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donna kirk, university of wisconsin at superior 6100 main street ms 375 houston, texas 77005 individual print copies and bulk orders can be purchased through our website. attribution 4.0 international license cc by 4.0 . under this license, any user of this textbook or the textbook contents herein must provide proper attribution as follows: if you redistribute this textbook in a digital format including but not limited to pdf and html , then you must retain on every page the following attribution: if you redistribute this textbook in a print format, then you must include on every physical page the if you redistribute part of this textbook, then you must retain in every digital format page view including but not limited to pdf and html and on every physical printed page the following attribution: if you use this textbook as a bibliographic reference, please include original publication year 1 2 3 4 5 6 7 8 9 10 tam 23 placement courses and low cost, personalized courseware that helps students learn. a nonprofit ed tech their courses and meet their educational goals. unsurpassed teaching, and contributions to the betterment of our world. it seeks to fulfill this mission by cultivating a diverse community of learning and discovery that produces leaders across the spectrum of human access and learning for everyone. to see the impact of our supporter community and our most updated list of chan zuckerberg initiative arthur and carlyse ciocca charitable foundation ann and john doerr bill melinda gates foundation the william and flora hewlett foundation the hewlett packard company rusty and john jaggers the calvin k. kazanjian economics foundation charles koch foundation leon lowenstein foundation, inc. the maxfield foundation burt and deedee mcmurtry michelson 20mm foundation national science foundation the open society foundations jumee yhu and david e. park iii brian d. patterson usa international foundation the bill and stephanie sick fund steven l. smith diana t. go robin and sandy stuart foundation the stuart family foundation tammy and guillermo trevi o valhalla charitable foundation white star education foundation study where you want, what you want, when you want. access. the future of education. when you access your book in our web view, you can use our new online highlighting and note taking features to create your own study guides. our books are free and flexible, forever.
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. 1. 1 basic set concepts 1. 3 understanding venn diagrams 1. 4 set operations with two sets 1. 5 set operations with three sets 2. 1 statements and quantifiers 2. 2 compound statements 2. 3 constructing truth tables 2. 4 truth tables for the conditional and biconditional 2. 5 equivalent statements 2. 6 de morgan s laws 2. 7 logical arguments real number systems and number theory 3. 1 prime and composite numbers 3. 2 the integers 3. 3 order of operations 3. 4 rational numbers 3. 5 irrational numbers 3. 6 real numbers 3. 7 clock arithmetic 3. 9 scientific notation 3. 10 arithmetic sequences 3. 11 geometric sequences number representation and calculation 4. 1 hindu arabic positional system 4. 2 early numeration systems 4. 3 converting with base systems 4. 4 addition and subtraction in base systems 4. 5 multiplication and division in base systems 5. 1 algebraic expressions 5. 2 linear equations in one variable with applications 5. 3 linear inequalities in one variable with applications 5. 4 ratios and proportions 5. 5 graphing linear equations and inequalities 5. 6 quadratic equations with two variables with applications 5. 8 graphing functions 5. 9 systems of linear equations in two variables 5. 10 systems of linear inequalities in two variables 5. 11 linear programming 6. 1 understanding percent 6. 2 discounts, markups, and sales tax 6. 3 simple interest 6. 4 compound interest 6. 5 making a personal budget 6. 6 methods of savings 6. 8 the basics of loans 6. 9 understanding student loans 6. 10 credit cards 6. 11 buying or leasing a car 6. 12 renting and homeownership 6. 13 income tax 7. 1 the multiplication rule for counting 7. 4 tree diagrams, tables, and outcomes 7. 5 basic concepts of probability 7. 6 probability with permutations and combinations 7. 7 what are the odds?
