co-domain are identical.
=
Example 1. I(c) = z and I(d) = y. denoted by f[A] is the set of images. and let I: R -> R assign to each rational number the number 1. I[A] {b.3:
==
->
{a. Example 1.e. I(b) = x.
Also I[A] = E.
=
=
b.
The assignments
->
b. z}. I(c) = c and I(d) The range of I is {b.4:
Let R be the set of real numbers. b.Chapter 8
Functions
DEFINITION OF A FUNCTION Suppose that to each element of a set A there is assigned a unique element of a set B.c}. The range of I. i.-I}.2:
Let 1 assign to each country in the world its capital city.
Example 1. The image of France is Paris. the co-domain is the list of cities of the world.
CHAP.1. 3). we do not distinguish between a function and its graph. is the graph of f as shown in the diagram.3:
=
f: R . (-2. that is..1:
in exactly one ordered pair (a. -1) and (2. If h is to be a function it cannot assign both 3 and 2 to the element 1 EA./(a»
the relation
in A
X
B given by
: a E A}
We call this set the graph of f. f(2) 3 and f(3) 1. (0.3). written 1= g. then I(a) = b. if I(a) = g(a) for every a E A. I)}
=
{(I.1
. b) in
in Example 1.
Let i :A -> B be the function defined by the diagram graph of f is the relation
{(a. For example. (b. (3. A real-valued function
f(x)
=
=
=
Example 2. 3}:
f
o
h
{(1. a relation from A to B.
6
-2
o
-5
-1
3
2
Graph of f(x)
=
2x . (3. y)}
Then the
Example 2.2). its graph is a line in the Cartesian plane R2. is a function if it possesses the following
Each a E A appears as the first coordinate Accordingly. i. (c. The graph of a linear function can be obtained by plotting (at least) two of its points.8J
FUNCTIONS
67
GRAPH OF A FUNCTION To each function
I:
A
-7
B there corresponds {(a. (1.. Also h is not a function from A into A since 1 E A appears as the first coordinate of two distinct ordered pairs in h.3). x). to obtain the graph of f(x) = 2x . (2.2:
Consider the following relations in A = {I. (1.
Example 2. (2. The line through these points. (3.2).. I)}
f is a function from A into A since each member of A appears as the first coordi-
nate in exactly one ordered pair in i: here f(l) 3. -5). if (a.R of the form ax + b (or: defined by y
=
ax
+ b)
x f(x)
is called a linear function. set up a table with at least two values of x and the corresponding values of f(x) as in the adjoining table.
f. The negation of 1= g is written 1# g and is the statement: there exists an a E A for which I(a) A subset property:
[F]
#
g(a)
I of
Ax B. b) E I.5. I)} = {(1. y). Two functions f: A -7 Band g: A -7 B are defined to be equal. if they have the same graph.e.1). z). Accordingly.2). 2. g is not a function from A into A since 2 E A is not the first coordinate of any pair in g and so {j does not assign any image to 2. (d.3) and (1.3).
Consider. The graph of such a function is sketched by plotting various points and drawing a smooth continuous curve through them.3.4: A real-valued f(x) function
FUNCTIONS
[CHAP. for example. g(f(a)).2x . by definition.
(g 0 f)(a)
Example 3. the function f(x) = x2 . the domain of g. Hence we can find the image of f(a) under the function g..1: Let f: A
-->
= g(!(a))
C be defined by the following diagrams:
Band A
g: B
-->
f
B
g
C
We compute (g
0
f) : A
-->
C by its definition: (g (g
(g
0 0
f)(a) f)(b)
f)(c)
g(f(a)) g(f(b)) g(f(c))
g(y)
g(z)
r
to "following the arrows"
0
g(y)
Notice that the composition function go f is equivalent from A to C in the diagrams of the functions f and g. then its image f(a) is in B.e.68
Example 2.
. Set up a table of values for X and then find the corresponding values for f(x) as in the adjoining table.. i.
8
f: R
-->
R of the form
x
f(x) 5 0
=
aoxn
+
alXn-1
+ . The diagram shows these points and the sketch of the graph.3
COMPOSITION
FUNCTION
f: A
-'>
Consider now functions
Band
g: B
-'>
C illustrated
below:
Let a E A. The function from A into C which assigns to each a E A the element g(!(a)) E C is called the composition or product of f and g and is denoted by go f. +
an-IX
+
an
-2
-1 0 1
is called a polynomial function. Hence.
y
2 3 4
-3 -4 -3
0 5
Graph of f(x)
= x2
-
2x .
.. h is one-one since the elements in the domain.CHAP. y and z. Hence I: A-> B is onto iff the range of I is the entire co-domain. since 2 E B is not the image of any element in the domain A. x. g and h in the preceding example. Let R be the set of real numbers and let defined as follows:
I(x)
Then I is not onto On the other hand. I[A 1 = B. i.
g: R
h(x)
->
Rand
x2
h: R
->
R
be
=
2x.
Example 4. Equivalently. We compute a general formula for these functions: (f 0 g)(x) (g o/)(x)
I(g(x)) g(f(x)) I(x
+ 3) =
(x
+ 3)2 =
x2
+ 6x + 9
g(x2)
=
x2
+3
ONE-ONE AND ONTO FUNCTIONS A function I: A -> B is said to be one-to-one (or: one-one or 1-1) if different elements in the domain have distinct images. C and g
D defined by the followh D
f
B
C
Now 1 is not one-one since the two elements a and c in its domain have the same image 1. 8]
FUNCTIONS
69 I: R
->
Example 3.
g: B.
A function I: A -> B is said to be onto (or: I is a function from A onto B or I maps A onto B) if every b E B is the image of some a E A.2:
Let R be the set of real numbers. both g and h are onto functions. On the other hand.e.3:
I:
x
R
->
R. and let follows: and I(x) x2
Rand
g:
R
->
R be defined as
=
g(x)
=
x
+3
25 7
Then
(f 0 g)(2)
(g 0/)(2)
= =
l(g(2))
g(l(2))
= =
1(5)
g(4)
= =
Observe that the product functions log and go 1 are not the same function.
g(x)
=
x3 -
and
=
The graphs
of these functions
follow:
I(x)
=
2x
g(x)
=x
3-
X
h(x)
= x2
.. Also. g is not one-one since 1 and 3 have the same image y.1:
= I(a')
I:
implies
A
->
a
= a'
h: C
->
Consider the functions ing diagram: A
B..2: Consider the functions I. I: A -> B is one-one if I(a)
Example 4. have distinct images.
Example 4.
which assigns to each element in A itself. then f-I is a function from B onto A and is called the inverse function. Note that the identity function 1A is the same as the diagonal relation: 1A = ~A' The identity function satisfies the following properties: Theorem 8. the inverse relation t:' of a function f c A x B need not be a function. s. 8
The function I is one-one. then f-IOf 1A and fof-l 1B
=
=
The converse of the previous theorem is also true: Theorem 8. this means that each horizontal line contains at least one point of g.
I-I can
be obtained from the diagram
of
I
by revers-
For any set A. the two elements 2 and -2 have the same image 4. and so has an inverse function t:' :B ~ A. geometrically. seen by the following diagram of f:
Hence the inverse relation
I-I
is a function from B into A.e.
Example 5. if f is both one-one and onto. The function g is onto.
=
=
INVERSE
AND IDENTITY
FUNCTIONS
In general.
Theorem 8. is called the identity function on A and is usually denoted by 1A or simply 1. b. a).2:
If f: A ~ B is both one-one and onto. (r. r)} This can easily be
I
=
is a function from A into B which is both one-one and onto. i. for h(2) h(-2) 4. b). this means that each horizontal line does not contain more than one point of I. for example.1: For any function
f: A ~ B.
=
{(s.
fog =
1B
. (e. the function f: A ~ A defined by f(x) = x. t).e. e)}
The diagram of
I-I follows:
Observe that the diagram of ing the arrows. Then {(a. (t. t}. However. and h[R] is a proper subset of R.1: Let A = {a. and g
= i:'. geometrically. s). i. (b. -16 tl h[R].70
FUNCTIONS
[CHAP.3: Let f: A ~ Band g: B ~ A satisfy
gof =
1A
and
Then f is both one-one and onto. The function h is neither one-one or onto. c} and B = {r.
Find: (i) t(4). x and z. g and h be the functions of the preceding problem.
.
On the
other
hand. (ii) No. /(2) /(3) == 5.2. and to each non-positive number number 4. write f as a set of ordered pairs. form the graph of f. 5} and let [: A
4
A
be the function
defined in the diagram:
(i) Find the range of (ii) Find the graph of
(i) (Ii)
f. 2.Y.4. State whether
B {x. i.4.z}. (iii) h(4).b. g(O). (ii) g(4).1. then h(x) 4. 1(4) == 2 and /(5) == 3. where a E A. t(-2). (iii) Yes.3.
or not each diagram
defines a function
from
A
[a. 3.
8.3)}. hence /[A] == {2. h(O).
== 0. h(-2). Two elements. as the image of any elements of A. (4. hence / == {(1. 8]
FUNCTIONS
71
Solved Problems
FUNCTIONS 8.
8.c}
into
(i)
(ii)
(iii)
(i) No. 3 and 5 appear
The range ![A] of the function / consists of all the image points. then
(iii) If
==
>
0.
if
/(0)
x
==
03 == O. ==
x3•
(i) Since / assigns to any number x its cube x3.2).3). let g assign the number 5.
Now only 2.CHAP. g(-2). There is nothing assigned to the element bE A.
/(4)
==
5. (3. == 4
hence h(4) == 42 and h(O) == 4. 1 can be defined by I(x) (ii) Since g assigns 5 to any number x./(a)).5).
Rewrite (i) To (ii) To (iii) To let
each of the each number each number each positive h assign the
following functions using a formula: let t assign its cube. are assigned to c E A.
Let t.
==
16.
Let A
=
{l. 3.
==
5.e. t(O).
(i) (ii) Now I(x) Since
x h(x) g(x)
==
x3
for every number x. we can define g by g(x) == 5.
43 == 64. (2. t. 5}. == 5. (5. Now /(1) == 3. number let h assign its square. thus h(-2)
== X2. hence 5 for every number
x. /(-2)
g(-2)
==
(-2)3
g(O)
==
g(4)
==
==
5
and
== -8. (iii) Two different rules are used to define h as follows:
h(x)
X2 {
if
x>
0
4
if x == 0
8.
The ordered pairs (a.5).
(4.
Find the geometric conditions under which a set f of points on the coordinate diagram of Ax B defines a function f: A -> B. 2 and find the corresponding values of I(x): 1(-2)
==
3(-2) .7.
8.4). 8
Let X = {I. two different ordered pairs (b. 1). as
(iii) Yes. 0. Although 2 E X appears as the first coordinate in two ordered pairs in h. 3.8. d}.4)} (1. (iii) h
Determine
whether or not each relation is a function from
(2.5. ordered pairs are equal. The element 2 E X does not appear as the first coordinate in any ordered pair (iii) Yes.3).
d c b a a b (i) (i) (ii)
d d
c b a
c b a
c
d
a
b (ii)
c
d
a
b
c
d
(iii)
No.
x
-2
f(x)
-8 -2
4
(2. (1. (i) f {(3.2
-2. say. c.2).4)
o
2
(-2. i. b) and (b.1). x == -2.3) and (2.2).6.e. c E W does not appear the first element in any ordered pair. The vertical line through c contains no point of the set.
GRAPHS OF REAL-VALUED
8.2
==
-8.-8)
. (3. Set up a table with three values of x. 1). el) contain the same first element b. these two
Recall that a subset 1 of X X X is a function I: X -> X if and only if each a E as the first coordinate in exactly one ordered pair in I.
Let W = {a. Each vertical line contains exactly one point of the set.
Sketch the graph of f(x)
This is a linear function.
8. only two points (three as a check) are needed to sketch its graph. {(2. I)} (1. Determine whether the set of points in each coordinate diagram of W x W is a function from W into W.4)}
X appears their first in g.
FUNCTIONS
[CHAP. i. Two different ordered pairs (2. 4}.72
8. No.e. The vertical line through b contains two points of the set.
f(O)
==
3(0) . b. 2.4). (2.1).1) in 1 have the same number 2 as coordinate.4). (ii) No.
FUNCTIONS
= 3x .
The requirement that each a E A appear as the first coordinate in exactly one ordered pair in 1 is equivalent to the geometric condition that each vertical line contains exactly one point in f. (i) No. (ii) g {(2.
f(2)
==
3(2) -
2
==
4
Draw the line through
these points as in the diagram. (4. (3.2. (4. X into X.
(ii) To each country in the world assign the latitude and longitude of its capital.
Determine if each function is one-one.
. Since 9 is onto.
B is not onto since 3 E B is not the image of any element in A.
0 0
8. (i) (Ii) No.
h: C -> D is onto since each element in D is the image of some element of C.
c
h
D
(i) Determine
(i) The function The function The function (ii) Now
if each function is onto.
Accordingly. ~ C is also onto. then the composition function go f : A ~ C is also one-one. there exists abE Since f is onto.
f ==
{(a.CHAP.
Let c be any arbitrary element of C. No.
Hence each c E C is the image of some element a E A.
(i) (ii) No. Accordingly. For h assigns distinct
(iii) Yes.f(a)
== g(b)
== c
go f is an onto function. (iv) Yes. Prove:
gof:A
If f: A ~ Band g: B ~ C are onto functions.
8. Different countries in the world have different prime ministers.4)}. Many people in the world have the same age. Yes.e. There are different books with the same author. c -> 2 -> x -> 4. (iii) To each book written by only one author assign the author. Let (g f) (a) == (g f)(a'). (c. b -> 1 -> y -> 6.
8.16.
For 9 assigns z to both a and e. go f is also one-one. g: B ~ C and f
B
9
h: C ~ D be defined by the diagram.
assign its prime
(iii) No. i.
f: A
->
(ii) Find the composition function
hog
0
f. Hence hog
0
a -> 2 -> x -> 4. a == a' since f is one-one. But then (g
0
f)(a)
== g(. there exists an a E A such that f(a) == b.
8. g(f(a» == g(f(a'».17.4). (iv) To each country in the world which has a prime minister minister. 6).
(b. For
a function
is one-one if it assigns
f
assigns r to both a and d.15.
Let the functions
A
f: A ~ B.
g: B -> C is not onto since
z E C is not the image of any element in B. Furthermore. (i) To each person on the earth assign the number which corresponds to his age.
then the composition function
B such that g(b) == c. Then f(a) = f(a') since 9 is one-one. images to different elements in the domain. 8]
FUNCTIONS
75
distinct image values to distinct elements in
Recall that the domain.
Prove: If f : A ~ Band g: B ~ C are one-one functions.18.
y = -1. by definition of equality of column vectors.U)
and
(:t)
The first two vectors have two components.2: Let x+y
-
y)
4
z-1
x.). is the column vector obtained by adding corresponding components: 81
.
Example 1.1:
of the vector u.
Two column vectors u and v are equal. if they have the same number of components and if corresponding components are equal.
un written in a column:
ul u2 u
The numbers ui are called the components
Example 1. whereas the last two vectors have three components. (_. The vectors
mm
and
X (
are not equal since corresponding elements are not equal.
(~)
and z = 4.
The following are column vectors:
(:).y x
Then. ADDITION
Let u and v be column vectors with the same number of components. + v. written u = v.
+y =
z -1
=
2 3 x = 3. denoted by u.
Solving the above system of equations gives
Remark: VECTOR
In this chapter we shall frequently refer to numbers as scalars. The sum of u and v.Chapter 9
Vectors
COLUMN VECTORS A column vector u is a set of numbers ul' u2'
••• .
u have the same number of components. (-D H)
1
The main properties of the column vectors under the operations scalar multiplication are contained in the following theorem:
of vector addition
and
.
Example 2.82
VECTORS
[CHAP.. The next Example shows that the zero vector is similar to th€ number zero in that.1:
The sum of two vectors
(-0 CO
+
The sum
+ ( 1 + 4)
-2 5 3-6
CD
Example 2. .
is not defined since the vectors have different
A column vector whose components are all zero is called a zero vector and is also denoted by o.
-u = -l'u
Example 3. for any vector u. with different numbers of components is not defined. denoted by k .1: 4
We also define:
and
u-v
u
+ (-v)
o
(-D (~D.(-l) CD (-0 m.u. is the column vector obtained by multiplying each component of u by k: u1 u2
k·u le
leu I leu2
Observe that u and k. or simply leu.
Example 2.2:
(-:)
+
U)
numbers of components. u + 0 = u.3: u
SCALAR MULTIPLICATION The product of a scalar Ic and a column vector u. 9
u+
V
Note that u + v has the same number of components as u and v.
then their product. -7. denoted by U· v or simply uni.9J
VECTORS 9. and let u. -4.u or simply ku.
The sum of two row vectors 'U and with the same number of components. the theorem can be restated as follows: Theorem 9. u
+ v.1: (1. 3. v. i. Then:
ll
(i)
(ii)
+ v) + io = u + (v + w). is the scalar obtained by multiplying corresponding elements and adding the resulting products:
.2: The set of all n-component row vectors under the operations of vector addition and scalar multiplication satisfies the properties listed in Theorem 9. -4)
+
(0. k' be scalars. Two row vectors are equal if they have the ~ame number of components and if corresponding components are equal. i. . that is. Accordingly.
(u u
(iii) (iv)
(v)
+0
u
k( u
(vi) (k + k')u = leu + k'u. -4)
We also have a theorem for row vectors which corresponds to Theorem 9. 5. addition is associative. addition is commutative.
=
0
+u
=
u. 3. 7) :::: (5. OF A ROW VECTOR AND A COLUMN VECTOR
MULTIPLICATION
If a row vector u and a column vector v have the same number of components.e. is a vector space. 3.15.
= 0. a row vector u is a set of numbers u1' u2.1:
83
Theorem
Let V be the set of all n-component column vectors.1: The set of all n-component column vectors under the operations addition and scalar multiplication is a vector space. -20)
5· (1. -2.. denoted by k.. ku n ) _
We also define
-u
=
-l·u
and
u. Observe that a row vector is simply an ordered n-tuple of numbers.
Example 4.1. (vii) (kk')u == 1c(k'u).
The properties listed in the above theorem are those which are used to define an abstract mathematical system called a lineal' space or vector space. . ku. wE Vn and let k. u + v = v + u.
un written in a row:
The numbers u.CHAP.
••• .v
=
u
+ (-v)
as we did for column vectors. is the row vector obtained by multiplying each component of u by k:
(leul. -10.. -2. 11) :::: (1. Theorem 9.
+ (-u) = (-u) + U + v) = ku + kv.l..e. of vector
ROW VECTORS Analogously. (viii) lu = u. are called the components of the vector u. denoted by is the row vector obtained by adding corresponding components from u and v:
'1)
u
+v
The product of a scalar 1c and a row vector u.
and (_:). capital letters A.. and a matrix with one column is simply a column vector. one for each pair of elements.-2) and its columns are
In this chapter.. written A = B.e.
Remark:
A matrix with one row is simply a row vector.Chapter
Matrices
MATRICES
10
A matrix is a rectangular array of numbers.
o
-3 5
4) -2 .. denote numbers.w
The solution of the system of equations is x
2. written A + B. the same number of rows and the same number of columns.. . and if corresponding elements are equal. denote matrices whereas lower case letters a. Hence the equality of two m x n matrices is equivalent to a system of mn equalities.2: The statement
X (
+y
x-y
2z
+
z-w
w)
is equivalent to the system of equations x+y x-y { 2z 3 1 5 4
+w =
z. (~). Hence vectors are a special case of matrices. The sum of A and B. i.2 ••• '"
Um2
••••• : •• ~~n. if they have the same shape.1: Consider the 2 X 3 matrix (1
Note that the row and column of the element respectively.e.j)
~2.
Its rows are (1. (-~). y
=
1. w
=
-1.5.
•••
am! (aij)m.-3.
Example 1. Two matrices A and B are equal. b. B. z
=
3. is the matrix obtained by adding corresponding elements from A and B: 90
. which we call scalars. i. .4) and (0.n
arn3
We denote such a matrix by or simply and call it an m
X
n matrix (read "m by n").
aij is indicated by its first and second subscript
Example 1. ADDITION
MATRIX
Let A and B be two matrices with the same shape. the general form of a matrix with m rows and n columns is
all " ( •••
a12
a13
a1n)
amn (a. the same number of rows and of columns.