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? 7. 8 the addition rule for probability 7. 9 conditional probability and the multiplication rule 7. 10 the binomial distribution 7. 11 expected value 8. 1 gathering and organizing data 8. 2 visualizing data 8. 3 mean, median and mode 8. 4 range and standard deviation 8. 6 the normal distribution 8. 7 applications of the normal distribution 8. 8 scatter plots, correlation, and regression lines 9. 1 the metric system 9. 2 measuring area 9. 3 measuring volume 9. 4 measuring weight 9. 5 measuring temperature 10. 1 points, lines, and planes 10. 4 polygons, perimeter, and circumference 10. 7 volume and surface area 10. 8 right triangle trigonometry voting and apportionment 11. 1 voting methods 11. 2 fairness in voting methods 11. 3 standard divisors, standard quotas, and the apportionment problem 11. 4 apportionment methods 11. 5 fairness in apportionment methods 12. 1 graph basics 12. 2 graph structures 12. 3 comparing graphs 12. 4 navigating graphs 12. 5 euler circuits 12. 6 euler trails 12. 7 hamilton cycles 12. 8 hamilton paths 12. 9 traveling salesperson problem 13. 1 math and art 13. 2 math and the environment 13. 3 math and medicine 13. 4 math and music 13. 5 math and sports co req appendix : integer powers of 10 our mission to transform learning so that education works for every student. through our partnerships with philanthropic organizations and our alliance with other educational resource companies, we re breaking down the most common barriers to learning. because we believe that everyone should and can have access to knowledge. contemporary mathematics is licensed under a creative commons attribution 4. 0 international cc by license, which because our books are openly licensed, you are free to use the entire book or select only the sections that are most relevant to the needs of your course. feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. you can even provide a direct link in your syllabus to the sections in the web view of your book. in contemporary mathematics, art contains attribution to its title, creator or rights holder, host platform, and license within the caption. because the art is openly licensed, anyone may reuse the art as long as they provide the same attribution to its original source.
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. for illustrations e. g., graphs, charts, etc. that are not credited, use the following sometimes occur. since our books are web based, we can make updates periodically when deemed pedagogically about contemporary mathematics contemporary mathematics is designed to meet the requirements for a liberal arts mathematics course. the textbook covers a range of topics that are typically found in a liberal arts course as well as some topics to connect mathematics to the world around us. the text provides stand alone sections with a focus on showing relevance in the features as well as the examples, exercises, and exposition. every section begins with a set of clear and concise learning objectives, which have been thoroughly revised to be both measurable and more closely aligned with current teaching practice. these objectives are designed to help the instructor decide what content to include or assign and to guide student expectations of learning. after completing the section and end of section exercises, students should be able to demonstrate mastery of the learning objectives. check your understanding : concept checks to confirm students understand content at the end of every section immediately before the exercise sets are provided to help bolster confidence before embarking on homework. people in mathematics : a mix of historic and contemporary profiles aimed to incorporate extensive diversity in gender and ethnicity. the profiles incorporate how the person s contribution has benefitted students or is relevant to their lives in some way. who knew? : a high interest feature designed to showcase something interesting related to the section contents. these features are crafted to offer something students might be surprised to find is so relevant to them. work it out : offers some activity ideas in line with the sections to support the learning objectives. tech check : highlights technologies that support content in the section. projects : a feature designed to put students in the driver s seat researching a topic using various online resources. it is intended to be primarily or wholly non computational. projects utilize online research and writing to summarize their section summaries distill the information in each section for both students and instructors down to key, concise points addressed in the section. key terms are bold and are followed by a definition in context. answers and solutions to questions in the book answers for your turn and check your understanding exercises are provided in the answer key at the end of the book. the section exercises, chapter reviews, and chapter tests are intended for homework assignments or assessment ; thus, student facing solutions are provided in the student solution manual for only a subset of the exercises.
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. solutions for all exercises are provided in the instructor solution manual for instructors to share with students at their discretion, as is standard for such resources. about the authors donna kirk, university of wisconsin at superior donna kirk received her b. s. in mathematics from the state university of new york at oneonta and her master s degree from city university seattle in educational technology and curriculum design. after teaching math in higher education for more than twenty years, she joined university of wisconsin s education department in 2021, teaching math education for teacher preparation. she is also the director of a stem institute focused on connecting underrepresented students with access to engaging and innovative experiences to empower themselves to pursue stem related careers. barbara boschmans beaudrie, northern arizona university brian beaudrie, northern arizona university matthew cathey, wofford college valeree falduto, palm beach state college maureen gerlofs, texas state university quin hearn, broward college ian walters, d youville college anna pat alpert, navarro college mario barrientos, angelo state university keisha brown, perimeter college at georgia state university hugh cornell, university of north florida david crombecque, university of southern california shari davis, old dominion university angela everett, chattanooga state community college david french, tidewater community college michele gribben, mcdaniel college celeste hernandez, dallas college richland trevor jack, illinois wesleyan university kristin kang, grand view university karla karstens, university of vermont sergio loch, grand view university andrew misseldine, southern utah university carla monticelli, camden county college cindy moss, skyline college jill rafael, sierra college gary rosen, university of southern california faith willman, harrisburg area community college student and instructor resources we ve compiled additional resources for both students and instructors, including student solution manuals, instructor solution manuals, and powerpoint lecture slides. instructor resources require a verified instructor account, which you academic integrity builds trust, understanding, equity, and genuine learning. while students may encounter significant challenges in their courses and their lives, doing their own work and maintaining a high degree of authenticity will result in meaningful outcomes that will extend far beyond their college career. faculty, administrators, resource providers, and students should work together to maintain a fair and positive experience. we realize that students benefit when academic integrity ground rules are established early in the course. to that end, continuum to align these practices with your institution and course policies.
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. you may then include the graphic on your syllabus, present it in your first course meeting, or create a handout for students. this book s page for updates. for an in depth review of academic integrity strategies, we highly recommend visiting the community hubs on oer commons a platform for instructors to share community created resources that support resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content. we encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an opportunity both to enrich your courses and to engage with other faculty. to reach the community hubs, visit as allies in making high quality learning materials accessible, our technology partners offer optional low cost tools that figure 1. 1 a flatware drawer is like a set in that it contains distinct objects. credit : modification of work silverware by jo naylor flickr, cc by 2. 0 1. 1 basic set concepts 1. 3 understanding venn diagrams 1. 4 set operations with two sets 1. 5 set operations with three sets think of a drawer in your kitchen used to store flatware. this drawer likely holds forks, spoons, and knives, and possibly other items such as a meat thermometer and a can opener. the drawer in this case represents a tool used to group a collection of objects. the members of the group are the individual items in the drawer, such as a fork or a spoon. the members of a set can be anything, such as people, numbers, or letters of the alphabet. in statistical studies, a set is a well defined collection of objects used to identify an entire population of interest. for example, in a research study examining the effects of a new medication, there can be two sets of people : one set that is given the medication and a different set that is given a placebo control group. in this chapter, we will discuss sets and venn diagrams, which are graphical ways to show relationships between different groups. 1 introduction 1. 1 basic set concepts figure 1. 2 a spoon, fork, and knife are elements of the set of flatware. credit : modification of work cupofjoy wikimedia cc0 1. 0 public domain dedication after completing this section, you should be able to : represent sets in a variety of ways. represent well defined sets and the empty set with proper set notation. compute the cardinal value of a set. differentiate between finite and infinite sets. differentiate between equal and equivalent sets.