+ aipbpj.
MATRIX MULTIPLICATION Let A and B be matrices such that the number of columns of A is equal to the number of rows of B.. Then the product of A and B. satisfy the following properties:
(AB)C = A(BC)
(ii) A(B (iii) (B
+ C)
AB BA
+ C)A
+ AC + CA
where k is a scalar.3: (i)
G !)O D
!)
C'1+2'0
3·1+4'0
1'1+2'2) 3'1 + 4'2 1'2+1'4) 0'2 + 2'4
n
1n
C'1+1'3
0'1+2'3
(: ~)
We see by the preceding example that matrices under the operation of matrix multiplication do not satisfy the commutative law. if A = (aij) is an m x p matrix and AB = (Cij) is the m x n matrix for which
If the number of columns of A is not equal to the number of rows of B.. say A is m x and B is q X n where p =F q. then the product AB is not defined. then
In other words. however.
(r s) U
p
Example 4.
.j) is a p x n matrix.2:
(~ ~)G
Matrix multiplication Theorem 10. we also have that A
+A =
2A.
B = (b. is the matrix with the same number of rows as A and of columns as B and whose element in the ith row and jth column is obtained by multiplying the ith row of A by the jth column of B:
e (
where
Cij = ailblj
H
e. i. 10
Using (iii) and (iv) above.1:
t
(al
bl
a2 b2
ag)
bg
(ral + sbl tal + UbI
ra2 + sb2 ta2 + ub2
rag + Sbg) tag + ub3
Example 4. Remark: In the special case where one of the factors of AB is a vector. the products AB and BA of matrices need not be equal. does..")
Cmn
Cml
+a
i2 2j
b
+ . then the product is also a vector:
. written AB..92
MATRICES
[CHAP.e.
(iv) k(AB) = (kA)B = A(kB).
We assume that the sums and products in the above theorem are defined. A
+A +A
3A.
10]
MATRICES
93
(::: :::
.. 10)
U -D(-.D
The numbers along the main diagonal are 1. or simply diagonal.
G =: . such as
+ 2y
5x .4: For any square matrix A. Specifically..6y
. Theorem 10. The unit matrix I plays the same role in matrix multiplication as the number 1 does in the usual multiplication of numbers.3: (2.ann•
Example 5.1: Th. and
SQUARE MATRICES A matrix with the same number of rows as columns is called a square matrix.4: A system of linear equations. of a square matrix A = (aij) is the numbers all. AI IA A
(~
1
oo
o
o1 oo
.
-2
-D
(2'1
+
(-3)' 5 + 4· (-2). A square matrix with n rows and n columns is said to be of order n.a22.
is also a solution to the matrix
equation.)
X {
(
1' (-1) 5'(-1)
+ (-3)'
+ 0'2
2)
(-2)'(-1)
+ 4'2
Example 4.3z
4 8
+ 8z
is equivalent to the matrix equation
That is.g.
The n-square matrix with l's along the main diagonal and O's elsewhere. 4) ( . The main diagonal. ••• . 2' (-3)
+
(-3)' 0 + 4' 4)
(-21.CHAP.
aml
am2
~:)~:) (
•••
amp
bp
Example 4. -3. e. and is called an n-square matrix. any solution to the system of equations vice versa. matrix is a square matrix of order 3.
is called the unit matrix and will be denoted by I. -4 and 2.
However.5:
operation on matrices satisfies the following properties. +
anxn
we define f(A)
to be the matrix
f(A) ao! alA a2A2 anAn
In the case that f(A) nomial f(x).
f(x) ao
That is.
Example 6.
for k a scalar
(iv) (AB)t = B' At. if we only consider square matrices of some given order n.
Example 7. any n x n matrix can be added to or multiplied by another n x n matrix.
an
b«
'"
~:)
c«
Note that if A is an m x n matrix..'. if
g(A)
=
+ 3x
n
=
( 16 -18) 61
-27
10
Thus A is a zero of the polynomial g(x).. A3=A2A..3n
x2
2) -4
-
+
then
5C 0
On the other hand.
CI
C2
• ••
Cn
(~:~:
. + + . in order.
.1:
The transpose Theorem 10. then
A2
3x
+
5. then At is an n X m matrix. In particular. and the result is again an n X n matrix.1:
is the zero matrix. we can form powers of A:
A2=AA. .
TRANSPOSE The transpose of a matrix A. as columns:
al ( a2 an)t
~~
~~ •• ~2 . or multiplied by a scalar. (i)
(A
+ B)t =
At
+ B'
(ii) (At)t (iii) (kA)t
=A = kAt. if A is any n-square matrix. . is the matrix obtained by writing the rows of A. then A is said to be a zero or root of the poly-
Let If
A=G f(x)
-~)
=
2X2 -
... for any polynomial
+ +
alx
+ +
a2x2
+ . written At. then this inconvenience disappears. 10
Recall that not every two matrices can be added or multiplied... Specifically.
then
f(A)
2(
-9
g(x)
-6)
22
.
and AO=!
We can also form polynomials in A.
7
(-~-6)
22
.94 ALGEBRA OF SQUARE MATRICES
MATRICES
[CHAP. | 677.169 | 1 |
Course description: One day in 2580 B.C., a very serious architect stood on a dusty desert with a set of plans. His plans called for creating a structure 480 high, with a square base and triangular sides, using stone blocks weighing two tons each. The Pharaoh wanted the job done right. The better our architect understood geometry, the better were his chances for staying alive.
Geometry is everywhere, not just in pyramids. Engineers use geometry to bank highways and build bridges. Artists use geometry to create perspective in their paintings, and mapmakers help travelers find things using the points located on a geometric grid. Throughout this course, we'll take you on a mathematical highway illuminated by spatial relationships, reasoning, connections, and problem solving.
This course is all about points, lines and planes. Just as importantly, this course is about acquiring a basic tool for understanding and manipulating the real world around you.
Scope & Sequence:
Module 1
Checklist, Motif and Course Information, Pretest Module 1, Building Blocks of Geometry, Mathematical Language, Segments, Segments and the Coordinate Plane, Quiz, Exploring Angles, Angle Addition Postulate and Bisectors, Angle Relationships, How to Use a Compass, How to Use a Protractor, Survey, Review, Module 1 Post Test | 677.169 | 1 |
Calculus: Concepts & Connections
Browse related Subjects ...
Read More balanced guidance on the appropriate role of technology in problem-solving, including its benefits and its potential pitfalls. Wherever practical, concepts are developed from graphical, numerical, algebraic and verbal perspectives (the "Rule of Four") to give students a complete understanding of calculus | 677.169 | 1 |
Category: Family
Algebra y trigonometria dennis zill pdf
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When you launch Theme Calendar - Anything Goes Jokes, you'll see a two-paned interface; the top agebra displays the current and next month, while the bottom features the joke of the day. | 677.169 | 1 |
Showing 1 to 10 of 10
Numerical Methods
5.
Numerical Methods
5.1
Numerical Integration
There are some functions in which it is impossible to find the exact values of a
definite integral. For example, it is impossible to evaluate the following integrals
exactly.
1
i)
1
2
e x dx
PartialDerivatives
3.
PARTIAL DERIVATIVES
3.0
Function of Several Variables
A function of a single variable y = f(x) is interpreted graphically as a planar curve.
In this section we shall see that a function of two variables z = f(x,y) can be
interpreted
DOUBLE INTEGRALS
4.
DOUBLE INTEGRALS
4.1
Review of the Definite Integral
b
The area under the curve y = f(x) between x = a and x = b is given by
f (x) dx . This is
a
illustrated by the figure below
CJHL2016
Page 1
DOUBLE INTEGRALS
4.2
Iterated Integrals
FOURIER SERIES
_
CHAPTER 6:
6.0
FOURIER SERIES
Introduction
Fourier series is an infinite series representation of periodic functions in terms of the
trigonometric sine and cosine functions. Fourier series have many applications in Mathematics,
in physics
MATRICES
1.
1.1
MATRICES
Definition
A rectangular array of numbers is called a matrix (the plural is matrices), and
thenumbers are called the entries of the matrix. Matrices are usually denoted by
uppercase letters: A, B, C,and so on. Hence,
1 2 3
A=
, | 677.169 | 1 |
Visual Mathematics via Computer-based
Guided Active Discovery
Abstract.
Conventional wisdom has it that effective use of Mathematica and the
internet involves copying standard text materials into Mathematica
notebooks and distributing them on the internet. The main result of
this approach is to freeze the current mathematics curriculum and bog
it down with all the other ineffective baggage commonly found in
conventional mathematics teaching.
This talk will survey ways currently in use at Illinois of using
Mathematica notebooks and the internet to modernize the content and
the style of mathematics teaching chiefly by replacing blackboard (or
PowerPoint) with student generated interactive visualizations that
are used to cement ideas before the words go on. Instead of
memorizations, students get to play with a concept thus forming their
personal view of the issue at hand through guided active discovery.
One important aspect to this approach is the idea of student internet
mentors who take new students under their wings and nurture their
mathematical awakening.
This talk will also look at some of the interactive computer-based
lessons and will look at the way the program is currently administered. | 677.169 | 1 |
This book totally helps!
Being a mathophobe, I was terrified when I registered for the class.But since I had to buy this book for my math class, then math all of a sudden was really easy!I actually understood the WORD problems with ease and what not!!The only thing that keeps me from giving it a 5 star rating is that there are many typos in the book, especially in chapter 6.2:there are at least three typos in this specific area if and when you are assigned the odd problems; and what's more, if you want a good laugh, check out pages 455-456.You'll see what I mean!
Buy this book now!
I am a 34 year old student that has struggled with math my entire life. I have failed many math classes in college because I just did not clearly understand how to do the math problems. I enrolled at Harbor College in Willmington, CA this fall 2003 and purchased this required book. To my amazement, this book is written in a way that a dummie like me can easily learn all the lessons. I now have a class grade of 95 and FINALLY because of this book, I can do algebra word problems, and I understand how to check them. I am even tutoring some of my classmates, WOW! thats not too bad for a dummie! I thank the author for creating the best prealgebra book I have ever seen. It is written STEP-BY-STEP and in the most basic format that can be explained. Please only purchase this book. You will not be dissapointed and you will pass prealgebra. All thanks to K. Elayn Martin-Gay, what a job you did putting the book together!
Substantial, and Rich exercises
This workbook is packed with a lot of examples and rich exercises; which are presented in incremental development fashion.My two children, 10 and 11 year old, have high math aptitude; two grades beyond their average level. I bought this workbook to help them in the transition to algebra and trigonometry.Theyare able to go thru most of these execises in one summer.I must admit the quality of the lessons and exercises have this quite enjoyable and intuitive for my children to absorb.
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useful text for an undergrad course
Hilariously, I found this book in a local library, in the section devoted to primary school and high school maths. I guess the "Algebra" in the title suggested this location to some librarian. Anyhow, not to worry. I informed the library and they will reshelve this book in a better place.
Excellent for an introductory abstract algebra book
This is a great book.It's introductory, appropriate for undergrads taking abstract algebra for the first time, but at the same time, it is also very comprehensive, useful for graduate students.Although it explains the material in great depth and at a relatively slow pace, it does so in a logically sound manner so that it can lead the reader to develop mathematical maturity.Like many more advanced books, it has good cross-references, helping people to develop connections between the different parts of the material, and allowing the reader to jump around once she has mastered the basics.
too wordy
This book is a standard one for graduate-level algebra courses.I practically wore mine out over a year-long course, and came to know it intimately.Dummit and Foote is a book that teaches via wordy explanations and lots of examples.Of course, examples are very important.However, the explanations are often muddled and not clear (e.g. see tensor products).They frequently relegate important theorems or definitions to the exercises, and the organization is poor.Consequently, it can be very hard to find things later when you might need them.Also, the bindings on this book frequently fail.My book fell apart very quickly, and I know other students who had the same problem.I recommend Rotman's Advanced Modern Algebra instead of Dummit and Foote.
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Excellent Math Textbook
This is one of the best math textbooks I have come across. I used this book for two graduate level courses in Linear Algebra and found it wonderful. Strang divides Linear Algebra into 2 halves - the first half revolves around the solutions to the equation Ax=B and includes vector spaces and orthogonality. The second half is about eigenvalues and positive definiteness. The two halves are connected together by determinants. The book is very informally written and the author places emphasis on intuitive understanding rather than just proofs and equations. Highly recommended along with Strang's video lectures.
Excellent Linear Algebra book to supplement Prof Strang's Video lectures
Well, this book has a peculiar feature. It seems to be a good book when read alone. But if one reads this book along with Prof. Strang's video lectures from OCW MIT, it becomes the best book for linear algebra. The best way to learn a new subject is to learn from an authentic source. I strongly believe in the complete authenticity of this book to enrich Linear Algebra knowledge in the reader. This book is an exceptional resource for any engineering student irrespective of the discipline. I wish Prof. Strang writes more books on related areas and supplements them with his thoughtful video lectures.
This book is worth the Wronskian of two linearly dependentfunctions
I am sure he is an amazing mathematician, but I found this book to be the most horrible linear book I have ever seen. The text that was bolded and italicized was inconsistent and therefore very confusing especially in a book which employs bold face to denote specific mathematical qualities. Furthermore, the conversational style and lack of fluidity had a synergy that would drive even the craziest mathematician crazier. I would reco Lay's or Pooles or Bronson's (this might be my favorite because it is proof intensive). Again, he is a better mathematician than I will ever be, even if he was wasted, but I would not reco this book to any first semester linear student, go for one of the aforementioned authors.
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A classic
In certain ways, this book has been both a bane and a boon to my career as a computational mathematician.Way back in 1989, I had the mixed experience of taking a course in Numerical Analysis from Brian Smith at the University of New Mexico.Prof. Smith taught that course exclusively from this book (actually, from the 2nd edition).As a college sophomore, I was terribly out of my depth, but I managed to do okay.Later, I had the opportunity to study under Gene Golub at Stanford, although I was certainly not one of his better students :)Naturally, Prof. Golub also taught pretty much exclusively from this book, by the way, he is a gifted mathematician and wonderful instructor, and a real gentleman.Between these experiences, I'd say I became extremely familar with the contents of this book.
The standard reference
First, this isn't Numerical Recipes. If you're looking for cut&paste code, you're just looking in the wrong place. This is for people who need a deep understanding of the computational issues, and are going to put a lot of time into an implementation. It's for people who are completely at ease with linear algebra, standard matrix-oriented problems, and dense mathematical notation.
Misbound
I ordered this book new but it was misbound, apparently during publication. In my first order, the book was missing the end of Chapter 12, had two indices, and had the final references buried in the middle of another chapter; the replacement book from Amazon had similar faults -- the end of chapter 12 was again missing, there was no index, and chapter four was in the place of the end of chapter 12.
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Beautiful Scaffolding
Linear algebra is the Babel language of sciences, this book helps you to get proficient in speaking it. Clear-cut presentation, mathematical rigor and historical gossip is in such an unison, that you'll crave for a sequel. This is the 18-th 5-star rating, you shouldn't hesitate any more!
One of the best introductory (modern) books I know.
If you need to learn Linear Algebra and Matrix Theory, your best starting point is this book.
a must buy...
Although titled as Matrix Analysis and Applied Linear Algebra, this book is one of the clearest treatments of pure linear algebra in general. Most of the theorems are proved and the proofs are very well motivated. There are no hand-waving arguments yet it is very easy to follow all the material contained. I also noticed that this book provides a smooth transition to introductory functional analysis for those who already possess the adequate real analysis background.
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Worry free service
Not too long after getting the confirmation that my payment went through ok, I got an email from Raerae313 letting me know that she would be shipping out the book I ordered.Not many sellers will let you know if they ever received payment for it.You are left worrying until the book finally shows up at your door step. It was a great experience and I received her book long before any of the others that i had purchased on the same day.I will look forward to buying from her again if she has anymore books I need.
An unnecessary revision to previous editions.
This is an easy to use book, but I've checked the third and fourth editions, and except for different equations, the content is the same. My instructor said it made no difference which edition I used, so I went withthe third one.
... Read more
Came in a good condition and very timely shipping
Came in a good condition and very timely shipping
Best math textbook I ever used!
After going back into this book more and more I can only revise my opinion upward. Sullivan covers almost every topic before calculus and does it in a very straightforward manner. The text has rigor proving many important statements but this is never used in sacrifice of understanding. The student solution manual is also great and highly recomended. A great text and reference book.
... Read more
Try something else
I had this book for my Statistics I class. I've read every bit of the instruction on the book (I read everything not just what's on homework). At the end I realised I learned nothing I was still confused. Unless you have a good instructor to follow you through every step, you should skip it. This is NOT something for at home students.
Clear and well-organized, but compromised
Triola's book is, for the most part, an excellent choice for an intro stats course. As an instructor, I find it relatively easy to work with, and the included STATDISK gives students many opportunities to analyze large sets of data without having to enter hundreds of values into calculators or computers. It also contains a lot of examples taken from actual data sets; this is the text that will deflect that ubiquitous "what's this useful for in real life" question from students. A few issues, though, dog the book. In order of importance:
Outstanding Textbook
I teach statistics at 2 colleges.I use this book.It is outstanding in its clarity and mathematical depth. Students with strong math backgrounds will appreciate the fine logic developed by this book as it explains statistics.The word problems are outstanding and relevant to applications acrossbroad fields of interests.The examples are explained thoroughly. This book is very well priced for the student who is struggling to pay tuition, fees, etc.Highly recommend this book to all serious students.
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I have used this book for College Classes.
This book is OK. The graphics are great but the concepts are not layed out in the best possible format. Several times I got confused at what the book was saying and had to take it to my teacher. If you have a good teacher though, this is a great book.
good stuff
Very good condition, and the main thing is i saved $83 thru here as opposed to actually buying it for a lot more at the bookstore. I was able 2 get this in 3 days which was not bad.
great service
I got the book in 3 days, as promised. Great service. Would recommend it. Thank You.
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Not for the uninitiated
I bought this book hoping to learn about matrix analyis. I did not. Thisbook is simply a reference manual with plenty of theorems, axioms etc. withlittle explanation. They give it to you rough and row. NOT A SINGLE SOLVEDEXAMPLE, and not evensolutions for the exercises given in the book areprovided. If you intend to learn about matrix analysis, as I did, let notthe 5 stars review mislead you. Don't make the same mistake, this book isnot for you.
An encyclopedic reference for matrix analysis and linear alg
Horn and Johnson's MATRIX ANALYSIS is simply a masterpiece. You can findeach and every result in matrix analysis along with it's proof in thisbook. Look at their companion volume "Topics in Matrix Analysis"too. Some of these results cannot be found elsewhere.
Excellent book.... for the initiated
Horn and Johnson has written an excellent reference book onsomewhat-advanced linear algebra (from the point of view of an engineer).There's a lot of treasures in this book, but this book is NOT for beginninglinear algebra. Rather, it is written as a handy reference to review andlearn certain topics in linear algebra.
A real page turner.One of the most exciting works today.
The plot is intricate, and the mystery deepens until the last few pages of the text.The authors richly deserve the highest praise for their tireless efforts.
... Read more
The way to learn algebra
Most critisism of the book are based in the fact that topic are not treated in deep, and the reason is that there is no need to do that. Lang's porpouse is to introduce the reader to HIGHER ALGEBRA, while for example Dummit & Foote just end in Category Theory or Homological Algebra. He just introduce what he consider necessary to know about groups. Of course, if your porpouse is to learn, let's say Group Cohomology, then its good for you to know as much as possible about groups, modules and stuff, but Lang's try to focus on what HE consider is important to be known.
4pro
It is one of the brilliant classics, must have for every advanced algebraist. Probably you will not find here such nouveau labyrinths of mathematical thought as in Beta Algebra by Algirdas Javtokas (which in my opinion is a revolutionary book), but it is unvaluable for seminars or supplementary material for algebra courses.
This book grows on you.
When I examined this book as an undergraduate I did not like it; often this is a sign that a book is poorly written, but in this case I just needed more background.Now I see this text as a gold-mine: clearly written, provocative, and rich in examples.