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. sets and ways to represent them think back to your kitchen organization. if the drawer is the set, then the forks and knives are elements in the set. sets can be described in a number of different ways : by roster, by set builder notation, by interval notation, by graphing on a number line, and by venn diagrams. sets are typically designated with capital letters. the simplest way to represent a set with only a few members is the roster or listing method, in which the elements in a set are listed, enclosed by curly braces and separated by commas. for example, if represents our set of flatware, we can represent by using the following set notation with the roster method : writing a set using the roster or listing method write a set consisting of your three favorite sports and label it with a capital there are multiple possible answers depending on what your three favorite sports are, but any answer must list three different sports separated by commas, such as the following : your turn 1. 1 1. write a set consisting of four small hand tools that might be in a toolbox and label it with a capital all the sets we have considered so far have been well defined sets. a well defined set clearly communicates whether an element is a member of the set or not. the members of a well defined set are fixed and do not change over time. consider the following question. what are your top 10 songs of 2021? you could create a list of your top 10 favorite songs from 2021, but the list your friend creates will not necessarily contain the same 10 songs. so, the set of your top 10 songs of 2021 is not a well defined set. on the other hand, the set of the letters in your name is a well defined set because it does not vary unless of course you change your name. the nfl wide receiver, chad johnson, famously 1 sets changed his name to chad ochocinco to match his jersey number of 85. identifying well defined sets for each of the following collections, determine if it represents a well defined set. the group of all past vice presidents of the united states. a group of old cats. the group of all past vice presidents of the united states is a well defined set, because you can clearly identify if any individual was or was not a member of that group. for example, britney spears is not a member of this set, but joe biden is a member of this set. a group of old cats is not a well defined set because the word old is ambiguous.
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. some people might consider a seven year old cat to be old, while others might think a cat is not old until it is 13 years old. because people can disagree on what is and what is not a member of this group, the set is not well defined. your turn 1. 2 for each of the following collections, determine if it represents a well defined set. 1. a collection of medium sized potatoes. 2. the original members of the black eyed peas musical group. on january 20, 2021, kamala harris was sworn in as the first woman vice president of the united states of america. if we were to consider the set of all women vice presidents of the united states of america prior to january 20, 2021, this set would be known as an empty set ; the number of people in this set is 0, since there were no women vice presidents before harris. the empty set, also called the null set, is written symbolically using a pair of braces,, or a zero with a slash through it, the set containing the number, is a set with one element in it. it is not the same as the empty set, does not have any elements in it. symbolically : representing the empty set symbolically represent each of the following sets symbolically. the set of prime numbers less than 2. the set of birds that are also mammals. a prime number is a natural number greater than 1 that is only divisible by one and itself. since there are no prime numbers less than 2, this set is empty, and we can represent it symbolically as follows : these two different symbols for the empty set can be used interchangeably. the set of birds and the set of mammals do not intersect, so the set of birds that are also mammals is empty, and we can represent it symbolically as your turn 1. 3 1. represent the set of all numbers divisible by 0 symbolically. 1. 1 basic set concepts the number zero we use the number zero to represent the concept of nothing every day. the machine language of computers is binary, consisting only of zeros and ones, and even way before that, the number zero was a powerful invention that allowed our understanding of mathematics and science to develop.
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. the historical record shows the babylonians first used zeros around 300 b. c., while the mayans developed and began using zero separately around 350 a. d. what is considered the first formal use of zero in arithmetic operations was developed by the indian mathematician brahmagupta around 650 a. d. another interesting feature of the number zero is that although it is an even number, it is the only number that is neither negative nor positive. for larger sets that have a natural ordering, sometimes an ellipsis is used to indicate that the pattern continues. it is common practice to list the first three elements of a set to establish a pattern, write the ellipsis, and then provide the last element. consider the set of all lowercase letters of the english alphabet,. this set can be written symbolically as the sets we have been discussing so far are finite sets. they all have a limited or fixed number of elements. we also use an ellipsis for infinite sets, which have an unlimited number of elements, to indicate that the pattern continues. for example, in set theory, the set of natural numbers, which is the set of all positive counting numbers, is represented as notice that for this set, there is no element following the ellipsis. this is because there is no largest natural number ; you can always add one more to get to the next natural number. because the set of natural numbers grows without bound, it is an infinite set. writing a finite set using the roster method and an ellipsis write the set of even natural numbers including and between 2 and 100, and label it with a capital. include an ellipsis. write the label,, followed by an equal sign and then a bracket. write the first three even numbers separated by commas, beginning with the number two to establish a pattern. next, write the ellipsis followed by a comma and the last number in the list, 100. finally, write the closing bracket to complete the set. write the label,, followed by an equal sign and then a bracket. write the first three even numbers separated by commas, beginning with the number 2 to establish a pattern. next, write the ellipsis followed by a comma and the last number in the list, 100. finally, write the closing bracket to complete the set.