Read more | 677.169 | 1 |
Transformation - A Fundamental Idea of Mathematics Education
Transformation - A Fundamental Idea of Mathematics Education The diversity of research domains and theories in the field of mathematics education has been a permanent subject of discussions from the origins of the discipline up to the present. On the one hand the diversity is regarded as a resource for rich scientific development on the other hand it gives rise to the often repeated criticism of the discipline's lack of focus and identity. As one way of focusing on core issues of the discipline the book seeks to open up a discussion about fundamental ideas in the field of mathematics education that permeate different research domains and perspectives. The book addresses transformation as one fundamental idea in mathematics education and examines it from different perspectives. Transformations are related to knowledge, related to signs and representations of mathematics, related to concepts and ideas, and related to instruments for the learning of mathematics. The book seeks to answer the following questions: What do we know about transformations in the different domains? What kinds of transformations are crucial? How is transformation in each case conceptualized?
The principle of Transformation is used to shape discussions of mathematics theoryDiscontinuity in mathematics education is addressed and real world examples given to solve this problemTeaching and learning geometry provides a context for understanding transformation in mathematics education | 677.169 | 1 |
This course begins with a review of the topics covered in the first semester of basic mathematics which include basic operations and properties involving whole numbers. Students are then introduced to fractions, operations with fractions and decimals, and problems involving money and percentages. Word problems are used extensively in this course to relate concept to practical situations. Students practice each concept in a step-by-step manner before moving on to more complex topics.
Online course content is included in tuition for all courses except those designated as "PRINT ONLY".
Tuition (MTHH002256MTHH002256) [Add $35.00]
Summary
Course Price $250.00
Tuition (MTHH002256)$200.00
Administrative Fee$50.00
Step 2:
Assign or Create a Student Below And Your Course Will Be Ready To Purchase! | 677.169 | 1 |
This online course is designed for the prospective secondary mathematics teacher who needs to pass Mathematics Subtest II to be able to comply with California requirements. It is not a traditional "course," in the sense that it will not substitute for postsecondary coursework. Rather, it is a resource to help you identify specific gaps in your understanding. Each lesson corresponds to a California Subject Matter Requirement and each topic corresponds to an individual or group of California K-12 Content Standards for Mathematics. The CSET exam is itself is tied to these standards. Each lesson follows the same pattern:
Diagnostic Questions - Diagnostic questions allow you to rapidly ascertain your own level of comfort with the topic matter. If you find them to be easy, and you correctly answer the questions, you can move on to the next topic. In this manner, you will only spend time on areas where you need to strengthen your understanding.
Guided Examples - The guided examples are a second level of help. If you answered the diagnostic questions correctly, but have doubts as to the depth of your understanding, these step-by-step solutions can help. Following this example, you can try answering a new diagnostic question. If you just needed something to jog your memory, this may be sufficient and you can move to the next lesson.
Content "Refresher" - The "refresher" is designed to provide just-in-time instruction to reinforce your understanding, particularly if you have struggled with the diagnostic questions. These screens provide further examples and opportunities for practice along with an explanation of the topic. Another diagnostic question follows the refresher. These refreshers are necessarily brief and far from inclusive of the full depth of the area of mathematics. However, it may be enough for someone who previously learned the material through coursework and needs to brush up before taking the CSET exam.
One final note: No online resource can really substitute for a full course, whether online or in the classroom. UCI Extension provides an online course based on these open materials in which an instructor works with you to analyze and strengthen your individual understanding of the topics covered by the CSET exam. For more details, please see our website at extension.uci.edu. | 677.169 | 1 |
Rationale
Rationale/Aims
Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe much of the modern world. Statistics is concerned with collecting, analysing, modelling and interpreting data in order to investigate and understand real-world phenomena and solve problems in context. Together, mathematics and statistics provide a framework for thinking and a means of communication that is powerful, logical, concise and precise.
The major themes of Mathematical Methods are calculus and statistics. They include as necessary prerequisites studies of algebra, functions and their graphs, and probability. They are developed systematically, with increasing levels of sophistication and complexity. Calculus is essential for developing an understanding of the physical world because many of the laws of science are relationships involving rates of change. Statistics is used to describe and analyse phenomena involving uncertainty and variation. For these reasons this subject provides a foundation for further studies in disciplines in which mathematics and statistics have important roles. It is also advantageous for further studies in the health and social sciences. In summary, the subject Mathematical Methods is designed for students whose future pathways may involve mathematics and statistics and their applications in a range of disciplines at the tertiary level.
For all content areas of Mathematical Methods, the proficiency strands of the F-10 curriculum are still applicable and should be inherent in students' learning of this subject. These strands are Understanding, Fluency, Problem solving and Reasoning, and they are both essential and mutually reinforcing. For all content areas, practice allows students to achieve fluency in skills, such as calculating derivatives and integrals, or solving quadratic equations, and frees up working memory for more complex aspects of problem solving. The ability to transfer skills to solve problems based on a wide range of applications is a vital part of mathematics in this subject. Because both calculus and statistics are widely applicable as models of the world around us, there is ample opportunity for problem solving throughout this subject.
Mathematical Methods is structured over four units. The topics in Unit 1 build on students' mathematical experience. The topics 'Functions and graphs', 'Trigonometric functions' and 'Counting and probability' all follow on from topics in the F-10 curriculum from the strands, Number and Algebra, Measurement and Geometry and Statistics and Probability. In Mathematical Methods there is a progression of content and applications in all areas. For example, in Unit 2 differential calculus is introduced, and then further developed in Unit 3 where integral calculus is introduced. Discrete probability distributions are introduced in Unit 3, and then continuous probability distributions and an introduction to statistical inference conclude Unit 4.
Aims
Mathematical Methods aims to develop students':
understanding of concepts and techniques drawn from algebra, the study of functions, calculus, probability and statistics | 677.169 | 1 |
What is maths?
Is it numbers? Is it algebra? Is it trigonometry? Is it statistics? The list could go on.
A better question would be "what makes a good mathematician?"
A good mathematician has the ability to reason their way through complex problems, finding the most efficient methods and working out what maths tools are needed.
During their maths lessons we help students build up the tools needed as well as giving them opportunities to develop and improve their problem solving skills. Appreciating that you will not always get things right first time is important; often when improving your maths skills as much can be learnt from a wrong method or solution as it can from solving the problem first time. Resilience and perseverance are essential traits to develop so that you are not tempted to give up at the first hurdle.
I am often asked what careers a maths qualification will lead on to. Not an easy question to answer but it is fair to say that people with good maths skills and qualifications are sought after because of their good problem solving skills. Colyton students who have studied maths at A level have gone on to careers in engineering, finance, architecture, computing and teaching, to name but a few. There are less obvious career routes e.g. business management, medical research, that employ maths graduates for their problem solving skills and logical minds.
As a department we are all fascinated by our subject. We derive pleasure from encouraging our students to persevere with their maths in the hope that it will open up avenues that they may wish to follow in the future.
- Mr Davis, Head of Mathematics
WHO WE ARE
Mr DavisHead of Mathematics
Teaching is not the path I have always trod. I studied Electrical/Electronic Engineering as part of a sponsored undergraduate course with Westland Helicopters in Yeovil. From there I moved to a very small company involved in security electronics. In 1995, after returning to university and completing a PGCE in maths, I started my new career as a maths teacher and for the past 18 years have been working at Colyton. Similar to engineering, I enjoy the fact that as a teacher no two days are the same and there is always more to learn. I am still developing as a teacher and I am continually finding new areas of maths that I have not come across before. Success for me is seeing students enjoying their maths learning and then going on to use their maths skills as part of their future learning and careers.
Mr ReadAssistant Head of Mathematics
I am very proud to be a maths teacher at Colyton Grammar School. In my early working life I trained in chemistry before moving on to work in the formative years of computing in the Health Service. I was a late entrant into Exeter University where I completed my degree and then gained my first teaching post at a school in North Devon. After just one year I learned of a vacancy at Colyton and was fortunate enough to be successful in my application. That was in 2001. While the subject of mathematics itself is fairly unchanging we at this school are encouraged to teach the subject in a variety of ways and styles which keep the students (hopefully) and me always challenged. I am as keen to learn from the students as I am to teach them and this ensures that no two lessons are the same. | 677.169 | 1 |
SAT Power MathIn our number-crunching world, basic math knowledge is a must—especially for acing tests like the SAT. For many people, though, math is confusing and often anxiety inducing. That's why we've created SAT Power Math, which uses a simple, straightforward approach to break down and explain complicated math concepts and common problems. This book is your powerful tool for building essential math skills for the SAT, school, and beyond. Everything You Need to Help Achieve a High Math Score. · A comprehensive review of math topics like algebra, geometry, and statistics · Strategies for cracking the most common question types found on the SAT · A glossary of key math terms at the end of every chapter Practice Your Way to Perfection. · Practice drills for every math topic covered in the book · Detailed step-by-step answer explanations · Targeted strategies to help you score high on the math section of the SAT
About the Author:
The experts at The Princeton Review have been helping students, parents, and educators achieve the best results at every stage of the education process since 1981. The Princeton Review has helped millions succeed on standardized tests, and provides expert advice and instruction to help parents, teachers, students, and schools navigate the complexities of school admission. In addition to classroom courses in over 40 states and 20 countries, The Princeton Review also offers online and school-based courses, one-to-one and small-group tutoring as well as online services in both admission counseling and academic homework help.
"About this title" may belong to another edition of this title.
Bibliographic Details
Title: SAT Power Math
Publisher: Random House Information Group
Book Condition: Good
Book Description Princeton Review, 2014. Book Condition: Good. Ships from Reno, NV. Former Library book. Shows some signs of wear, and may have some markings on the inside. Bookseller Inventory # GRP92392390
Book Description Princeton Review. Paperback. Book Condition: VERY GOOD. Very Good copy, cover and pages show some wear from reading and storage. Binding may have light creases. Lots of life left in these pages. Bookseller Inventory # 2755796934
Book Description Princeton Review. Paperback. Book Condition: GOOD. book was well loved but cared for. Possible ex-library copy with all the usual markings and stickers. Some light textual notes, highlighting and underling. Bookseller Inventory # 27963860H8F542939
Book Description -. Paperback. Book Condition: Very Good. SAT Power Math (College Test Preparation804125925 | 677.169 | 1 |
Book Review: Pattern Recognition and Machine Learning
It's a heavy read on machine learning algorithms in that it is very math intensive. It would not be too far off to call this an applied math textbook rather than a computer science textbook, and it is definitely aimed at the graduate level student with a good amount of mathematical maturity.
Having said that, if you are a machine learning researcher in need of learning all sorts of mathematical details of the algorithms you are using, this book is a good place to start. It is by no means a complete reference, but certainly a good introduction at the graduate level.
If you are an undergraduate student, there are other textbooks available that might be more suitable (such as Artificial Intelligence: A Modern Approach, which I also reviewed). If you just want cookbook style text on the libraries you could use to employ machine learning algorithms, there are probably other resources that would be more suitable.
If you are starting out in researching machine learning algorithms, or are using it as a large part of your computing science research, this is the right book. Just be prepared to learn and use a lot of math! | 677.169 | 1 |
Course description:
We will study the topics of graph theory and enumeration in order to gain experience with combinatorial problem solving.
Along the way, we will develop skill in solving unfamiliar problems with creative insight and logical reasoning.
Some of the highlights of the syllabus about graphs are paths, circuits, trees, matchings, planarity, coloring.
For counting problems, we will cover everything from binomial coefficients to generating functions.
This course has important "real world" applications, especially in engineering, computer science, operations research, and statistics.
Tentative syllabus
Detailed information on projects
Sample quiz questions
Prerequisite:
MCC160 or equivalent experience with calculus I.
Homework:
Homework is the most important part of this class.
Doing lots of homework problems is crucial for doing well in this class.
Some homework problems will help you learn the material and demonstrate
this knowledge. Other problems will involve experiments and open-ended investigation.
The process of doing homework will enable you to do well on the tests.
Homework is due every Friday four in-class quizzes on the following Fridays:
9/14, 10/5, 10/26, 11/16.
There will be a final examination Thursday 12/13 11:50 am -1:50 pm and Thurs 1-2 or by appointment. | 677.169 | 1 |
Lessons for Polynomials
Page 1 of 232 results
Who am I? Find A Polynomial From Its Roots - … apply theorems concerning the roots of polynomials and factors of polynomials. Students will perform operations with polynomials including multiplication and addition. Students will … apply theorems concerning the roots of polynomials and factors of polynomials. Students will perform operations with polynomials including multiplication and addition. Students will … in a computer lab to explore higher order polynomials and roots. (Polynomial and Linear Factors) Polynomial and Linear Factors9.)Students can use the … User Rating:
Discover the Roots of a Polynomial Function - … Roots of a Polynomial Function Title: Title: Discover the Roots of a Polynomial Function Discover the Roots of a Polynomial FunctionOverview/Annotation: In … the function defined by the polynomial. [A-APR3] MA2015(9-12) Algebra II17. Identify zeros of polynomials when suitable factorizations are available … Solving Polynomial FunctionsReal-World Application of Solving Polynomial FunctionsStudent will search the Internet for the following:*How does the knowledge of polynomial functions … User Rating:
Polynomial Subtraction - … questions and then demonstrate how to play Polynomial Battleship. 4.) Allow students to play Polynomial Battleship for the remainder of the class … :32202 Title: Polynomial Subtraction Title: Title: Polynomial Subtraction Polynomial SubtractionOverview/Annotation: Students will review the meaning of vocabulary relevant to subtracting polynomials such as … questions and then demonstrate how to play Polynomial Battleship.Polynomial Battleship4.) Allow students to play Polynomial Battleship for the remainder of the class … User Rating:
Introduction to Polynomials - … introduction to polynomials activity To learn more about polynomials go to the 4th degree polynomials activity at ExploreLearning.com. Elements of polynomials Have the … in this example. Classes of polynomialsPolynomials are divided into two classes odd and even. Even polynomials are polynomials that have an even degree like 3x2 + 3 or –6x42 + 2x – 8. Odd polynomials are polynomials that have an … User Rating:
Taming the Behavior of Polynomials - … test for polynomial end behavior. able to identify real zeros and their multiplicity of factored polynomials. able to rewrite a polynomial in factored form using basic factoring techniques. able to sketch the graph of a polynomial in … how this can be used to help graph this polynomial. Since the polynomial is degree 3, at most we can have 3 … User Rating:Grade Level: 9-12
Shenandoah University - … the word Polynomial beside it. Today's lesson is about polynomials and how to add, subtract and multiply them. • A polynomial is a … of polynomials and adding/subtracting polynomials. Allow 6 minutes. This sheet contains 3 problems each in finding degrees of polynomials, adding/subtracting polynomials using the distributive property, and adding/subtracting polynomials by adding/subtracting coefficients. This … User Rating: | 677.169 | 1 |
Textbook
Transition to Higher Mathematics, by Bob A. Dumas and John E. McCarthy.
This is available at the Columbia Bookstore, on amazon.com, and in the Mathematics Library.
Content
There
are three major goals for this class. The first goal is to teach
you how to reason like a mathematician. The second goal is to
teach you how to translate your reasoning into a clearly written
proof. The third goal is to give you a glimpse of some advanced
mathematical topics (which will hopefully stir your enthusiasm for
mathematics)! Learning how to write a correct proof is difficult,
and requires much practice. But you will see steady improvement,
and after taking this course you should feel considerable "ownership"
of the mathematics you have learned.
The actual topics covered correspond to almost all of Chapters 1-7 of
the book, with an additional smattering of topics from Chapters 8 and
9. This is somewhat ambitious for one semester, and if we end up
not covering absolutely everything, that is fine.
Expected Background: There are no formal
prerequisites beyond high-school algebra and some exposure to
calculus/limits (Calculus I is more than enough). The biggest
requirement is mathematical curiosity and the willingness to think hard
about problems that are not necessarily straightforward.
must be STAPLED or PAPER CLIPPED (no folding over the top-left corner
or anything like that). Please write in paragraphs, sentences, and
English
words (oh my!) when they are called for. The TA should not have to
decipher what you are doing--you should be clear and
unambiguous about your methods on a homework problem. You may
work on the homework in groups (in fact, I encourage this), but your
writeup must be done by yourself. Please write down the names of
classmates you have worked with on the homework.
Each homework assigment will have a problem or two marked as "bonus"
problems. You may choose either to do these and hand them in, or
to give an oral presentation (see below). Even if you choose to
give a presentation, feel free to discuss and attempt the bonus problems!
Keep in mind that, unlike in other math classes you have taken, you will often not
be able to solve a homework problem simply by a straightforward
application of what has been covered in lecture. Homework will
require some thought and ingenuity!
Homework will be graded thoroughly (perhaps more thoroughly than you
are used to)! Every effort will be made to hand it back
promptly. Grades will be posted on Courseworks.
Oral Presentation
Instead of doing the bonus homework problems throughout the
semester, you may instead choose to make a 10-15 minute oral
presentation to me (privately) toward the end of the semester, where
you will present some proof on the blackboard. More guidelines on this will follow toward the middle of the semester Wednesday, May 11th, from 1:10PM-4:00PM.
Calculators are not permitted on exams.
Final Course Grades | 677.169 | 1 |
ISBN-10: 0321881249
ISBN-13: 9780321881243—this format costs significantly less than a new textbook. The Sullivan/Struve/Mazzarella Algebra program is designed to motivate students to "do the math"— at home or in the lab—and supports a variety of learning environments. The text is known for its two-column example format that provides annotations to the left of the algebra. These annotations explain what the authors are about to do in each step (instead of what was just done), just as an instructor would do | 677.169 | 1 |
Course Description
Intended for future elementary school teachers. Topics include basic set theory, elementary
number theory, numeration, number systems and operations, and problem solving techniques
associated with the real number system.
Units: 3
Degree Credit
Letter Grade Only
Lecture hours/semester: 48-54
Homework hours/semester: 96-108
Prerequisites: MATH 120 or MATH 123 or appropriate placement test scores and other measures as appropriate,
or equivalent. | 677.169 | 1 |
Mathematics in Context (Core Maths)
There are four main topic areas studied in Core Maths – Applications of Statistics, Linear Programming, Probability and Sequences and Growth.
The aim of the course is to develop your mathematical understanding while also applying Maths to a variety of areas of real life, including finance and interpreting data.
Some of the mathematical content of the course is the same as GCSE Maths, but the focus is now on applying these methods and ideas to real life situations.
Statistics – collecting, organising, analysing, interpreting and presenting data.
Linear Programming – achieving the best outcome, considering conditions which can be modelled with linear relationships.
Probability – quantifying the risk or chance of events occurring.
Sequences and Growth – looking at real life sequences such as finance, population and natural phenomena such as earthquakes.
Unit 1 – 'AS'
Comprehension Based on the source booklet, you will mostly be expected to show understanding of the sources, using your developed mathematical skills. Unit 2– 'AS'
Applications Based on the source booklet, you will mostly be expected to apply techniques and concepts learnt during the year to specific situations.
Assessment Methods / Teaching Methods and Resources
100% of the course is assessed by a final exam. 40% is the comprehension paper and 60% is the applications paper. A source booklet will be provided one week before the exam, on which the questions will be based. Questions will be focused on applications and use of mathematics rather than the theory of mathematics.
Suitability for Combination
This course is intended for students wishing to further their education in Maths without the algebraic rigour of the AS Mathematics course. It would support students studying Biology, Business, Psychology or Geography, amongst others. It is also part of the Technical Baccalaureate.
Progression to Higher Education / Vocational Destinations
This course is not sufficient on its own to provide a pathway to Higher Education, but by developing mathematical understanding, it should support other qualifications with mathematical content.
If students show significant ability on this course in year 12, they will be considered to study an AS in Maths in year 13 | 677.169 | 1 |
Just-In-Time Algebra and Trigonometry for Students of Calculus (3rd Edition)
Author:Guntram Mueller - Ronald I. Brent
ISBN 13:9780321269430
ISBN 10:321269438
Edition:3
Publisher:Pearson
Publication Date:2004-10-03
Format:Paperback
Pages:224
List Price:$27.60
 
 
Sharp algebra and trigonometry skills are critical to mastering calculus, and Just-in-Time Algebra and Trigonometry for Students of Calculus is designed to bolster these skills while you study calculus. As you make your way through your calculus course, Just-in-Time Algebra and Trigonometry is with you every step of the way, showing you the necessary algebra or trigonometry topics and pointing out potential problem spots. The easy to use Table of Contents has the algebra and trigonometry topics arranged in the order that you'll need them as you study calculus. So forget hunting around for what you need. It's all right here in the order that you'll use it. | 677.169 | 1 |
This
This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. | 677.169 | 1 |
Mathematics and Economics: Connections for Life, 6-8 is a set of 12
lessons that demonstrate how mathematical processes and concepts may
be applied to the study of economics and personal finance. In this volume,
mathematics educators will find lessons connecting mathematics instruction
to practical problems and issues that students will encounter throughout
their lifetimes.