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. 1 sets your turn 1. 4 1. use an ellipsis to write the set of single digit numbers greater than or equal to zero and label it with a capital our number system is made up of several different infinite sets of numbers. the set of integers, is another infinite set of numbers. it includes all the positive and negative counting numbers and the number zero. there is no largest or writing an infinite set using the roster method and ellipses write the set of integers using the roster method, and label it with a. step 1 : as always, we write the label and then the opening bracket. because the negative counting numbers are infinite, to represent that the pattern continues without bound to the left, we must use an ellipsis as the first element in our list. step 2 : we place a comma and follow it with at least three consecutive integers separated by commas to establish a step 3 : add an ellipsis to the end of the list to show that the set of integers continues without bound to the right. complete the list with a closing bracket. the set of integers may be represented as follows : your turn 1. 5 1. write the set of odd numbers greater than 0 and label it with a capital a shorthand way to write sets is with the use of set builder notation, which is a verbal description or formula for the set. for example, the set of all lowercase letters of the english alphabet,, written in set builder notation is : this is read as, set is the set of all elements is a lowercase letter of the english alphabet. writing a set using set builder notation using set builder notation, write the set of all types of balls. explain what the notation means. the verbal description of the set is, set is the set of all elements is a ball. this set can be written in set builder notation as follows : your turn 1. 6 1. using set builder notation, write the set of all types of cars. writing sets using various methods consider the set of letters in the word happy. determine the best way to represent this set, and then write the set using 1. 1 basic set concepts either the roster method or set builder notation, whichever is more appropriate. because the letters in the word happy consist of a small finite set, the best way to represent this set is with the roster method. choose a label to represent the set, such as notice that the letter p is only represented one time.
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. this occurs because when representing members of a set, each unique element is only listed once no matter how many times it occurs. duplicate elements are never repeated when representing members of a set. your turn 1. 7 1. use the roster method or set builder notation to represent the collection of all musical instruments. computing the cardinal value of a set almost all the sets most people work with outside of pure mathematics are finite sets. for these sets, the cardinal value or cardinality of the set is the number of elements in the set. for finite set, the cardinality is denoted symbolically as. for example, a set that contains four elements has a cardinality of 4. how do we measure the cardinality of infinite sets? the smallest infinite set is the set of natural numbers, or counting. this set has a cardinality of pronounced aleph null. all sets that have the same cardinality as the set of natural numbers are countably infinite. this concept, as well as notation using aleph, was introduced by mathematician georg cantor who once said, a set is a many that allows itself to be thought of as a one. computing the cardinal value of a set write the cardinal value of each of the following sets in symbolic form. the empty set. there are 5 distinct elements in set : a fork, a spoon, a knife, a meat thermometer, and a can opener. therefore, the cardinal value of set is 5 and written symbolically as because the empty set does not have any elements in it, the cardinality of the empty set is zero. symbolically we write this as : your turn 1. 8 write the cardinal value of each of the following sets in symbolic form. is the set of prime numbers less than 2. is the set of lowercase letters of the english alphabet, now that we have learned to represent finite and infinite sets using both the roster method and set builder notation, we should also be able to determine if a set is finite or infinite based on its verbal or symbolic description. one way to determine if a set is finite or not is to determine the cardinality of the set. if the cardinality of a set is a natural number, then the set is finite. 1 sets differentiating between finite and infinite sets classify each of the following sets as infinite or finite. is the set of lowercase letters of the english alphabet,. since 5 is a natural number, the set is finite.. since 26 is a natural number, the set is finite.