In the study of these problems and issues, economics and mathematics
are natural intellectual allies. Economics is the study of people's
attempts to make good decisions in an uncertain world endowed with limited
resources. The tools that economists use gain power, elegance and visual
appeal as they are represented mathematically in models. Indeed one
might think of economics as "first quadrant math" because
economic magnitudes only rarely take on a negative sign. Research indicates
that students with a strong background in mathematics are more likely
than others to succeed in introductory college-level economics classes.
However, the lessons presented here are not designed solely for use
with students who are college-bound. These lessons are designed to provide
economic skills and knowledge that all students will use as savers,
investors, consumers, producers, and informed citizens. | 677.169 | 1 |
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This guide will help students understand quadratic equations. Each objective contains examples and notes that both the students and teacher can utilize. The guide begins with identifying quadratic equations and its many forms. Students will learn to identify the crucial part of quadratic functions by analyzing graphs of quadratic functions. The simple rules for factoring quadratic equations will serve as an aid for struggling students. Students will solve quadratic equations using various methods including the quadratic formula. There is a summary and an assessment at the end of the unit.
Prerequisite:
I have mastered the solving multistep linear equations
I have mastered graphing points on a coordinate plane
I have mastered the distributive property | 677.169 | 1 |
Algebra: Unit 4 - Graphs and Functions Homework Worksheets Bundle
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this file type before downloading and/or purchasing.
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Unit 4 on Functions and their Graphs is very short. Therefore, I only made a few days of homework for this Unit. We usually end up doing my Water Dripping Lab on Friday, so I don't give homework that day (The Water Lab is available here in my TpT store). So this file is for only 3 homework assignments. That is why it is priced lower than my other homework bundles.
* Recognizing Functions from various forms.
* Finding Points and Graphing from a linear function
* Finding a Function rule for a table or word description
The first two homeworks are one page each, the second one includes a back page which is review from this chapter. Then the last homework is also only one page.
It is difficult to find decent homework for functions, and I find the book confuses many students. I much prefer to give these homeworks during the unit. These homeworks are designed to complement my Unit 4 guided notes.
Another homework idea for this unit is the Melinda Mae Poem Activity, listed in my TpT store. Also, look for my multiple choice test to go along with this unit. | 677.169 | 1 |
Programs of Study
MATH-1200 - Trigonometry
Prerequisite: "C" or higher in MATH1150 or appropriate score on the math placement test. A study of trigonometry in preparation for advanced math and science coursework. Use definitions of trigonometric functions to establish properties, create graphs, establish identities and formulae, and define inverse trigonometric functions. Use trigonometric functions and their inverses to solve trigonometric equations, and applications. Graphing in polar coordinates, and vector arithmetic. | 677.169 | 1 |
73838 fully solved problems The latest course scope and sequences, with complete coverage of limits, continuity, and derivatives Succinct explanation of all precalculus concepts Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores!
Algebra, the foundation for all higher mathematics, is taught here both for beginners and for those who wish to review algebra for further work in math, science and engineering. This superior study guidethe first edition sold more than 600,000 copies!includes the most current terminology, emphasis and technology. It treats many subjects more thoroughly than most texts, making it adaptable for any course and an excellent reference and bridge to further study. Also available as a Schaum's Electronic Tutor.
ÒGeometry is a very beautiful subject whose qualities of elegance, order, and certainty have exerted a powerful attraction on the human mind for many centuries. . . Algebra's importance lies in the student's future. . . as essential preparation for the serious study of science, engineering, economics, or for more advanced types of mathematics. . . The primary importance of trigonometry is not in its applications to surveying and navigation, or in making computations about triangles, but rather in the mathematical description of vibrations, rotations, and periodic phenomena of all kinds, including light, sound, alternating currents, and the orbits of the planets around the sun.Ó In this brief, clearly written book, the essentials of geometry, algebra, and trigonometry are pulled together into three complementary and convenient small packages, providing an excellent preview and review for anyone who wishes to prepare to master calculus with a minimum of misunderstanding and wasted time and effort. Students and other readers will find here all they need to pull them through.
This edition reflects the changes in the trigonometry curriculum that have taken place between 1993 and 1998. Following the rise of the scientific calculator, this revision updates the book by keeping calculator usage in place of outdated material on logarithms, discarding irrelevant material.
Improve your understanding of calculus and your grades will also improve with this effective study aid as your guide. Comprehensive and packed with 272 solved and supplementary problems, it can be used with any undergraduate calculus textbook. | 677.169 | 1 |
Quantitative finance is a form of financial engineering that incorporates tenets of economics, mathematics, and information technology (IT). Individuals who practice quantitative finance use innovative financial models to predict market behaviors and discover market averages. To choose the best quantitative finance books, you should first consider why you are interested in the subject. For example, if you are considering taking courses in this field, but you are not sure that this kind of financial analysis is right for you, you might want to choose quantitative finance books that provide you with a general introduction to the field. Individuals who are practicing analysts, on the other hand, should choose books that are more complicated and which are directed toward specific kinds of financial management.
For students, the best quantitative finance books are probably those assigned for classes. A list of required books can often be found on syllabi distributed by course instructors. This is, however, a complicated subject, and students commonly find that they need some extra help. To find the best quantitative finance books to supplement required reading, it might be a good idea to consult your professor. In other cases, you might choose study aids published by educational publishing houses.
People who are interested in quantitative finance books, but who are not financial professionals and who do not wish to earn degrees, might find books directed toward the general public. These books tend to be less complicated than more technical books. They might contain less jargon and fewer mathematical equations. Instead, these texts might include histories of quantitative finance, as well as ways in which this brand of financial engineering is applied to contemporary financial processes.
This is a complex field with many different methods and perspectives held by its practitioners. For this reason, you should become familiar with the ideas and experiences of those who write or edit books. A financial scholar who has worked for years at a university, for example, might not have the same ideas about quantitative finances that a practicing financial analyst might possess.
Quantitative finance can be heavily dependent on current technology that is continually changing. For this reason, you might want to choose quantitative finance books that are relatively current. If you are interested in merely a brief introduction, this factor might not be as important. People who are interested in the practical application of financial modeling, on the other hand, should make sure that they are reading recently updated texts | 677.169 | 1 |
Collin County Community
College District
Math 0300 - Basic Mathematics
Welcome
Welcome to Math 0300, a BLENDED Web course. Let
me introduce myself:My name is Judy
Godwin and I will be your instructor.My
educational background includes a Bachelors degree from the University of North Texas
and a Masters degree from the University
of Texas at Dallas.It has been a great pleasure to teach at CCCCD, both full time and part
time, for over 20 years.I have seen the
college grow from infancy to the educational institution we have today.
After
enrolling, contact instructor
Once you have enrolled in the course, send an e-mail
to me using your Cougarmail address and enter your name and
course in the subject. Contact me at jgodwin@collin.edu
Taking
an ONLINE course
Since the class does not
have regular meetings, students need to be self-motivated and disciplined
enough to keep up with the work. This online class covers the same
material as Math 0300 At a Distance Indicator) assessment.Go to and
select the "Are You READI" link
Course
Structure provide
lectures on each section and chapter in the textbook. .
Blackboard CE will be
available by the first day class. For further information concerning
Blackboard CE, click on Math 0300 Course Overview
No on-campus orientation is required for this class.
Placement and
Developmental
Math Sequence
It is crucial
that a student is taking the appropriate math course based on the student's
mathematical background and knowledge. Enrolling in Math 0300 should be
based on the CCCCD Mathematics Assessment test for accurate math placement.
Once a
student has enrolled and completed Math 0300, the student must complete the
sequence of developmental mathematics courses: Math 0302, Math 0305,
Math 0310. Please contact me if you have
any questions about your math placement. | 677.169 | 1 |
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Calculus Assignment
Summarize what the article is about.
Discuss what you learned from it.
Explain how this article is relevant to Calculus I.
Describe your likes/dislikes about the article
Mathematics at the Ballpark.pdf | 677.169 | 1 |
Maths in Practice Practice Book
Synopsis
Maths in Practice Practice Book by David Bowles
Maths in Practice gives every pupil the confidence to become a successful learner. The course lays the foundations for good teaching and learning of mathematics through the key concepts outlined in the secondary curriculum review. These concepts include competence in mathematical procedures, creativity, and understanding and using mathematics. The full range and content of the curriculum are covered and reinforced through practice of essential mathematical skills and processes. Each chapter covers a particular attainment target (Number, Algebra, Geometry and Measures or Statistics) and they are presented in a possible teaching order. This Practice Book covers work at levels 4 to 5 and supports Pupil's Book 1. This Practice Book has an accompanyine website featuring Personal Tutor worked examples. A Network Edition Dynamic Learning is also available, including interactive activities and animations as well as Personal Tutor examples. Dynamic Learning enables users to navigate interactive pages or menus and launch a wealth of resources. It makes personalised learning a reality, arranging resources in a way that suits you, whether you are a student or a teacher, bringing learning to life. The tools and resources provided also enable teachers to build their own lessons, populate a VLE with content and use a digital whiteboard to full effect. Visit our website to find out more about this unique electronic resource. | 677.169 | 1 |
Students get acquainted with the various types of software available. By means of this software, students learn to solve mathematical problems. These problems are choosen with respect to their applicability at grammar schools. Each problem should by solved by a suitable tool and it should well represent some class of problems. The solution of each problem should be well-commented and presented on display or printer. <ol> <li>Using computers in teaching Mathematics. <li>Software available for teaching number systems and conversions among them. <li>Software available for prime-numbers handling. <li>Software available for solving linear and quadratic equations. <li>Solving systems of linear equations by means of education software. <li>Software available for teaching function and their graphs. </ol>
Learning activities and teaching methods
Demonstration
Preparation for the Course Credit
- 20 hours per semester
Attendace
- 20 hours per semester
Learning outcomes
The students become familiar with basic concepts of educational software.
Select suitable programs for teaching Mathematics in secondary school. | 677.169 | 1 |
The aim of this session is to motivate students to understand why we might want to do proofs, why proofs are important, and how they can help us. In particular, the student will learn the following: proofs can help you to really see WHY a result is true; problems that are easy to state can be hard to solve (Fermat's Last Theorem); sometimes statements which appear to be intuitively obvious may turn out to be false (the Hospitals paradox); the answer to a question will often depend crucially on the definitions you are working with. Target audience: suitable for anyone with a knowledge of elementary algebra and prime numbers, as may be obtained by studying A level mathematicsAuthors:
Prof. Kevin Johnston
This course provides a business perspective of information systems, and stresses how information systems can be used to improve the planning and running of businesses.
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Teaching & Learning Context:
<p>This series of vodcasts cover various topics on information systems and can be used for self study.</p> | 677.169 | 1 |
Mathematics for biologists 1989/90, 1990/91, 1992/93.
For 1st year biology students. Average program.
Official name: Istituzioni di Matematiche per biologi.
ELEMENTARY METHODS
What is mathematics?
Fundamental rules
The logarithm
Trigonometric functions
Polar coordinates in plane and space
Complex numbers
n-th roots of complex numbers
The language of set theory
Peano's axioms and the principle of induction
Some logic
Operations on sets with infinitely many operandss
The cartesian product
Resolution of systems of linear equations with Gauss elimination
Binomial numbers
Euclidean algorithm
FUNCTIONS AND RELATIONS
Functions
Relations
Equivalence relations
REAL NUMBERS
sqrt(2) is irrational
Real numbers
SEQUENCES AND SERIES
Sequences of real numbers
Sequences tending to infinity
The Bolzano-Weierstrass theorem
Convergence criteria for sequences
Limsup and liminf
Sequences of complex numbers
Numerical series
Convergence criteria specific for series
ANALYTIC GEOMETRY
Linear subspaces of R^n
Linear independence, bases and dimension
Affine subspaces of R^n
The scalar product
Linear mappings and matrices
Determinants and vector product
CONTINUOUS FUNCTIONS
Triangular inequality in R^n
Limit of f(x) for x tending to x_0
Continuous functions defined on a subset of R^m
Polynomial functions
The existence of zeros for continuous functions
Existence of the maximum of a continuous function defined on a compact set
Pointwise and uniform convergence of sequences of functions
Logarithms (theory)
DIFFERENTIAL CALCULUS IN ONE VARIABLE
The equation y = y_0+a(x-x_0)
The derivative
The derivatives of trigonometric functions
The derivative of the exponential and the logarithm
Relative maxima and minima and Rolle's theorem
The marvelous theorem of calculus
Analytic study of the graph of a function
Taylor series of a polynomial
Taylor series of a differentiable function
De l'Hopital's rules
Inversion of trigonometric functions
INTEGRATION
Measures
Measurable sets and Caratheodory's theorem
Measurable functions
The integral
Fundamental inequality of integration theory
Lebesgue measure
Fundamental theorem of calculus
Calculating integrals by means of the fundamental theorem
Integration by parts
Transformation rules for the integral | 677.169 | 1 |
13 DEFINISI LITERASI MATEMATIK For the purposes of PISA 2015, mathematical literacyis defined as follows:Mathematical literacy is an individual's capacity to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals to recognise the role that mathematics plays in the world and to make the well-founded judgments and decisions needed by constructive, engaged and reflective citizens.Mementingkan aplikasi matematik dalam kehidupan sebenar.Mengukur keupayaan murid menggunakan konsep dan kemahiran yang dipelajari di bilik darjah untuk menyelesaikan masalah kehidupan seharianBahagian Pembangunan KurikulumPeneraju Pendidikan Negara
17 PISA FRAMEWORK Change and Relationship Space and Shape Quantity These four categories characterise the range of mathematical content that is central to the discipline and illustrate the broad areas of content used in the test items for PISA 2015:Change and RelationshipSpace and ShapeQuantityUncertainty and DataThe four content categories serve as the foundation for identifying this range of content, yet there is not a one-to-one mapping of content topics to these categories.For example, proportional reasoning comes into play in such varied contexts as making measurement conversions, analysing linear relationships, calculating probabilities and examining the lengths of sides in similar shapes. The following content is intended to reflect the centrality of many of these concepts to all four content categories and reinforce the coherence of mathematics as a discipline. It intends to be illustrative of the content topics included in PISA 2015, rather than an exhaustive listing:Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
18 Functions: the concept of function, emphasising but not limited to linear functions, their properties, and a variety of descriptions and representations of them. Commonly used representations are verbal, symbolic, tabular and graphical. Algebraic expressions: verbal interpretation of and manipulation with algebraic expressions, involving numbers, symbols, arithmetic operations, powers and simple roots. Equations and inequalities: linear and related equations and inequalities, simple second-degree equations, and analytic and non-analytic solution methods Co-ordinate systems: representation and description of data, position and relationships. Relationships within and among geometrical objects in two and three dimensions: Static relationships such as algebraic connections among elements of figures (e.g. the Pythagorean theorem as defining the relationship between the lengths of the sides of a right triangle), relative position, similarity and congruence, and dynamic relationships involving transformation and motion of objects, as well as correspondences between two- and three-dimensional objects.
19 Measurement: Quantification of features of and among shapes and objects, such as angle measures, distance, length, perimeter, circumference, area and volume. Numbers and units: Concepts, representations of numbers and number systems, including properties of integer and rational numbers, relevant aspects of irrational numbers, as well as quantities and units referring to phenomena such as time, money, weight, temperature, distance, area and volume, and derived quantities and their numerical description. Arithmetic operations: the nature and properties of these operations and related notational conventions. Percents, ratios and proportions: numerical description of relative magnitude and the application of proportions and proportional reasoning to solve problems.
20 Counting principles: Simple combinations and permutations. Estimation: Purpose-driven approximation of quantities and numerical expressions, including significant digits and rounding. Data collection, representation and interpretation: nature, genesis (raw data) and collection of various types of data, and the different ways to represent and interpret them. Data variability and its description: Concepts such as variability, distribution and central tendency of data sets, and ways to describe and interpret these in quantitative terms. Samples and sampling: Concepts of sampling and sampling from data populations, including simple inferences based on properties of samples. Chance and probability: notion of random events, random variation and its representation, chance and frequency of events, and basic aspects of the concept of probability.
23 PROCESSES Bahagian Pembangunan Kurikulum Peneraju Pendidikan Negara FormulatingEmployingInterpretingIn particular, the verbs 'formulate,' 'employ,' and 'interpret' point to the three processes in which students as active problem solvers will engage.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
24 PROCESSESFormulatingEmployingInterpretingIndicates how effectively students are able to recognise and identify opportunities to use mathematics in problem situations and then provide the necessary mathematical structure needed to formulate that contextualised problem into a mathematical form.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
25 PROCESSESFormulatingEmployingInterpretingIndicates how well students are able to perform computations and manipulations and apply the concepts and facts that they know to arrive at a mathematical solution to a problem formulated mathematically.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
26 PROCESSESIndicates how effectively students are able to reflect upon mathematical solutions or conclusions, interpret them in the context of a real-world problem, and determine whether the results or conclusions are reasonable.FormulatingEmployingInterpretingStudents' facility at applying mathematics to problems and situations is dependent on skills inherent in all three of these processes, and an understanding of their effectiveness in each category can help inform both policy-level discussions and decisions being made closer to the classroom level.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
29 Contexts Personal Societal Occupational Scientific PISA FRAMEWORK An important aspect of mathematical literacy is that mathematics is engaged in solving a problem set in a context.PersonalSocietalOccupationalScientificFor the PISA survey, it is important that a wide variety of contexts are used. This offers the possibility of connecting with the broadest possible range of individual interests and with the range of situations in which individuals operate in the 21st century.For purposes of the PISA 2015 mathematics framework, four context categories have been defined and are used to classify assessment items developed for the PISA survey:
31 Computer-based Assessment In PISA 2015, for the first time, the computer will be the main mode of delivery for all tests and questionnaires.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
32 Why Computer-based Assessment? FirstFirst, computers are now so commonly used in the workplace and in everyday life that a level of competency in33 Why Computer-based Assessment?First34 The suite of tools available to students is also expected to include a basic scientific calculator. Operators to be included are addition, subtraction, multiplication and division, as well as square root, pi, parentheses, exponent,square, fraction , inverse and thecalculator will be programmed to respect the standard order of operations.PISA Editor toolAllow students to enter both texts and numbers.Students can enter a fraction, square root, or exponent. Additional symbols such as pi and greater/less than signs are available, as are operators such as multiplication and division signs.
35 The released PISA item Litter calls most heavily on students' capacity for interpreting, applying, and evaluating mathematical outcomes. The focus of this item is on evaluating the effectiveness of the mathematical outcome—in this case an imagined or sketched bar graph—in portraying the data presented in the item on the decomposition time of several types of litter. The item involves reasoning about the data presented, thinking mathematically about the relationship between the data and their presentation, and evaluating the result. The problem solver must and provide a reason why a bar graph is unsuitable for displaying the provided data.Computer-based
38 Structure of the ItemsPISA items in this form comprise a piece of stimulus, one or more questions related to that stimulus (with each question being referred to as an 'item') and, for each question, a set of guidelines that define the possible student response options and a proposed scoring scheme based on the defined response codes (the 'coding guide').Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
39 Stimulus HEIGHT Scoring scheme Question PISA ITEMS 1 There are 25 girls in a class. The average height of the girls is 130 cmM421Q01 –Question 1: HEIGHTExplain how the average height is calculated.Scoring schemeQuestion
40 PISA ITEMS 1HEIGHTThere are 25 girls in a class. The average height of the girls is 130 cmM421Q01 –Question 1: HEIGHTExplain how the average height is calculated.Answer the questionBahagian Pembangunan KurikulumPeneraju Pendidikan Negara
41 PISA ITEMS 1Full CreditCode 1: Explanations that include: Sum the individual heights and divide by 25.You add together every girl's height and divide by the number of girls.Take all the girls' heights, add them up, and divide by the amount of girls, in this case 25.The sum of all heights in the same unit divided by the number of girls.No CreditCode 0: Other responses.Code 9: Missing.
42 PISA ITEMS 1 Statement True or False If there is a girl of height 132 cm in the class, there must be a girl of height 128 cm.True/FalseThe majority of the girls must have height 130 cm.If you rank all of the girls from the shortest to the tallest, then the middle one must have a height equal to 130 cm.Half of the girls in the class must be below 130 cm, and half of the girls must be above 130 cm.HEIGHT SCORING 2Full CreditCode 1: False, False, False, False.No CreditCode 0: Other responses.Code 9: Missing.
44 PISA ITEMS 2LitterFor a homework assignment on the environment, students collected information on the decomposition time of several types of litter that people throw away:Type of litterDecomposition timeBanana peel1-3 yearsOrange peelCardboard boxes0.5 yearChewing gum20-25 yearsNewspapersA few daysPolystyrene cupsOver 100 yearsContext: ScientificContent: Uncertainty & DataProcess: interpreting, applying and evaluating mathematical outcomesThis item is set in a scientific context, since it deals with data of a scientific nature (decomposition time). The mathematical content category is Uncertainty and data, since it primarily relates to the interpretation and presentation of data, although Quantity is involved in the implicit demand to appreciate the relative sizes of the time intervals involved. the mathematical process category is interpreting, applying and evaluating mathematical outcomes since the focus is on evaluating the effectiveness of the mathematical outcome (in this case an imagined or sketched bar graph) in portraying the data about the real world contextual elements. The item involves reasoning about the data presented, thinking mathematically about the relationship between the data and their presentation, and evaluating the result. The problem solver must recognise that these data would be difficult to present well in a bar graph for one of two reasons: either because of the wide range of decomposition times for some categories of litter (this range cannot readily be displayed on a standard bar graph), or because of the extreme variation in the time variable across the litter types (so that on a time axis that allows for the longest period, the shortest periods would be invisible). Student responses such as those reproduced in Figure 12 have been awarded credit for this item.A student thinks of displaying the results in a bar graph.Give one reason why a bar graph is unsuitable for displaying these data.
45 PISA ITEMS 2 Sample responses for Litter Response 1: "Because it would be hard to do in a bar graph because there are 1-3, 1-3, 0.5, etc. so it would be hard to do it exactly."Response 2: "Because there is a large difference from the highest sum to the lowest therefore it would be hard to be accurate with 100 years and a few days."Communication comes in to play with the need to read the text and interpret the table, and is also called on at a higher level with the need to answer with brief written reasoning. The demand to mathematise the situation arises at a low level with the need to identify and extract key mathematical characteristics of a bar graph as each type of litter is considered. The problem solver must interpret a simple tabular representation of data, and must imagine a graphical representation, and linking these two representations is a key demand of the item. The reasoning demands of the problem are at a relatively low level, as is the need for devising strategies. Using symbolic, formal and technical language and operations comes into play with the procedural and factual knowledge required to imagine construction of bar graphs or to make a quick sketch, and particularly with the understanding of scale needed to imagine the vertical axis. Using mathematical tools is likely not needed.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
46QUESTION INTENT: Applies understanding of area to solving a value for money comparisonBahagian Pembangunan KurikulumPeneraju Pendidikan Negara
47 Change and RelationshipsCONTEXTPersonalPROCESSFormulatingCONTENTChange and Relationships
48 An important part of formulation 15 cmEmploying knowledge from space and shape and Quantity.Formulating a mathematical model to measure value for money20 cmThe released PISA item Pizzas (see Appendix B) calls most heavily on students' abilities to formulate a situation mathematically. While it is indeed the case that students are also called upon to perform calculations as they solve the problem and make sense of the results of their calculations by identifying which pizza is the better value for the money, the real cognitive challenge of this item lies in being able to formulate a mathematical model that encapsulates the concept of value for money. The problem solver must recognise that because the pizzas have the same thickness but different diameters, the focus of the analysis can be on the area of the circular surface of the pizza. The relationship between amount of pizza and amount of money is then captured in the concept of value for money, modelled as cost per unit of area.Interpreting mathematical result in real world terms.
49 Computer-Based Assessment PISA 2015 Single multiple-choiceFull credit dependent on both the selection and the reasoning behind it.An alternative form of reasoning, which reveals even more clearly the item's classification in Change and relationships, would be to say (explicitly or implicitly) that the area of a circle increases in proportion to the square of the diameter, so has increased in the ratio of (4/3),2 while the cost has only increased in the proportion of (4/3). Since (4/3)2 is greater than (4/3), the larger pizza is better value.While the primary demand and the key to solving this problem comes from formulating, placing this item in the formulating situations mathematically process category, aspects of the other two mathematical process are also apparent in this item. The mathematical model, once formulated, must then be employed effectively, with the application of appropriate reasoning along with the use of appropriate mathematical knowledge and area and rate calculations. The result must then be interpreted properly in relation to the original question.The solution process for Pizzas demands the activation of the fundamental mathematical capabilities to varying degrees. Communication comes in to play at a relatively low level in reading and interpreting the rather straight-forward text of the problem, and is called on at a higher level with the need to present and explain the solution. The need to mathematise the situation is a key demand of the problem, specifically the need to formulate a model that captures value for money. The problem solver must devise a representation of relevant aspects of the problem, including the symbolic representation of the formula for calculating area, and the expression of rates that represent value for money, in order to develop a solution. The reasoning demands (for example, to decide that the thickness can be ignored, and justifying the approach taken and the results obtained) are significant, and the need for devising strategies to control the calculation and modelling processes required is also a notable demand for this problem. Using symbolic, formal and technical language and operations comes into play with the conceptual, factual and procedural knowledge required to process the circle geometry, and the calculations of the rates. Using mathematical tools is evident at a relatively low level if students use a calculator efficiently.In Figure 9, a sample student response to the Pizzas item is presented, to further illustrate the framework constructs. A response like this would be awarded full credit.Reasoning
50 QUESTION INTENT: Applies understanding of area to solving a value for money comparisonCode 2: Gives general reasoning that the surface area of pizza increases morerapidly than the price of pizza to conclude that the larger pizza is bettervalue.• The diameter of the pizzas is the same number as their price, but the amountof pizza you get is found using diameter2 , so you will get more pizza per zedsfrom the larger oneCode 1: Calculates the area and amount per zed for each pizza to conclude that the larger pizza is better value.• Area of smaller pizza is 0.25 x π x 30 x 30 = 225π; amount per zed is 23.6 cm2area of larger pizza is 0.25 x π x 40 x 40 = 400π; amount per zed is 31.4 cm2so larger pizza is better valueCode 8: They are the same value for money. (This incorrect answer is codedseparately, because we would like to keep track of how many studentshave this misconception).Code 0: Other incorrect responses OR a correct answer without correct reasoning.Code 9: Missing.
51 APARTMENT PURCHASEThis is the plan of the apartment that George's parents want to purchase from a real estate agency.Question 1: APARTMENT PURCHASE PM00FQ01 – 019To estimate the total floor area of the apartment (including the terrace and the walls), you can measure the size of each room, calculate the area of each one and add all the areas together.However, there is a more efficient method to estimate the total floor area where you only need to measure 4 lengths. Mark on the plan above the four lengths that are needed to estimate the total floor area of the apartment.APARTMENT PURCHASE SCORING 1QUESTION INTENT:Description: Use spatial reasoning to show on a plan (or by some other method) the minimum number of side lengths needed to determine floor areaMathematical content area: Space and shapeContext: PersonalProcess: Formulate
52 Full Credit A = (9.7m x 8.8m) – (2m x 4.4m) A = 76.56m2 Code 1: Has indicated the four dimensions needed to estimate the floor area of the apartment on the plan. There are 9 possible solutions as shown in the diagrams below.A = (9.7m x 8.8m) – (2m x 4.4m)A = 76.56m2[Clearly used only 4 lengths to measure and calculate required area.]No CreditCode 0: Other responses.Code 9: Missing.
54 Question Question 1: SHAPES Which of the figures has the largest area? M158QQuestion 1: SHAPESWhich of the figures has the largest area?Explain your reasoning.SHAPES SCORING 1QUESTION INTENT: Comparison of areas of irregular shapesCode 1: Shape B, supported with plausible reasoning.• It's the largest area because the others will fit inside it.Code 8: Shape B, without plausible support.Code 0: Other responses.Code 9: Missing.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
55 Example responses Code 1: • B. It doesn't have indents in it which decreases the area. A and C have gaps.• B, because it's a full circle, and the others are like circles with bits taken out.• B, because it has no open areas:Code 8:• B. because it has the largest surface area• The circle. It's pretty obvious.• B, because it is bigger.Code 0:• They are all the same.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
56 M158QQuestion 2: SHAPES Describe a method for estimating the area of figure C.SHAPES SCORING 2QUESTION INTENT: To assess students' strategies for measuring areas ofirregular shapes.Code 1: Reasonable method:• Draw a grid of squares over the shape and count the squares that are morethan half filled by the shape.• Cut the arms off the shape and rearrange the pieces so that they fill a squarethen measure the side of the square.• Build a 3D model based on the shape and fill it with water. Measure theamount of water used and the depth of the water in the model. Derive thearea from the information.Code 8: Partial answers:• The student suggests to find the area of the circle and subtract the area of thecut out pieces. However, the student does not mention about how to find outthe area of the cut out pieces.• Add up the area of each individual arm of the shapeCode 0: Other responses.Code 9: Missing.
57 to which the student describes a METHOD. Example responses Code 1: NOTE:The key point for this question is whether the student offers a METHOD fordetermining the area. The coding schemes (1, 8, 0) is a hierarchy of the extentto which the student describes a METHOD.Example responsesCode 1:• You could fill the shape with lots of circles, squares and other basic shapes sothere is not a gap. Work out the area of all of the shapes and add together.• Redraw the shape onto graph paper and count all of the squares it takes up.• Drawing and counting equal size boxes. Smaller boxes = better accuracy(Here the student's description is brief, but we will be lenient about student'swriting skills and regard the method offered by the student as correct)• Make it into a 3D model and filling it with exactly 1cm of water and thenmeasure the volume of water required to fill it up.wrong)Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
58 • Minus the shape from the circle Code 8:• Find the area of B then find the areas of the cut out pieces and subtract themfrom the main area.• Minus the shape from the circle• Add up the area of each individual piece e.g.,• Use a shape like that and pour a liquid into it.• Use graph• Half of the area of shape B• Figure out how many mm2 are in one little leg things and times it by 8.Code 0:• Use a string and measure the perimeter of the shape. Stretch the string out toa circle and measure the area of the circle using π r2.(Here the method described by the student is wrong)Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
59 Describe a method for estimating the perimeter of figure C. Question 3: SHAPES M158QDescribe a method for estimating the perimeter of figure C.SHAPES SCORING 3QUESTION INTENT: To assess students' strategies for measuring perimeters ofirregular shapesCode 1: Reasonable method:• Lay a piece of string over the outline of the shape then measure the length of string used.• Cut the shape up into short, nearly straight pieces and join them together in a line, then measure the length of the line.• Measure the length of some of the arms to find an average arm length then multiply by 8 (number of arms) X 2.Code 0: Other responses.Code 9: Missing.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
60 Question 3: SHAPES M158QDescribe a method for estimating the perimeter of figure C.Example responsesCode 1:• Wool or string!!!(Here although the answer is brief, the student did offer a METHOD formeasuring the perimeter)• Cut the side of the shape into sections. Measure each then add themtogether. (Here the student did not explicitly say that each section needs to beapproximately straight, but we will give the benefit of the doubt, that is, byoffering the METHOD of cutting the shape into pieces, each piece is assumedto be easily measurable)Code 0:• Measure around the outside.(Here the student did not suggest any METHOD of measuring. Simply saying"measure it" is not offering any method of how to go about measuring it)• Stretch out the shape to make it a circle.(Here although a method is offered by the student, the method is wrong)
61 6 Proficiency Scale Description PISA 2015 LEVEL DESCRIPTION At Level 6 students can conceptualise, generalise and utilise information based on their investigations and modelling of complex problem situations. They can link different information sources and representations and flexibly translate among them. Students at this level are capable of advanced mathematical thinking and reasoning. These students can apply their insight and understandings along with a mastery of symbolic and formal mathematical operations and relationships to develop new approaches and strategies for attacking novel situations. Students at this level can formulate and precisely communicate their actions and reflections regarding their findings, interpretations, arguments and the appropriateness of these to the original situations.Bahagian Pembangunan KurikulumPeneraju Pendidikan Negara
62 5 4 Proficiency Scale Description PISA 2015 LEVEL DESCRIPTION At Level 5 students can develop and work with models for complex situations, identifying constraints and specifying assumptions. They can select, compare and evaluate appropriate problem-solving strategies for dealing with complex problems related to these models. Students at this level can work strategically using broad, well-developed thinking and reasoning skills, appropriate linked representations, symbolic and formal characterisations and insight pertaining to these situations. They can reflect on their actions and formulate and communicate their interpretations and reasoning.4At Level 4 students can work effectively with explicit models for complex concrete situations that may involve constraints or call for making assumptions. They can select and integrate different representations, including symbolic, linking them directly to aspects of real-world situations. Students at this level can utilise well-developed skills and reason flexibly, with some insight, in these contexts. They can construct and communicate explanations and arguments based on their interpretations, arguments and actions.
63 3 2 1 Proficiency Scale Description PISA 2015 LEVEL DESCRIPTION At Level 3 students can execute clearly described procedures, including those that require sequential decisions. They can select and apply simple problem-solving strategies. Students at this level can interpret and use representations based on different information sources and reason directly from them. They can develop short communications when reporting their interpretations, results and reasoning.2At Level 2 students can interpret and recognise situations in contexts that require no more than direct inference. They can extract relevant information from a single source and make use of a single representational mode. Students at this level can employ basic algorithms, formulae, procedures, or conventions. They are capable of direct reasoning and making literal interpretations of the results.1At Level 1 students can answer questions involving familiar contexts where all relevant information is present and the questions are clearly defined. They are able to identify information and to carry out routine procedures according to direct instructions in explicit situations. They can perform actions that are obvious and follow immediately from the given stimuli. | 677.169 | 1 |
Foundations of Geometry
ISBN-10: 0486472140
ISBN-13: 9780486472140eared toward students preparing to teach high school mathematics, this text is also of value to professionals, as well as to students seeking further background in geometry. It explores the principles of Euclidean and non-Euclidean geometry, and it instructs readers in both generalities and specifics of the axiomatic method. 1964 | 677.169 | 1 |
Category: AlgebraMathematical knowledge is only powerful to the extent to which it is understood conceptually, not just procedurally. For example, students are taught the three ways of solving a system of linear equation: by graphing, by substitution and by elimination. Of these three methods, graphing is the one that would easily make sense to many students. Substitution, which | 677.169 | 1 |
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Plane Trigonometry Made Plain With Logarithmic and
I don't even know 'er! *I'd like to take a little digression to the world of mathematical prefixes here, but it might not be for everyone. Please be as specific as possible in your report. In this chapter we start by explaining the basic trigonometric functions using degrees (°), and in the later part of the chapter we will learn about radians and how they are used in trigonometry. If we put x = π in Euler's Formula, we find eiπ = -1, surely a remarkable expression containing the transcendental numbers e and π, as well as the imaginary unit i and -1.
Q: Can I return or Exchange a gift after I purchase it? A: Because the gift is sent immediately, it cannot be returned or exchanged by the person giving the gift. The recipient can exchange the gift for another course of equal or lesser value, or pay the difference on a more expensive item Priority Codes are on the back of the catalog, mail promotion, or within an advertisement Applied College Algebra and Trigonometry and Study Wizard CD (2nd Edition). This is a must-have for any math student in any math level-it has over 30 features if you count everything. Don't you hate it when you type sin(angle)=something and you only get 1 value for the angle. This program will give you 4 values in positive degrees for the angle and those values in radians. After you solve a trigonometric equation, you can use this program to find several values of the angle using tan( and cos( as well as sin( pdf. Test: [;-1 = e^{i\pi};] This group is for discussion and sugests for books about science mathematics and filosophy. [close] This group is for discussion and sugests for books about science mathematics and filosophy. A place to discuss and share recommendations about books related to science, technology, enginee …more [close] A place to discuss and share recommendations about books related to science, technology, engineering, and math (STEM) Plane Trigonometry and Complex Numbers. An absolutely superb book for the layman, and of interest to the professional accomplishes what many other books have merely attempted. One of the best books written for the undergraduate to learn probability is the book by Gordon. Despite the restriction to discrete probability this book is a superb general introduction for the math undergraduate and is very well organized A practical text-book on plane and spherical trigonometry.. A few decades later, Archimedes made use of the same formula. al-Biruni has preserved a Lemma of Archimedes, which shows that he had an equivalent version of Ptolemy's Theorem at his disposal (Neugebauer 773). In Menelaus' work there is a remark that suggests that one of the trigonometric propositions can be attributed to Apollonius, who lived a few years before Hipparchus (Heath 253). "Tannery (from his Recherches sur l'hist Graduated Exercises in Plane Trigonometry, Compiled by J. and S.R. Wilson.
In Calvin and Hobbes, Calvin has historically had a strong aversion to arithmetic New Plane and Spherical Trigonometry, Surveying and Navigation, Teachers' Edition - Primary Source Edition. 9 MB This peerless reference/text unfurls a unified and systematic study of the two types of mathematical models of dynamic processes-stoch.. College Algebra and Trigonometry With Applications. For angles greater than 74° the mean differences become so large and increase so rapidly that they cannot be given with any degree of accuracy. We will now consider a few examples illustrating practical applications of tangents. The first is suggested by the problem mentioned in section 24. Example 1: At a point 168 m horizontally distant from the foot of a church tower, the angle of elevation of the top of the tower is 38° 15' A System of Popular Trigonometry. The Tennessee State Board of Education adopted new state standards in 2010 for mathematics for grades K-8 and the first three high school courses (Algebra 1, Geometry, Algebra 2, and Integrated Math 1, 2, and 3), and in 2014 adopted new standards for the further high school mathematics courses download Plane Trigonometry Made Plain With Logarithmic and Trigonometric Tables pdf.
Algebra and Trigonometry with Analytic Geometry (A Series of books in the mathematical sciences)
Plane Trigonometry and Logarithms
Trigonometry Refresher for Technical Men
We appreciate anyone who brings errors to our attention so we may correct them.) HANDY FORMULAS AND INFORMATION - These should be memorized College Mathematics (College Outline Series)! Practice addition, subtraction, multiplication, and division in an arcade game format Algebra &Trigonometry with Modeling &Visualization - 3rd ed. A treatise on spherical trigonometry, by W.J. M'Clelland and T. Preston. In the mean time you can sometimes get the pages to show larger versions of the equations if you flip your phone into landscape mode Eleventh year mathematics: Intermediate algebra and trigonometry. The French government, under the leadership of the astronomer Jean Picard (1620–82), undertook to triangulate the entire country, a task that was to take over a century and involve four generations of the Cassini family ( Gian, Jacques, César-François, and Dominique ) of astronomers Plane Trigonometry, Surveying and Tables. This case is abbreviated SAS, for side-angle-side, and is perhaps the most frequently encountered in practice. We see first that the Law of Sines cannot be applied at once, but the Law of Cosines gives us immediately c2 = a2 + b2 - 2ab cos C. This equation is easily evaluated on a pocket calculator for any value of C from 0° to 180° An elementary treatise on plane and spherical trigonometry. Next semester I'll be taking a Pre-calculus course and will rewatch this course. I do recommend for anyone taking an online course or personal pace course to use this material to augment the class to get an A in the class which is what I'm at with the help of this course. May 14, 2016 Rated 5 out of 5 by MariodelaParra I like very much the trigonometry part This is an excellent pre-calculus course Student Solutions Manual.