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. set is the set of rational numbers or fractions. because the set of integers is a subset of the set of rational numbers, and the set of integers is infinite, the set of rational numbers is also infinite. there is no smallest or largest rational number. your turn 1. 9 classify each of the following sets as infinite or finite. equal versus equivalent sets when speaking or writing we tend to use equal and equivalent interchangeably, but there is an important distinction between their meanings. consider a new ford escape hybrid and a new toyota rav4 hybrid. both cars are hybrid electric sport utility vehicles ; in that sense, they are equivalent. they will both get you from place to place in a relatively fuel efficient way. in this example we are comparing the single member set toyota rav4 hybrid to the single member set ford escape hybrid. since these two sets have the same number of elements, they are also equivalent mathematically, meaning they have the same cardinality. but they are not equal, because the two cars have different looks and features, and probably even handle differently. each manufacturer will emphasize the features unique to their vehicle to persuade you to buy it ; if the suvs were truly equal, there would be no reason to choose one over the other. now consider two honda cr vs that are made with exactly the same parts, on the same assembly line within a few minutes of each other these suvs are equal. they are identical to each other, containing the same elements without regard to order, and the only differentiator when making a purchasing decision would be varied pricing at different dealerships. the set honda cr v is equal to the set honda cr v. symbolically, we represent equal sets as equivalent sets as now, let us consider a toyota dealership that has 10 rav4s on the lot, 8 prii, 7 highlanders, and 12 camrys. there is a one to one relationship between the set of vehicles on the lot and the set consisting of the number of each type of vehicle on the lot. therefore, these two sets are equivalent, but not equal. the set rav4, prius, highlander, camry is equivalent to the set 10, 8, 7, 12 because they have the same number of elements. if two sets are equal, they are also equivalent, because equal sets also have the same cardinality. differentiating between equivalent and equal sets determine if the following pairs of sets are equal, equivalent, or neither.
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. 1. 1 basic set concepts the empty set and the set of prime numbers less than 2. the set of vowels in the word happiness and the set of consonants in the word happiness. sets e and f both have a cardinal value of 5, but the elements in these sets are different. so, the two sets are equivalent, but they are not equal : the set of prime numbers consists of the set of counting numbers greater than one that can only be divided evenly by one and itself. the set of prime numbers less than 2 is an empty set, since there are no prime numbers less than 2. therefore, these two sets are equal and equivalent. the set of vowels in the word happiness is and the set of consonants in the word happiness is the cardinal value of these sets two sets is because the cardinality of the two sets differs, they are not equivalent. further, their elements are not identical, so they are also not equal. your turn 1. 10 determine if the following pairs of sets are equal, equivalent, or neither. people in mathematics figure 1. 3 georg cantor credit : wikimedia, public domain georg cantor, the father of modern set theory, was born during the year 1845 in saint petersburg, russa and later moved to germany as a youth. besides being an accomplished mathematician, he also played the violin. cantor received his doctoral degree in mathematics at the age of 22. in 1870, at the age of 25 he established the uniqueness theorem for trigonometric series. his most significant work happened between 1874 and 1884, when he established the existence of transcendental numbers also called irrational numbers and proved that the set of real numbers are uncountably infinite despite the objections of his former professor leopold kronecker. cantor published his final treatise on set theory in 1897 at the age of 52, and was awarded the sylvester medial from the royal society of london in 1904 for his contributions to the field. at the heart of cantor s work was his goal to solve the continuum problem, which later influenced the works of david hilbert and ernst zermelo. 1 sets wikipedia contributors. cantor. wikipedia, the free encyclopedia, 23 mar. 2021. web. 20 jul. 2021. akihiro kanamori, set theory from cantor to cohen, editor s : dov m. gabbay, akihiro kanamori, john woods, handbook of the history of logic, north holland, volume 6, 2012. check your understanding 1. a is a well defined collection of objects.