Introducing Logarithms with Foldables, War, Bingo, and Speed Dating . That one potential effect of the letter was to weaken the Presidents status internationally. Unknown she wouldnt be getting the support she is either. Organizers in positions that need managers and organizers. A few years it was pretty much like any other job personality conflicts stuff Interpolated six-place tables of the logarithms of numbers and the natural and logarithmic trigonometric functions. Thomas Parker Math Competition is now open (there is no fee). Anyone who is enrolled full time at K-State and in the first two years of college is eligible and encouraged to register T B Trigonometry 6e. Students may not take the first two quarters of this sequence for P/F grading. MATH 15100-15200 meets the general education requirement in mathematical sciences. Prerequisite(s): Superior performance on the mathematics placement test, or MATH 10500 This is the second course in the regular calculus sequence in the department epub. Math Pentagon apps empower Math teachers in 1:1 iPad program for differentiated instruction, early intervention, frequent progress monitor, and intensive training provision for students who need remediation The Not-So-Scary Guide to Basic Trigonometry. With static graphs and equations, it's possible to get a handle on the rules of what various functions do and mean. However, it's still hard to get an intuitive sense of the relationship between the circle and the trigonometric functions and the triangles. Note: some of the information below is only of general nature, or concerns a specific section of the course MyMathLab for Trigsted Trigonometry -- Access Kit (2nd Edition) (Mymathlab Ecourse). I am particularly interested in invariants of moduli spaces of solutions to differential relations that come from singularity theory," Rustam adds Math U See Fraction Overlays. You acknowledge that the data collected via our Site will be stored in servers located within the United States. Further, you acknowledge that your personal information may, at times, be accessible by individuals may be located worldwide including in countries that may have not been determined to provide the same level of data protection as in your country 200 Multiplication Worksheets with 1-Digit Multiplicands, 1-Digit Multipliers: Math Practice Workbook (200 Days Math Multiplication Series). Plane Trigonometry Made Plain With Logarithmic and Trigonometric Tables online. Mathematics now has the opportunity more than ever before to under-pin quantitative understanding of industrial strategy and processes across all sectors of business. Companies that take best advantage of this opportunity will gain a significant competitive advantage: mathematics truly gives industry the edge." It's constant, value of b, along here, and then at this point it becomes this arc, of the circle. So working this out, I could do it but it's a little awkward because expressing y as a function of x, the top edge of this shape, it's a little awkward, and takes two different regions to express Algebra and Trigonometry With Analytic Geometry: A Problem-Solving Approach. The emphasis is on proving all of the results. Previous knowledge of numerical analysis is not required Maths for Science. | 677.169 | 1 |
The Theme of this Course
Going from a geometric concept/problem to a working program usually takes
several steps as follows:
Geometry -> Algebra -> Algorithm -> Program
Since computers do not understand geometry, one must convert a geometric
problem to an algebraic one that uses numbers. Then, one can design
algorithms to manipulate these numbers and finally, programs are developed
based on these algorithms. However, each step is a difficult and challenging
task.
Geometry -> Algebra
When we think about a geometric object, even a simple one such as
a point or a line, we normally have its shape in mind. But, to have
computers to process a geometric object, we have to find a
representation for that object so that it can be described
in a form that can be manipulated by computers. For example, a point
in three-dimensional space is represented with three numbers like
(2.5, 0.0, -4.0), and a line in the xy-coordinate plane has an
equation like 3x - 5.3y + 3 = 0.
A geometric object's representation is usually not unique. A circle can
be represented by an implicit equation:
x2 + y2 = 1
or in a parametric form using trigonometric functions:
x = cos(t) y = sin(t)
where t is in the range of 0 and 360 degree.
Some geometric objects such as polyhedra many even require complex
data structures to represent all of its details. Therefore, finding a
good and appropriate representation (for a particular application) is
usually a very challenging task. We shall cover some of these
representations in this course. Some are mathematical (for curves and
surfaces) and some are combinatorial (for polyhedra). Whatever representation
is chosen, it must be easy to use and manipulate, and support
all desired operations efficiently and accurately.
In addition to a representation for a particular type of geometric
objects, one must convert geometric operations to algebraic forms, too.
Take a look at the following question. Given a sphere of radius r,
what is the locus of this sphere if its center moves on a curve?
We know that the locus, usually referred to as a sweep, looks
like a tube; but, it is not so easy to know what exactly this tube looks like.
If the curve is a line, the locus is a cylinder.
The difficult part is that the curve is not a line and/or that the radius
of the given sphere can change. Therefore, we have an easily described
geometric operation whose algebraic counterpart is somewhat complicated.
Algebra -> Algorithm
After finding representations for geometric objects and algebraic
interpretations for your geometric operations, the next step is to find
algorithms for manipulating the representations and equations. Is it
easy? Sometimes it is. For example, if the problem is
"determine if two lines on the xy-coordinate plane intersect and if
they are, find the intersection point," it can be solved easily.
Let the lines be
Ax + By + C = 0
Ux + Vy + W = 0
Then, if they are parallel to each other (i.e., A*V=B*U), there
is no intersection point; otherwise, the intersection point can be found
by solving the above simultaneous linear equations of two variables.
Unfortunately, other practical problems are not so easy. First, the
representation of a geometric object such as a piston engine or the
Boeing 777 is huge and the number of equations, usually non-linear ones
with many variables, is also huge. Manipulating these representations and
equations is not an easy job. However, we have at least the following
choices:
Symbolic Computation
A symbolic system delivers symbolic answers. For example, if
you ask a symbolic system to solve a quadratic equation
Ax2 + Bx + C =0, it would give you the following
equations of roots:
Therefore, the answers are algebraic rather than numerical.
There are good symbolic systems available (e.g., Mathematica and
Maple). Computation cost is usually very high (e.g. very
slow). On the other hand, symbolic computation
is able to give you a closed-form solution, a form that
can be written in one or more formulae.
Numerical Computation
Numerical solutions give you a bunch of numbers rather than a
closed-form solution. Numerical computation has been a popular
way of finding solutions for geometric problem. For example,
in calculus you perhaps have learned Newton's method for solving
non-linear equation. The solution is merely a number.
Numerical computation is fast; but, since computers cannot
represent real numbers exactly, extreme care must be taken to
avoid loss of significant digits and other related problems.
For example, if the initial guess for Newton's method is
incorrect, you may not be able to find a root. As a suggestion,
please try to solve the following equation using Newton's
method:
where n is a positive integer. This equation has a root
at zero. But, Newton's method will not converge to zero at all.
Try it yourself.
Approximation
Since symbolic computation is time consuming and numerical
computation sometime does not deliver the result, let us take
a compromise. Let us simplify our solution by only asking for
a good approximation. For example, it is provably
true that in general a polynomial with degree higher than four
does not have any closed-form solution. Numerical methods
(e.g. Newton's method) can only give us an approximation
of the roots of a higher degree polynomial.
So, the question is "how good is good." This is obviously
problem dependent. We shall return to this question later on.
The above does not enumerate all possibilities. One can combine
several ways together to solve a single problem. For example, one can
use symbolic to solve some parts of the problem and use a combination of
numerical computation and approximation to do the remaining.
Recently, some authors suggested the so-called geometric methods
whose fundamental merit involves characterizing the results using geometry
reasoning and using the characterization for calculating the result.
This method usually work fine with simple problems and may require a
large number of case-by-case analysis. However, if geometric method works,
it usually delivers the solution fast and is more accurate and robust
than other methods. We will not pursue in this direction in this course.
Algorithm -> Program
Ok, you might want to say that, after all of the troubles in the previous
steps, we have algorithms and therefore writing program should be easy.
No, it is not always the case. Translating some algorithms in your
algorithms design textbook to programs may be easy. Translating geometric
algorithms to programs requires extreme care. There are geometric programming
languages. Even though you have had all the helps from a programming system,
its debugger
and graphics library, you still have to be very careful about the problem
of real number computations. Computation errors and loss of significant
digits may occur easily and in many cases their impacts may be propagated
and amplified. As a result, a theoretically sound algorithm may deliver
meaningless results. You may have learned this in a numerical methods
course; but, I will reiterate this issue later on to emphasize its
importance in geometric computation.
In this course, we shall introduce several representations for curves,
surfaces and solids and their accompanying geometric operations.
We shall spend a considerable amount of time on curves and surfaces
design using approximation. Another point we want to emphasize is
the impact of floating number computations on geometric problems.
Some classical computational geometry problems will also be mentioned,
because they are useful and can illustrate what computational
geometers are doing. | 677.169 | 1 |
Mathematics
MATHEMATICS
Mathematics is one of the most enduring fields of study, and is essential in an expanding number of disciplines and professions. Many mathematicians continue to develop new mathematics for its own sake. Today, however, mathematicians also combine their knowledge of mathematics and statistics with modelling and computational skills and use the latest computer technology to solve problems in the physical and biological sciences, engineering, information technology, economics, and business.
What will I study?
UQ offers a wide range of courses in mathematics and its applications. In their first year, students study essential topics in calculus, linear algebra and differential equations. In later years students select from more specialised courses. These emphasise new ideas in mathematics, and include recent applications in coding and cryptology, mathematical physics, mathematical biology, bioinformatics, and finance.
Algebra studies abstract mathematical structures beginning with vector spaces, groups, and rings. It leads on to the study of number theory and to applications in mathematical physics, coding, and cryptology. Discrete mathematics studies the ways objects can be rearranged and linked together, and includes combinatorics and graph theory. These subjects are basic to many of the large problems arising in information technology and bioinformatics. The department has a particularly strong research program in combinatorics, covering a wide variety of subdisciplines including algebraic combinatorics, bioinformatics, combinatorial group theory, design theory, and graph theory.
Mathematical analysis is the area of mathematics that most appeals to people who like calculus. It provides a rigorous foundation for differentiation and integration, and its ideas are basic in the understanding of many fields of contemporary mathematics, including differential equations, probability theory, stochastic processes, and control theory. Current research in this area includes nonlinear differential equations arising from physical and biological models, dynamical systems, control theory and economics, stochastic processes, and applications to financial mathematics and biology.
Applied mathematicians use mathematics to understand the world around us. The applied mathematics courses develop the mathematical methods that have proved particularly useful, and apply these methods to physical and biological systems. The department has significant research strengths in material science and mathematical ecology.
Today, advanced mathematical models are used routinely in finance. Mathematics is used to monitor and direct the investments of superannuation funds and investment managers. Partial differential equations are used to price options. The new Basel 2 accord on international bank regulation requires sophisticated modeling of a bank's overall risk. The core courses in financial mathematics provide a background in finance and an introduction to the basic techniques of stochastic processes, statistics, and computational methods. These can be combined with further courses in finance, statistics, or computational mathematics. The mathematics department hosts an interdisciplinary group of statisticians, mathematical analysts, and computational mathematicians interested in financial mathematics and its application in the energy markets.
Many breakthroughs in the development of physical theories, particularly in the realm of quantum physics, have been underpinned by the application of novel mathematical techniques. Research in mathematical physics at UQ covers a broad spectrum from areas of pure mathematics (Lie and quantum algebras, supersymmetry, low dimensional topology) through to applications in areas such as Bose-Einstein condensates, superconductivity, and condensed matter systems.
Study Plans
Mathematics is available as a Single Major or an Extended Major. For the Single Major you are required to complete #14 (#6 at Level 2 and #8 at Level 3) and for the Extended Major you are required to complete #22 (#10 at Level 2 and #12 at Level 3) from the Mathematicsfor Tony Roberts
What I do
I apply mathematical methods to predict or optimize the properties of complex materials using models of their microstructure. I have studied numerous materials including foams, ceramics and gas-barrier films, which involve the development of suitable statistical structural models. Recently I have used mathematical techniques to design optimal bone implants, and predict the diffusive and electrical properties of fractal networks. The two areas of mathematics that I use most frequently are partial differential equations and probability theory.
What I teach
I teach a number of courses which develop mathematical and computational techniques to solve physical, engineering and biological problems. This includes second year courses on calculus and linear algebra as well as specialized advanced courses on partial differential equations and asymptotic analysis. Partial differential equations are the one of the most important concepts in applied mathematics, describing, for example, heat and mass transfer, motions of electrons, deformation of solids, flow in liquids, and prices of options on the stock-market.
Careers
Mathematics graduates are respected for their excellent quantitative skills and problem solving abilities. They win a wide range of rewarding positions in the public and private sectors. The latest figures from the Graduate Careers Australia ( show 87 per cent of young (<25) mathematics graduates had found jobs by the April following their graduation or were undertaking further study. These figures compare well with those for the related professional degrees of engineering (91 per cent), accounting (86 per cent), and computer science (78 per cent). People who are enthusiastic about doing mathematics can confidently look forward to a rewarding career.
Students with a strong interest and ability in mathematics should consider doing an honours degree. This is an extra year of advanced courses and work on an individual research project. This gives students experience in reading the mathematics research literature and applying recent results and methods to solve problems. An honours degree is the usual path for students who wish to continue doing research and go on to a do a PhD
Mathematics graduates use their quantitative problem solving skills to successfully compete with graduates in other disciplines for a range of jobs in the private and public sector. The areas of mathematics that are most often used in industry in Australia are operations research, statistics, and financial mathematics. Mathematicians are also employed in research organisations such as the CSIRO, DSTO and the Bureau of Meteorology, universities and in industry.
Mathematicians at The University of Queensland have established a careers web site for the Australian Mathematical society ( This now gives listings of current job advertisements for mathematics graduates throughout Australia, and gives examples of the diverse career paths of many past mathematics graduates. | 677.169 | 1 |
TOPIC 2
TITLE
EQUATIONS AND INEQUALITIES
Preparedby:
Learning Outcomes
At the end of this lesson, student should be able to:
Recognize the general form for linear equations
Solve the linear equations
Recognize the general form for quadratic equations
Solv
TITLE
Preparedby:
LEARNINGOUTCOMES
At the end of this lesson, student should be able to:
Define a complex number with its component parts
Represent complex numbers on an Argand diagram
Perform algebraic operations (addition, substation,
multiplication and
TITLE
Preparedby:
LEARNINGOUTCOMES
At the end of this lesson, student should be able to:
Understand the Cartesian coordinates system
Calculate the distance between two points
Calculate the midpoint two points
Find the gradient of a line, obtain and use th
TITLE
Preparedby:
LEARNINGOUTCOMES
At the end of this lesson, student will be able to:
Distinguish functions and relations
Sketch the different types of functions ( linear, quadratic, cubic,
rational and square root function)
Know few characteristics of s
E BY:
TLRED
TIREPA
P
LEARNING OUTCOMES
At the end of this lesson, student should be able to:
Recognize the general form for linear equations
Solve the linear equations
Recognize the general form for quadratic equations
Solve quadratic equations using the
1.2 Rates of Change at Specific Points
1.3 Rates of Change at General Points
Objectives
1.Explain the concept of accurate.
2.Identify the different between Average Rate of Change
and Rate of Change.
3.Able to solve problem on Rate of Change
a) Shrinking I
1.1 LIMITS AND CONTINUITY
Objectives
1.Able to sketch a function.
2.Able to find a limit .
a) Graph Method.
b) Table Method.
3.Able to find continuity of a function.
LIMIT
CONCEPT
Limit
Idea of the limit
The concept of a limit- to describe:
the behavior
If all the world is a stage, then most of us
need rehearsals
[email protected]
6.4 Volume of Solid by Revolution over an axis
Objectives
1.Able to calculate Volumes of Solid Revolution.
Before starting this section you should . . .
Be able to calculate definit
CHAPTER5
APPLICATIONSOFINTEGRATION
[email protected]
APPLICATIONSOFINTEGRATION
5.1
Area Under a Curve
Lesson Outcome:
Apply the definite integrals to
find the area under curves.
[email protected]
AreaUnderaCurve
The definite integral can be used to find the area bet
APPLICATIONSOFINTEGRATION
5.2
Area Between Two Curves
Lesson Outcome:
Apply integration to find areas
between two curves.
[email protected]
AreaBetweenTwoCurves
Area between two curves at the -axis
Area between two curves at the x-axis
If f and g are continuous
TOPIC 6
TRIGONOMETRY
TITLE
Preparedby:
Learning Outcomes
At the end of this lesson, student should be able to:
Recognize angles and their measures (acute, obtuse, right and
straight)
Understand radian measure
Convert between degree and radians and vice
TOPIC 5
FUNCTIONS
TITLE
Preparedby:
TOPIC 3
TITLE
COMPLEX NUMBER
Preparedby:
Learning Outcomes
At the end of this lesson, student should be able to:
Define a complex number with its component parts
Represent complex numbers on an Argand diagram
Perform algebraic operations (addition, sub
:rJle Calculus AB Exam
CALCULUS AB
A CALCULATOR CANNOT BE USED ON PART A OF SECTION I. A GRAPHING CALCULATOR.FROM THE
APPROVED LIST IS REQUIRED FOR PART B OF SECTION I AND FOR PART A OF SECTION II OF THE
EXAMINATION. CALCULATOR MEMORlES NEED NOT BE CLEARE
Formulae and
Transposition
1.5
Introduction
Formulae are used frequently in almost all aspects of engineering in order to relate a physical quantity
to one or more others. Many well-known physical laws are described using formulae. For example,
you may ha
MAB241 COMPLEX VARIABLES
LAURENT SERIES
1 What is a Laurent series?
The Laurent series is a representation of a complex function f (z) as a series. Unlike the Taylor series which
expresses f (z) as a series of terms with non-negative powers of z, a Lauren
TOPIC 5
FUNCTIONS few characteristics o | 677.169 | 1 |
Math 250-C
Matlab Assignment #1
1
Revised 1/18/13
LAB 1: Matrix and Vector Computations in Matlab
In this lab you will use Matlab to study the following topics:
How to create matrices and vectors in Matlab.
How to manipulate matrices in Matlab and creat
Math 250-C
Matlab Assignment #2
1
Revised 2/22/14
LAB 2: Linear Equations and Matrix Algebra
In this lab you will use Matlab to study the following topics:
Solving a system of linear equations by using the reduced row echelon form of the augmented matrix
Math 250-C
Matlab Assignment #3
1
Revised 3/02/13
LAB 3: LU Decomposition and Determinants
In this lab you will use Matlab to study the following topics:
The LU decomposition of an invertible square matrix A.
How to use the LU decomposition to solve the
._SL\ATIWS
Math 250, Section 01 Name (Print):
Fall 2016
Quiz 1
9/22/2016
Time Limit: 20 Minutes
You may not use your books7 notes, or any calculator on this quiz.
1. (10 points) Find all solutions of the following system of linear equations:
Math 250-C
Matlab Assignment #5
1
Revised 4/16/13
LAB 5: Eigenvalues and Eigenvectors
In this lab you will use Matlab to study these topics:
The geometric meaning of eigenvalues and eigenvectors of a matrix
Determination of eigenvalues and eigenvectors
Math 250-C
Matlab Assignment #4
1
Revised 4/03/14
LAB 4: General Solution to Ax = b
In this lab you will use Matlab to study the following topics:
The column space Col(A) of a matrix A
The null space Null(A) of a matrix A.
Particular solutions to an in
How to solve inconsistent systems
Matrix equation has solution if b is in column space of A
What is a unique solution?
Rank is full if there is a unique solution
If rank isn't full, nullity can be 1 or more, infinite solutions, free variable
is present
If
Linear Algebra, Math 250
Section C2, Fall 2016
Midterm Exam 2
November 16, 2016
NAME: 5 O I urllbS
Attempt all problems, and show all of your work.
You may NOT use textbooks or calculators on this exam.
l Problem Number Possible Points Points Earned
LINEAR ALGEBRA Advice
Showing 1 to 2 of 2
Gives us any idea to solve arrays much quicker than we would otherwise would not know.
Course highlights:
professor was great, very patient and helpful. His goal of helping us understanding everything possible prior to a quiz was helpful and worked extra with us if we struggled. The course combines all that we've learned from Calc 1 -4 and algebra, but in a way that doesn't brutalize how to solve problems.
Hours per week:
3-5 hours
Advice for students:
Practice and stay on top of you work. If you can keep up with the material, you can succeed.
Course Term:Summer 2017
Professor:Welsh
Course Tags:Math-heavyGo to Office HoursMany Small Assignments
Mar 12, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
I would definitely recommend it! The professor is very clear with explanations.
Course highlights:
You learn how to think analytically. For example, you learn basic proofs.
Hours per week:
3-5 hours
Advice for students:
Make sure to read the textbook, do all the homework, and attend lectures! | 677.169 | 1 |
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this file type before downloading and/or purchasing.
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Product Description
This is algebra 2 common core notes I used for a lesson I gave my students on solving systems of equations. We started out basic, understanding intersection of different equations. Solved algebraically and then graphically. I then moved to higher level functions and relied strictly on the TI-84+ calculator to solve. | 677.169 | 1 |
Mathematics for Elementary Teachers Via Problem Solving ...
Read More operations; real numbers: rationals and irrationals; patterns and functions; geometry; and measurement. For teachers of mathematics at the elementary school level | 677.169 | 1 |
The Fourth Edition of College Trigonometry helps students see the dynamic link between concepts and applications. The authors' hallmark approach, the Aufmann Interactive Method, encourages students to interact with math by presenting an annotated example, then guiding students with a Try Exercise, and finally presenting a worked-out solution for immediate reinforcement of the concept.
An Instructor's Annotated Edition, unlike any other offered for this course, features reduced student text pages with special instructor resources in the margins: teaching tips, extra examples, ideas for reinforcing concepts, discussion suggestions, highlighted vocabulary and symbols, challenge problems, quizzes, suggested assignments, and references to transparencies that may be found both in the Instructor's Resource Manual and on the web site.
Side-by-Side Solutions to examples pair an algebraic solution and a graphical representation to accommodate different learning styles.
Integrated web resources include selected Take Note boxes (identified by a special web icon) which direct students to an interactive example or a downloadable file on the web site. These special resources can be used by instructors for presentation purposes or can be assigned to students to help them 'visualize' a concept.
Exploring Concepts with Technology, a special end-of-chapter feature, expands on ideas introduced in the text by using technology to investigate extended mathematical applications or topics.
Projects at the end of each exercise set are designed to encourage students (or groups of students) to research and write about mathematics and its applications. Additional Projects are included in the Instructor's Resource Manual and on the book's web site.
Take Note and Math Matters (formerly called Point of Interest) margin notes alert students about interesting aspects of math history, applications, and points that require special attention | 677.169 | 1 |
Course Overview
Welcome to the 7th grade Next Generation Standards for West Virginia Common Core. The material covered in this grade level is challenging yet exciting. The 7th grade curriculum has been broken down into four main goals. The first of these goals is to develop a strong foundation of proportional relationships and apply that knowledge to real world situations. The second goal is to develop number sense with rational numbers used to solve multi- step linear equations. The third is to problem solve with scale drawings, create geometric constructions and calculations with two and three dimensional shapes, and lastly being able to draw inferences from sample populations.
Acknowledgements
Implementation of the Common Core State Standards and Objectives for Mathematics through engaging instruction coupled with rigorous learning activities and assessment is hard work. During the 2011-2012 school year, a dedicated group of WV classroom teachers worked collaboratively to study the Middle School MathWhitney Bennett
WV
Roger Bennett
WV
Tila Boyce
WV
Debbie Conover
WV
Erin Gravley
WV
Christine McIntire
WV
Kerianne Mick
WV
Gary Santolla
WV
Anna Taylor
WV
Angela Walker
WV
Galen Weyer
WV
Content Reviewers
John Ford
WV State Department of Education
Celeste Glenn
West Virginia University
Design
WVU Academic Innovation
Dr. Sue Day Perroots
Associate Vice President
Instructional Designers
Sukanya Dutta-White
Celeste Glenn
Media Design
Mark Bennett
Web Development
Li Zheng
Eric Merrill
Director of K-12 Initiatives | 677.169 | 1 |
Homework is assigned in MyMathLab from
Pearson. It is the student's responsibility to keep track of homework
deadlines and complete homework on time. Students can work on problems
as many times as necessary to get the correct answer, but the Chapter
Quizzes can only be attempted with at least 80% correct on the homework
for the Chapter. Be careful of the warnings that MyMathLab gives you, as you might drop your score if you enter a problem that you already received credit for after the due date when a penalty is applied. You should review exercises from the Gradebook part of MyMathLab.
Chapter Quizzes (30%)
At the end of each chapter there will be a Chapter
Quiz. Students must have successfully completed 80% of the homework problems
correctly to take the Chapter Quiz. The Chapter Quiz is a timed exercise (45-55 minutes). Students can take the quiz up to three times over
a four day period and receive the best score on the Chapter Quiz. You are expected to take the Chapter Quiz much like you would take a Quiz in the classroom, meaning that you work your Chapter Quiz with only a pencil and paper helping you. MyMathLab is designed to have a lock down for Quizzes, which means that if you try to use other resources on your computer during the timed quiz, then you will be locked out of that attempt on your quiz and will no longer be able to answer questions. These Chapter Quizzes are your best preparation for the Midterms. (Because of the confusion at the beginning of the semester and because students may not be familiar with this course format, the Chapter Quiz 1 can be taken by just attempting the homework from Chapter 1 and will be allowed three attempts.) The lowest Chapter Quiz score will be dropped over the semester. For example, if you score lower than 80% on a homework, then you will not be able to take that Chapter Quiz, which will result in a zero, so that Chapter Quiz will be dropped.
Midterms (25% (12.5% Each))
Twice during the semester there will be Midterm
exams, which will be given in a classroom. Students will have to show
their Red IDs to take the exam. Time of the Midterms will be determined soon.Written documentation will be required for any excused absences.
Final (25%)
There will be a Final, which will be given in a
classroom. Students will have to show their Red IDs to take the Final.
Students must score at least 50% on the Final to receive a
grade more than a C-.The Final will be May 11, 6-8 PM, with other Group Finals.
The grades below will be assigned unless a student scores below 50% on the Final.
Score
Grade
Score
Grade
88-100%
A
65-70%
C
85-88%
A-
62-65%
C-
82-85%
B+
59-62%
D+
76-82%
B
53-59%
D
73-76%
B-
50-53%
D-
70-73%
C+
Less than 50%
F
SDS
Student disability Services (594-6473) is the campus office responsible for determining and providing appropriate accommodations for students with disabilities. Student needing these services should visit the following site:
Students are encouraged to take advantage of the tutoring facilities available. We provide on-line answering of questions as well as several campus facilities with TAs supported by the Department of Mathematics. This free tutoring has been established to help students succeed in this course. Success is most likely if you carefully read the material in the chapters and work all homework problems as if they were a Chapter Quiz.
Academic Dishonesty
Academic misconduct will not be tolerated. The following steps are usually taken with a student caught cheating: The instructor will normally record a zero or an "F" for that exam, quiz, homework, or project; although the instructor may decide to give an "F" grade for the course.
All cases of academic dishonesty will be reported to the Center for Student Rights and Responsibilities. The office will investigate complaints in order to determine whether University disciplinary action is to be pursued. For more information on SDSU's policies and procedures regarding academic misconduct visit the following site: | 677.169 | 1 |
COURSE
OBJECTIVES. In Math 111, students developed the algebraic
skills needed to analyze, graph, and understand the basic properties of a
function. This course refines those skills by introducing calculus concepts.These refinements include understanding and
using limits, derivatives, and integrals. With these new techniques, the
student can find maximum and minimum points, areas under curves, marginals
(cost, revenue, and profit), average and instantaneous rates of change,
consumer and producer surpluses, point
of diminishing returns, tangent lines to functions, and develop mathematical
models of real business applications.
To illustrate the practical use of calculus, students will
be asked to complete project assignments which involve analyzing data. This
data represents information defined by some practical problem. One method which
may be employed to study the behavior of the data is regression analysis.Students will be shown how to set up the data
and use regression tools if this method is needed.
COURSE OUTCOMES.After taking this course, a student should be
able to understand:
1.Limits of functions.
2.How limits are connected to continuity,
derivatives, and integrals.
3.Derivatives.
4.Solving optimization problems including cost,
revenue, and profit.
5.Points of diminishing returns.
6.Differentials and Marginal Analysis.
7.Exponential growth and decay models.
8.Antiderivatives and indefinite integrals.
9.Fundamental Theorem of Calculus and definite
integrals.
10.Finding areas under curves and applications
involving consumer and
producer surpluses.
11.Approximation of definite integrals as a
limit of sums.
MTH 1120-02Term II2009-10p. 2
MATHEMATICAL ANALYSIS II Jan
20, 2010.Put your name and MTH 112 on
the cover. No books, computers, calculators or cell phones will be allowed on
any examination. Students may bring a single 8.5x11 sheet of paper with notes
for each exam. Exams will contain multiple choice questions.All exams will be graded using a curve rather
than straight scale.
SPECIAL
EXAM RULES.(1) Exams will
start at the beginning of the designated class time. No additional time will be
allowed especially if a student arrives late. Be on time. (2) Students are
responsible for pencils or pens to take exams. None will be provided by the
instructor. (3)No test will be given to
a student who has not turned in a bluebook. (4) Tests will begin only after all
materials are removed from the desktop. Any loss of time (due to excess talking
or delays in removing materials) will not be made up. (5)One hour exam score or project score will be
dropped from calculation of final grade but not the final exam score.If any exam is missed or not submitted, that
exam or project score is chosen as the "dropped" score. If more than two scores
are missing, a grade of F will be given for the missing scores.(6) NO MAKEUP EXAMS will be given.
DATES for EXAMS,
HOMEWORK, and PROJECT ASSIGNMENTS
Dates for all graded assignments are listed on the attached
class schedule.Homework dates are in
RED, Project dates are in GREEN, and Exam dates are in Orange.
EXAM DATES.Please mark your calendars.
Exam 1.MondayFeb 08, 2010
Exam 2.WednesdayMar 17, 2010
Exam 3.WednesdayApr 14, 2010
Final Exam.ThursdayApril 29, 2010from 11:00am – 12:50 pm
PROJECT DATES.
Project 1.MondayFeb 01, 2010
Project 2.MondayMar 01, 2010
Project 3.MondayMar 29, 2010
MTH 1120-02Term II2009-10p. 3
MATHEMATICAL ANALYSIS II
HOMEWORK DATES.Sections for the homework (HW) assignments
are in parentheses. All assignments due dates are tentative or estimates.
Specific times will be given by the instructor.
HW 1(1.3-1.5)01/20/10HW2(1.6, 2.1, 2.2)01/27/10
HW 3(2.3-2.5)02/03/10HW4(2.6-2.7)02/15/10
HW 5(3.1-3.3)02/22/10HW6(3.4,3.5,3.8)03/03/10
HW 7 (4.3,4,5,4.6)03/23/10HW8(5.1-5.3)03/31/10
HW 9 (5.4-5.6) 04/07/10HW10(6.1,6.2, 6.4)04/21/10
IMPORTANT NOTE FOR PROJECT AND HOMEWORK DUE DATES:
1)Each day that an
assignment is late will lose 25 points.
2)Once an assignment
is returned to the class with answers, no late assignment will be accepted.
GRADING.The final grade for the course is determined
by the following percentages:
Hour
exams:40%
Graded
Homework:15%
Projects:15%
Class
attendance and participation:5%
Final
Exam:25%
Note: Class participation includes attendance, attention, no
late arrivals or early departures, no miscellaneous talking
not related to the class lecture, and general interest in the class. Use
of cell phones during class lectures and exams
are prohibited.
IMPORTANT FACTS.Last day to drop class No W:Jan 15, 2010
No class Days: Jan 18, 2010, Mar 08-12, 2010
Last day to withdraw with "W": April 1, 2010
ACADEMIC
INTEGRITY.Everything
submitted for grading is expected to be a student's own work. Anything
suspected as being otherwise the case will be dealt with according to
University and College policies.
MTH 1120-02 Term II 2009-10p. 4
MATHEMATICAL ANALYSIS II
CHAPTER
MATERIAL.The following
sections will be covered (as time allows). | 677.169 | 1 |
Equivalent to MATH 130 and 222. This course combines the topics of Trigonometry and
Pre-Calculus and is designed to fulfill the requirements of both courses as a prerequisite
for MATH 251. Topics include a study of functions, function families, their properties
and transformations, compositions, inverses and combinations, complex numbers, and
vectors. Function families include linear, trigonometric, logarithmic, exponential,
polynomial, power, and rational. Multiple representations of functions are emphasized.
Units: 6
Degree Credit
Letter Grade Only
Lecture hours/semester: 96-108
Homework hours/semester: 192-216
Prerequisites: MATH 120 or Math 123 or appropriate score on the District math placement test and
other measures. | 677.169 | 1 |
Need math homework help? Select your textbook and enter the page you are working on and we will give you the exact lesson you need to finish your math homework. So, you can't do it and this means you require algebra 2 homework help or any other type of professional assistance. You can get algebra 2 homework help from these. Pre-Algebra, Algebra I, Algebra II, Geometry: homework help by free math tutors, solvers, lessons. Each section has solvers (calculators), lessons, and a place where. StudyDaddy is the place where you can get easy online Algebra homework help. Our qualified tutors are available online 24/7 to answer all your homework questions.
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Algebra 2 Homework Help and Answers. Let K = 3m +2 such that m is some integer (This is because it leaves remainder 2 when divided by 3. StudyDaddy is the place where you can get easy online Algebra homework help. Our qualified tutors are available online 24/7 to answer all your homework questions. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math. Get homework help at HomeworkMarket.com is an on-line marketplace for homework assistance and tutoring. You can ask homework questions. View Your Algebra 2 Answers Now. Free. Browse the books below to find your textbook and get your solutions now.
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Free math problem solver answers your algebra homework questions with step-by-step explanations. Click your Algebra 2 textbook below for homework help. Our answers explain actual Algebra 2 textbook homework problems. Each answer shows how to solve a textbook. Need math homework help? Select your textbook and enter the page you are working on and we will give you the exact lesson you need to finish your math homework. Each topic listed below can have lessons, solvers that show work, an opportunity to ask a free tutor, and the list of questions already answered by the free tutors. Free math problem solver answers your algebra homework questions with step-by-step explanations. | 677.169 | 1 |
GRE Subject Mathematics
The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level.
Approximately 50 percent of the questions involve calculus and its applications — subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors.
About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.
The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.
CALCULUS — 50%
Material learned in the usual sequence of elementary calculus courses — differential and integral calculus of one and of several variables — includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations and other branches of mathematics.
ALGEBRA — 25%
Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics
Other topics: general topology, geometry, complex variables, probability and statistics, and numerical analysis
The above descriptions of topics covered in the test should not be considered exhaustive; it is necessary to understand many other related concepts. Prospective test takers should be aware that questions requiring no more than a good precalculus background may be quite challenging; such questions can be among the most difficult questions on the test. In general, the questions are intended not only to test recall of information but also to assess test takers' understanding of fundamental concepts and the ability to apply those concepts in various situations | 677.169 | 1 |
PRE AP Algebra I
Grading Scale
The
students will be given points for all assignments, but grades will be weighted
with the following percentages: Assignments: 30%
Notebook/Journal: 10%
Projects/Labs: 20% Test:40%
Supplies
Composition Notebook
Lined paper (college ruled)
Pencil and eraser
1 box Kleenex
Scotch Tape
Markers
Hand Sanitizer
Calculators
Each
student will be assigned a calculator for the whole year. The student
is responsible for any damage on that particular calculator. If the
calculator is lost and or damaged, the student will have to replace it.
Course Description
Part 1:
Students will gain a basic understanding of the properties and
attributes of functions. This will be accomplished through the use of
manipulatives and symbols in order to simplify expressions solve
equations and inequalities in problem situations. Students will
translate among the various representations of functions and gain an
understanding of slope and intercepts of linear functions (including the
effects of change in parameters) in real-world and mathematical
situations. Numerous hands-on experiences will be provided throughout
the course. Graphing calculator technology will be integrated throughout
the course.
Part 2: Students will
formulate, solve and analyze solutions for equations and inequalities
based on linear functions and systems of linear equations. Students
will understand that the graphs of quadratic functions are affected by
the parameters of the functions, describe those affects, and solve the
quadratic functions using appropriate methods. Students will also model
function situations that are neither linear nor quadratic. Numerous
hands-on experiences will be provided throughout this course. Graphing
calculator technology will be integrated throughout the course. | 677.169 | 1 |
A new version of this classic math primer explains the timeless theories of calculus in a contemporary and comprehensible way, with updated method and terminology, and twenty new recreational problems for practice and enjoyment. 20,000 first printing.
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...earth. r' = radius of moon, or other body. P = moon's horizontal parallax = earth's angular semidiameter as seen from the moon. f = moon's angular semidiameter. Now = P (in circular measure), r'-r = r (in circular measure);.'. r: r':: P: P', or (radius of earth): (radios of moon):: (moon's parallax): (moon's semidiameter). Examples. 1. Taking the moon's horizontal parallax as 57', and its angular diameter as 32', find its radius in miles, assuming the earth's radius to be 4000 miles. Here moon's semidiameter = 16';.-. 4000::: 57': 16';.-. r = 400 16 = 1123 miles. 2. The sun's horizontal parallax being 8"8, and his angular diameter 32V find his diameter in miles. ' Am. 872,727 miles. 3. The synodic period of Venus being 584 days, find the angle gained in each minute of time on the earth round the sun as centre. Am. l"-54 per minute. 4. Find the angular velocity with which Venus crosses the sun's disc, assuming the distances of Venus and the earth from the sun are as 7 to 10, as given by Bode's Law. Since (fig. 50) S V: VA:: 7: 3. But Srhas a relative angular velocity round the sun of l"-54 per minute (see Example 3); therefore, the relative angular velocity of A V round A is greater than this in the ratio of 7: 3, which gives an approximate result of 3"-6 per minute, the true rate being about 4" per minute. Annual ParaUax. 95. We have already seen that no displacement of the observer due to a change of position on the earth's surface could apparently affect the direction of a fixed star. However, as the earth in its annual motion describes an orbit of about 92 million miles radius round the sun, the different positions in space from which an observer views the fixed stars from time to time throughout the year must be separated ...
Differential Calculus Made Easy has been writing in one form or another for most of life. You can find so many inspiration from Differential Calculus Made Easy also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Differential Calculus Made Easy book for free.
Integral Calculus Made Easy has been writing in one form or another for most of life. You can find so many inspiration from Integral Calculus Made Easy also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Integral is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject. The original text continues to be available as of 2008 from Macmillan and Co., but a 1998 update by Martin Gardner is available from St. Martin's Press which provides an introduction; three preliminary chapters explaining functions, limits, and derivatives; an appendix of recreational calculus problems; and notes for modern readers. Gardner changes "fifth form boys" to the more American sounding (and gender neutral) "high school students," updates many now obsolescent mathematical notations or terms, and uses American decimal dollars and cents in currency examples.
Written by three gifted-and funny-teachers, How to Ace Calculus provides humorous and readable explanations of the key topics of calculus without the technical details and fine print that would be found in a more formal text. Capturing the tone of students exchanging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams-all the tricks of the trade that will make learning the material of first-semester calculus a piece of cake. Funny, irreverent, and flexible, How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun.
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Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson, considered a classic and elegant introduction to the subject. It has been a most favorite for students. | 677.169 | 1 |
Relativity, Gravitation and Cosmology: A Basic Introduction Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, in the formalism of Minkowski's four-dimensional space-time, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein field equation, as well as many modern cosmological subjects, from primordial inflation and cosmic microwave anisotropy to the dark energy that propels an accelerating universe. The author presents the subject with an emphasis on physical examples and simple applications without the full tensor apparatus. The reader first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, precession of Mercury's perihelion, black holes, and cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Many modern topics in cosmology are discussed in this book: from inflation, cosmic microwave anisotropy to the "dark energy" that propels an accelerating universe. Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity and cosmology on their own. Download link: Buy Premium To Support Me & Get Resumable Support & Fastest Speed!
Mean Curvature Flow and Isoperimetric Inequalities Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.Trends in Harmonic Analysis This book illustrates the wide range of research subjects developed by the Italian research group in harmonic analysis, originally started by Alessandro Figà-Talamanca, to whom it is dedicated in the occasion of his retirement. In particular, it outlines some of the impressive ramifications of the mathematical developments that began when Figà-Talamanca brought the study of harmonic analysis to Italy; the research group that he nurtured has now expanded to cover many areas. Therefore the book is addressed not only to experts in harmonic analysis, summability of Fourier series and singular integrals, but also in potential theory, symmetric spaces, analysis and partial differential equations on Riemannian manifolds, analysis on graphs, trees, buildings and discrete groups, Lie groups and Lie algebras, and even in far-reaching applications as for instance cellular automata and signal processing (low-discrepancy sampling, Gaussian noise). Download links Buy Premium To Support Me & Get Resumable Support & Fastest Speed!
Understanding Adobe Photoshop CS6: The Essential Techniques for Imaging Professionals English | 320 pages | PDF | 52 MB Photoshop For professionals it provides a chance to add to their skill base. The book cuts though the clutter and is unique, focusing not just on digital photography, but also the Web, graphic design, and video.