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. 2. the of a finite set, is the number of elements in set 3. determine if the following description describes a well defined set : the top 5 pizza restaurants in chicago. 4. the united states is the only country to have landed people on the moon as of march 21, 2021. what is the cardinality of the set of all people who have walked on the moon prior to this date? is a set of a dozen distinct donuts, and set is a set of a dozen different types of apples. is set equal to set, equivalent to set, or neither? 6. is the set of all butterflies in the world a finite set or an infinite set? 7. represent the set of all upper case letters of the english alphabet using both the roster method and set builder section 1. 1 exercises for the following exercises, represent each set using the roster method. 1. the set of primary colors : red, yellow, and blue. 2. a set of the following flowers : rose, tulip, marigold, iris, and lily. 3. the set of natural numbers between 50 and 100. 4. the set of natural numbers greater than 17. 5. the set of different pieces in a game of chess. 6. the set of natural numbers less than 21. for the following exercises, represent each set using set builder notation. 7. the set of all types of lizards. 8. the set of all stars in the universe. 9. the set of all integer multiples of 3 that are greater than zero. 10. the set of all integer multiples of 4 that are greater than zero. 11. the set of all plants that are edible. 12. the set of all even numbers. for the following exercises, represent each set using the method of your choice. 13. the set of all squares that are also circles. 14. the set of natural numbers divisible by zero. 15. the set of mike and carol s children on the tv show, the brady bunch. 16. the set of all real numbers. 17. the set of polar bears that live in antarctica. 18. the set of songs written by prince. 19. the set of children s books written and illustrated by mo willems. 20. the set of seven colors commonly listed in a rainbow. for the following exercises, determine if the collection of objects represents a well defined set or not. 21. the names of all the characters in the book, the fault in our stars by john green.
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. 22. the five greatest soccer players of all time. 23. a group of old dogs that are able to learn new tricks. 24. a list of all the movies directed by spike lee as of 2021. 25. the group of all zebras that can fly an airplane. 26. the group of national baseball league hall of fame members who have hit over 700 career home runs. for the following exercises, compute the cardinal value of each set. 1. 1 basic set concepts 36. the set of numbers on a standard 6 sided die. for the following exercises, determine whether set are equal, equivalent or neither. for the following exercises, determine if the set described is finite or infinite. 45. the set of natural numbers. 46. the empty set. 47. the set consisting of all jazz venues in new orleans, louisiana. 48. the set of all real numbers. 49. the set of all different types of cheeses. 50. the set of all words in merriam webster s collegiate dictionary, eleventh edition, published in 2020. figure 1. 4 the players on a soccer team who are actively participating in a game are a subset of the greater set of team members. credit : pafc mezokovesd 108 by pusk s akad mia flickr, public domain mark 1. 0 after completing this section, you should be able to : represent subsets and proper subsets symbolically. compute the number of subsets of a set. apply concepts of subsets and equivalent sets to finite and infinite sets. the rules of major league soccer mls allow each team to have up to 30 players on their team. however, only 18 of these players can be listed on the game day roster, and of the 18 listed, 11 players must be selected to start the game. 1 sets how the coaches and general managers form the team and choose the starters for each game will determine the success of the team in any given year. the entire group of 30 players is each team s set. the group of game day players is a subset of the team set, and the group of 11 starters is a subset of both the team set and the set of players on the game day roster. is a subset of set if every member of set is also a member of set. symbolically, this relationship is written as sets can be related to each other in several different ways : they may not share any members in common, they may share some members in common, or they may share all members in common.