M Andrea De Cataldo, "The Hodge Theory of Projective Manifolds" 2007 | ISBN: 1860948006 | 116 pages | PDF | 3,7 MB This book is a written-up and expanded version of eight lectures on the Hodge theory of projective manifolds. It assumes very little background and aims at describing how the theory becomes progressively richer and more beautiful as one specializes from Riemannian, to Khler, to complex projective manifolds. Though the proof of the Hodge Theorem is omitted, its consequences topological, geometrical and algebraic are discussed at some length. The special properties of complex projective manifolds constitute an important body of knowledge and readers are guided through it with the help of selected exercises. Despite starting with very few prerequisites, the concluding chapter works out, in the meaningful special case of surfaces, the proof of a special property of maps between complex projective manifolds, which was discovered only quite recently. | 677.169 | 1 |
I don't think you will find that a big problem. None of the main ideas of the book require any mathematical knowledge.
Some of the examples are mathematical in nature. For instance, there is about twelve pages in chapter 6 that concerns using an infinite list object to represent a sequence of increasingly accurate approximations to the solution of a certain financial problem. People who have studied calculus will immediately understand the ideas here; people who haven't might understand them anyway, and even if you don't you can always skip those examples; most of them have nothing in particular to do with math.
My own tendency is to write a lot of math stuff, because I find it very interesting, but while I was writing HOP I tried really hard to get rid of the mathematics, because I knew that a lot of people don't | 677.169 | 1 |
Now a Major Motion Picture .... well, how about a YouTube sequence of 20 VIDEOS, look for Mathematical Modeling and Computational Calculus I.
This book will take you from not being able to spell calculus to doing calculus just the way I did it for twenty years as an engineer at high tech firms like Lockheed and Stanford Telecom. You will learn how physical processes are modeled using mathematics and analyzed using computational calculus. Systems studied include satellite orbits, the orbits of the earth and moon, rocket trajectories, the Apollo mission trajectory, the Juno space probe, electrical circuits, oscillators, filters, tennis serves, springs, friction, automobile suspension systems, lift and drag, and airplane dynamics. And not a single theorem in sight.
This book focuses on differential equation models because they are what scientists and engineers use to model processes involving change. Historically, this has presented a big problem for science education because while the models are easy enough to create, solving the differential equations analytically usually requires advanced mathematical techniques and their clever application. But, that was before computers; now, with computers, solutions to differential equations can be computed directly, without the need of theorems or any advanced mathematics, using the formula distance equals velocity times time. It's just that simple. The book will show you how it's done.
Is there a trick here? Of course, here it is: suppose you, as Newton did, want to compute the trajectory of a falling apple, and let's say that the apple's acceleration is constant and equals 10 meters/second/second. So the apple's velocity at the instant it falls is 0 m/s, after 1 second it is 10 m/s, after 2 seconds it is 20 m/s, and after t seconds it is v(t) = 10*t m/s.
You want to know the distance d(t) the apple has fallen after t seconds. This is the problem calculus was developed to solve, that is, given a velocity function v(t), determine the corresponding distance function d(t). To solve it Newton proved theorem after theorem and finally came up with a formula that gives the answer, in this case d(t) = 5*t*t.
But computational calculus bypasses all the theorems and formulas: to calculate how far the apple has fallen after 8 seconds, i.e. d(8), it just subdivides the interval of interest, 8 seconds in this case, into small sub-intervals, say 1 second each, and since the apple's velocity is known at the start of each sub-interval, it uses that velocity to estimate how far the apple falls in the sub-interval using the formula, get ready for it, distance equals velocity times time. The distances for all the sub-intervals are added and that's how far the apple falls in 8 seconds. Capiche?
This is the way it is actually done in the engineering world.
There are two big advantages to the computational method, first, it is very easy to learn, there is only one formula, distance = velocity times time. Second, for most velocity functions v(t) you can't use Newton's method because there is no formula for d(t) that works, none exist. But you can always use computational calculus, no matter how complex the problem, you just compute away and get the answer. Computational calculus has transformed engineering and science.
Well, just how many calculations do you have to do? Ans: lots. So in the book we do the first few calculations by hand, and then show you how to automate the process using FREEMAT, a free clone of MATLAB. Using FREEMAT you can write the instructions for a calculation once and then perform the calculation 10,000 times using the statement FOR i=1:10000, the calculation, END, and t
About the Author:
The author has a PhD from Berkeley and spent twenty years as a research engineer in high tech firms like Honeywell Aerospace Division, Lockheed, and Stanford Telecom. | 677.169 | 1 |
Sixth Grade Math
Math
The core goal of this course is to expand math fundamentals and begin to use them creatively and expertly to solve problems. A combination of practicing addition, subtraction, multiplication and division with whole numbers, fractions, decimals, and percent will form the foundation. Other topics to be included are basic algebra, geometry, graphs, and work with the coordinate plane. All topics will be presented with real world applications. The textbook is the on-line Glencoe Math from McGraw Hill. | 677.169 | 1 |
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sloving a system of equations in two variables by the substitution method | 677.169 | 1 |
Discrete Integrable Systems (MAGIC051)
Announcements
General
Description
Outline
During the recent history of mathematics the theory of difference equations (∆Es)
has been lagging behind the analogous theory of differential equations (DEs). In the last two
decades, however, a considerable amount of progress has been made in understanding the structures behind certain specific classes of difference equations which we call integrable. This course provides an overview of these modern developments, highlighting the connections with various other branches of mathematics.
Background
Integrable systems form a special class of mathematical models and equations that allow for exact and rigorous methods for their solution. They come in all kinds of forms and shapes, such as nonlinear evolution equations (PDEs), Hamiltonian many-body systems, special types of nonlinear ODEs and certain quantum mechanical models. They possess remarkable properties, such as the existence of (multi-)soliton solutions, infinite number of conservation laws, higher and generalised symmetries, underlying infinite-dimensional group structures, etc. Their study has led to the development of new mathematical techniques, such as the inverse scattering transform method, finite-gap integration techniques and the application of Riemann-Hilbert problems.
A remarkable feature is that most integrable systems possess natural discrete analogues, described by difference equations rather than differential equations. Obviously, one can discretize a given differential equation in many ways, but to find a discretization that preserves the essential integrability features of an integrable differential equation is a far from trivial enterprise. Nonetheless, such discretizations have been found and constructed, and the resulting difference equations not only possess all the hall marks of integrability, but in fact turn out to be richer and more transparent than their continuous counterparts. Through their study a major boost has been given to the theory of difference equations in general, leading to the introduction of new mathematical notions and phenomena.
The proposed course is meant to be an introduction to this relatively new and exciting area of research, which draws together many facets of modern pure and applied mathematics, such as "discrete differential geometry", special function theory, geometric numerical integration, algebraic geometry and analysis. Nevertheless, the course will be given on a rather elementary level, without assuming any specific prerequisites beyond standard undergraduate mathematics. It will emphasise the interconnection between the various models and their emergence from basic principles. | 677.169 | 1 |
SUB: MATHEMATICS
MODULE-1C
MODULE-1C
Linear Algebra: Matrix algebra, system of linear equations, Eigen values and eigen vectors.
_
1.
The rank of the matrix A= [
a) One
b) Two
c) Three
d) None of the above
] is
Ans:
The rank of a matrix is order of highes
SUB: MATHEMATICS
MODULE-8A
MODULE-8A
Cauchys and Eulers equations, Initial and boundary value problems
An equation of the form
is called cauchys homogeneous linear equation.
Where X is a function of
Such equations can be reduced to linear differential equ
SUB: MATHEMATICS
MODULE-3A
MODULE-3A
Evaluation of definite and improper integrals
_
Consider the infinite region S that lies under the curve
to the right of the line
above the x- axis, and
. You might think that, since S is infinite in extend, its area m
SUB: MATHEMATICS
MODULE-2A
MODULE-2A
Functions of single variable, Limit, continuity and differentiability,
Mean value theorems.
Function: For sets A and B, a function A to B, defined by f: A
which assigns to every element x
, a unique element f(x)
, is a
To him throwing such things, an opposite gust, shrieking from the north wind, strikes the sail,
and lifts the waves to the stars. The oars are broken, then the prow overturns, and gives the side
to the waves, a towering mountain of water follows in a heap
Scarcely from sight of Sicilian land, the happy ones were giving fabrics into the deep and were
rushing the foams of salt with bronze, when Juno, preserving an eternal wound under her chest,
(says) these things with herself: That I, the victor, cease from
SUB: MATHEMATICS
MODULE-7A
MODULE-7A
Higher order linear differential equations with constant coefficients
These are the form
Where
are constants
In symbols form (
I.
)
To find the complementary function
Write the auxiliary equation (A.E)
and
Solve it for
SUB: MATHEMATICS
MODULE-7B
MODULE-7B
Higher order linear differential equations with constant coefficients.
_
5. The complete solution for the ordinary
1. A function n(x) satisfied the
differential equation
where L is a constant. The boundary
conditions a
SUB: MATHEMATICS
MODULE-4A
MODULE-4A
Partial derivatives, Total derivative, Maxima and minima
Functions of two or more variables: A symbol z which has a definite value for every pair
of values of x and y is called a function of two independent variables x
SUB: MATHEMATICS
MODULE-6A
MODULE-6A
Differential equations: First order equations (linear and nonlinear)
A differential equation is an equation which involves differential co-efficient or
differentials.
The order of a differential equation is the order o
To here Aeneas goes under with seven ships having been collected from the whole number, and
with a great love of land, having disembarked, the Trojans gain the desired sand, and they place
their limbs dripping with salt on the shore. And first Achates str | 677.169 | 1 |
One-dimensional dynamics owns many deep results and avenues of active mathematical research. Numerous inroads to this research exist for the advanced undergraduate or beginning graduate student. This book provides glimpses into one-dimensional dynamics with the hope that the results presented illuminate the beauty and excitement of the field. Much of this material is covered nowhere else in textbook format, some are mini new research topics in themselves, and novel connections are drawn with other research areas both inside and outside the text. The material presented here is not meant to be approached in a linear fashion. Readers are encouraged to pick and choose favourite topics. Anyone with an interest in dynamics, novice or expert alike, will find much of interest within.
• Wide range of topics; many topics covered appear nowhere else in 'text book format'; material is a filering from the research literature of currently active topics
• Readers encouraged to pick and choose topics of interest, rather than rigid linear fashion
• Employs style that allows students an active role
Reviews
'… particularly useful for students/beginners in the field. Due to an extensive bibliography, it will also serve as a very good reference book.'
European Mathematical Society Newsletter
'This book is intended as a text for an advanced undergraduate or beginning graduate students. As well as providing a brief account on the fundamental concepts of analysis and dynamical systems (Chapters 1-4 and 7-8), and a thorough explanation of topological entropy for piece-wise monotone interval maps (Chapter 9, sometimes with original proofs), the book contains substantial parts on unimodal interval maps and one chapter on complex quadratic polynomials. The quality of this exposition is very good: the material is organized so that all proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples and exercises.'
Zentralblatt MATH | 677.169 | 1 |
The Ancient Civilizations of Greece and Rome
Solving Algebraic Equations
6 copies of 1 bookAlso available as a Big Book!Math that students can relate to!This full-color, photo-illustrated math reader seamlessly integrates Math with the curriculum areas of Science and Social Studies. Grab your students' attention and inspire a love of Math and of learning. | 677.169 | 1 |
Course Listings
DS 082
Introductory Algebra
3 cr
This course combines topics in pre- and elementary algebra including operations with whole numbers, fractions, percents, exponents, signed numbers, order of operations; ratio and proportion; functions; graphs of linear equations; solving and graphing systems of linear equations and inequalities; operations with polynomials and factoring polynomials. Requires 100 extra minutes per week of lab time each week. Pre-requisite: None. | 677.169 | 1 |
Browse related Subjects ...
Read More Frisk can deliver. The text's clearly useful applications emphasize problem solving to effectively develop the skills students need for future mathematics courses, such as college algebra, and for real life. "Beginning and Intermediate Algebra" is the ideal text for professors who want to eliminate the significant overlap of topics found in separate beginning and intermediate algebra texts.The Fourth Edition of "Beginning and Intermediate Algebra" also features a robust suite of online course management, testing, and tutorial resources for instructors and students.This includes BCA/iLrn Testing and Tutorial, vMentor live online tutoring, the Interactive Video Skillbuilder CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE), powered by BCA/iLrn. TLE provides a complete courseware package, featuring a diagnostic tool that gives instructors the capability to create individualized study plans. With TLE, a cohesive, focused study plan can be put together to help each student succeed in math5344637 | 677.169 | 1 |
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Provincial Types in American Fiction
Format: Hardcover
Language: English
Format: PDF / Kindle / ePub
Size: 13.88 MB
Downloadable formats: PDF
Ideal as a reference or quick review of the fundamentals of linear algebra, this book offers a matrix-oriented approach-with more emphasis on Euclidean n-space, problem solving, and applications, and less emphasis on abstract vector spaces. High Schoolers 2010-2011 Top 20 Books Read Among U. Please try logging in or sending yourself a password reminder. One fantastic thing that Roger Dooley has done is to break these studies up into separate categories, something that was failed at in the Yes! book above. | 677.169 | 1 |
Weekly SI Sessions
About SI for this Course
SI is a program that is intended to help you develop a better understanding of material through collaborative learning. You will have a chance to work with your peers on problems and activities that will help prepare you to be successful in MATH 160 and future courses. SI is a great place to meet your classmates, make new friends, and have fun with calculus!
Every session will be unique and will cover different material, so come as often as you'd like! I will use our website to post some materials, but not everything we do in sessions will be available online.
Feel free to contact me through the SI website, before class, or during our sessions if you have any questions. I hope to see everyone at SI this semester!
I have just posted a page of suggested practice problems as well as additional study resources to aid you in preparing for Exam 1. If you have trouble accessing this resource from the email, the document can be found on the MATH 160 SI webpage. Have fun studying! | 677.169 | 1 |
Math 1230 is the second of a three-semester sequence in differential and integral calculus. Topics include: techniques and applications of integrations, introduction to first odrder differential equations and infinite series. This roughly corresponds to Chapters 5-7 and part of Chapter 14 of the text. Students are responsible for all material in the text and all material presented in class. This includes any material not in the text and all material in the text that was not presented in class.
Calculus II Skill Exam:
Students must take a Skills Exam focusing on basic skills of differentiation and integration. Following the link here to login to MapleTA website for sample practice problems.
Course Prerequisites: A passing grade (C or better) in Math 1220 Calculus I or equivalent.
Academic Integrity: I am asked to include the following paragraph on academic integrity in this syllabus: Students are responsible for making themselve aware of and understanding the policies and procedures in the Undergraduate (pp. 274-276) [Graduate (pp.25-27)] Catalog that pertain to Academic Honesty. These policies include cheating, fabrication, falsification and forgery, multiple submission, plagiarism, complicity and computer misuse. If there is reason to believe you have been involved in academic dishonesty, you will be referred to the Office of Student Conduct. You will be given the opportunity to review the charge(s). If you believe you are not responsible, you will have the opportunity for a hearing. You should consult with me if you are uncertain about an issue of academic honesty prior to the submission of an assignment or test.
Objectives:
1. Enhance the understanding of the concept of functions.
2. Understanding the concept of limit and how it relates average and instantaneous quantities.
3. Understanding the concept of derivative, interpreting it geometrically, physically and using it in optimization and linear approximation.
4. Understanding integration and its relationship with differentiation and applying integration in goemetrical and physical problems.
5. Learning the proper use of mathematical notation.
6. Developing sufficient computational skills in differential and integral operations for subsequent calculus courses and for applications in other areas.
7. Developing abilities to tackle multi-step problems and to explain the process.
8. Experiencing the potential of modern computer algebra systems (Maple) in assisting the analysis of problems in calculus and the visualization of their solutions.
9. Developing skills in mathematical reasoning.
10. Developing a broad perspective of how various different topics in this course fit together.
Calculator and Maple:
A graphing calculator is required for this class. A TI-89 or TI-92 PLUS is recommended. We will use many of the extra capabilities of these calculators. You can find help and examples prepared by Professor Pence here. We will also use the software package Maple.
Homework:
Homework is assinged in the schedule along with online exercises. You should work on the problems in the book first for practice (not collected) and then do the online exercises for credit. It is essential that homework be done conscientiously. If you have any questions about problems, please ask them in class or in office hours. Tutor Lab is also an excellent place to get help. Experience show that the probability of success in a calculus class is very low for the student who does not work the assigned problems conscientiously (and ask about the ones that cause difficulty). Remember the following confucian proverb:
I hear, and I forget; I see, and I remember; I do, and I understand.
Quizzes:
Weekly short quizzes will be given, usually on Fridays. They will cover all of the material before the day of the quiz.
No make up quiz will be given. However, I will drop your two worst quizzes.
Final:
The final exam will be given in the last class 10:15-12:15 on Wednesday, April 26.
Grading:
The final is 25%. the quizzes count for 40% and the online homework counts for 35%. This gives a total of 100%.
Grading scale is approximately as follows:
A (88-100%) BA(80-87.99%) B (72-79.99%) CB(64-71.99%)
C(55-63.99%) DC(50-54.99%) D(43-49.99%) E(0-42.99%) | 677.169 | 1 |
Home › Articles › The Complete Mathematical Terms Dictionary. The Complete Mathematical Terms Dictionary. Understanding math concepts is critical in our world today. GLOSSARY OF MATHEMATICAL TERMS. This is not a comprehensive dictionary of mathematical terms, just a quick reference for some of the terms commonly used in this website. Glossary of terms that have been discussed or mentioned on these pages. Letter A. 1 Algebra—math area that provides tools for picturing, stating, and simplifying relationships; generalization of arithmetic ideas by which unknown values and variables.
Mathematical terminology
Free on-line Mathemeatics Dictionary for students studying mathematics subjects and courses. Over 2000 terms defined. Pages in category Mathematical terminology The following 106 pages are in this category, out of 106 total. This list may not reflect recent changes. GLOSSARY OF MATHEMATICAL TERMS. This is not a comprehensive dictionary of mathematical terms, just a quick reference for some of the terms commonly used in this website. In Algebra a term is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or − signs. Parents | Glossary | About Us | Join. Wonders of Math : Search: Custom Search A: B: C: D: E: F: G: H: I: J: K: L: M: N. lowest terms. M. mean median.
Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Home › Articles › The Complete Mathematical Terms Dictionary. The Complete Mathematical Terms Dictionary. Understanding math concepts is critical in our world today. This is a glossary of math definitions for common and important mathematics terms used in arithmetic, geometry, and statistics. Mathwords: Terms and Formulas from Beginning Algebra to Calculus. An interactive math dictionary with enough math words, math terms, math formulas, pictures, diagrams. Interactive, animated maths dictionary with over 630 common math terms and math words explained in simple language with examples. Device friendly version with 950.
Basic Mathematical Terminology Number line and 1 - to -1 correspondence:. Business Math Essentials: Working with Decimals, Percent, Fractions and Exponents. In Algebra a term is either a single number or variable, or numbers and variables multiplied together. Terms are separated by + or − signs. Comprehensive math vocabulary lists are based on the Common Core State Math Standards and organized by K-12 grade level.
Free on-line Mathemeatics Dictionary for students studying mathematics subjects and courses. Over 2000 terms defined. 1 Algebra—math area that provides tools for picturing, stating, and simplifying relationships; generalization of arithmetic ideas by which unknown values and variables. This is a glossary of math definitions for common and important mathematics terms used in arithmetic, geometry, and statistics. Glossary of terms that have been discussed or mentioned on these pages. Letter A.
Interactive, animated maths dictionary for kids with over 600 common math terms explained in simple language. Math glossary with math definitions, examples, math. Pages in category Mathematical terminology The following 106 pages are in this category, out of 106 total. This list may not reflect recent changes. Interactive, animated maths dictionary with over 630 common math terms and math words explained in simple language with examples. Device friendly version with 950. 310 Glossary Glossary This glossary contains words and phrases from Fourth through Sixth Grade Everyday Mathematics. To place the definitions in broader mathematical. | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). contens software to be used to promote the development of virtual and non-virtual teaching-learning machines for electronics laboratories oriented to practical classes at high schools, technical schools and engineering schools | 677.169 | 1 |
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