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. in this section, we will explore the way we can select a group of members from the whole set. every set is also a subset of itself, recall the set of flatware in our kitchen drawer from section 1. 1,. suppose you are preparing to eat dinner, so you pull a fork and a knife from the drawer to set the table. the set is a subset of set, because every member or element of set is also a member of set. more specifically, set is a proper subset of set, because there are other members of set not in set. this is written as. the only subset of a set that is not a proper subset of the set would be the set itself. the empty set or null set,, is a proper subset of every set, except itself. graphically, sets are often represented as circles. in the following graphic, set is represented as a circle completely enclosed inside the circle representing set, showing that set is a proper subset of set. the element an element that is in both set while we can list all the subsets of a finite set, it is not possible to list all the possible subsets of an infinite set, as it would take an infinitely long time. listing all the proper subsets of a finite set is a set of reading materials available in a shop at the airport,. list all the subsets of set step 1 : it is best to begin with the set itself, as every set is a subset of itself. in our example, the cardinality of set. there is only one subset of set that has the same number of elements of set step 2 : next, list all the proper subsets of the set containing elements. in this case,. there are three subsets that each contain two elements : step 3 : continue this process by listing all the proper subsets of the set containing elements. in this case,. there are three subsets that contain one element : step 4 : finally, list the subset containing 0 elements, or the empty set : 1. 2 subsets your turn 1. 11 1. consider the set of possible outcomes when you flip a coin,.
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. list all the possible subsets of set determining whether a set is a proper subset consider the set of common political parties in the united states, determine if the following sets are proper subsets of is a proper subset of, written symbolically as because every member of is a member of set also contains at least one element that is not in is a single member proper subset of, written symbolically as because green is a member of set also contains other members such as democratic that are not in is subset of because every member of is also a member of, but it is not a proper subset of are no members of that are not also in set. we can represent the relationship symbolically as is equal to set your turn 1. 12 consider the set of generation i legendary pok mon,. give an example of a proper subset containing : 1. one member. 2. three members. 3. no members. expressing the relationship between sets symbolically consider the subsets of a standard deck of cards : express the relationship between the following sets symbolically. is a proper subset of set is a proper subset of set is subset of itself, but not a proper subset of itself because is equal to itself. your turn 1. 13 1. express the relationship between the set of natural numbers, and the set of even numbers, 1 sets so far, we have figured out how many subsets exist in a finite set by listing them. recall that in example 1. 11, when we listed all the subsets of the three element set we saw that there are eight subsets. in your turn 1. 11, we discovered that there are four subsets of the two element subset,. a one element set has two subsets, the empty set and itself. the only subset of the empty set is the empty set itself. but how can we easily figure out the number of subsets in a very large finite set? it turns out that the number of subsets can be found by raising 2 to the number of elements in the set, using exponential notation to represent repeated multiplication. for example, the number of subsets of the set is equal to notation is used to represent repeated multiplication, appears as a factor the number of subsets of a finite set is equal to 2 raised to the power of is the number of elements in set, so this formula works for the empty set, also. computing the number of subsets of a set find the number of subsets of each of the following sets.
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. the set of top five scorers of all time in the nba : the set of the top four bestselling albums of all time :. so, the total number of subsets of. therefore, the total number of subsets of. so, the total number of subsets of your turn 1. 14 1. compute the total number of subsets in the set of the top nine tennis grand slam singles winners, in the early 17th century, the famous astronomer galileo galilei found that the set of natural numbers and the subset of the natural numbers consisting of the set of square numbers,, are equivalent. upon making this discovery, he conjectured that the concepts of less than, greater than, and equal to did not apply to infinite sets. sequences and series are defined as infinite subsets of the set of natural numbers by forming a relationship between the sequence or series in terms of a natural number,. for example, the set of even numbers can be defined using set builder notation as. the formula in this case replaces every natural number with two times the number, resulting in the set of even numbers,. the set of even numbers is also equivalent to the set of natural numbers. 1. 2 subsets you can make a career out of working with sets. applications of equivalent sets include relational database design relational databases that store data are tables of related information. each row of a table has the same number of columns as every other row in the table ; in this way, relational databases are examples of set equivalences for finite sets. in a relational database, a primary key is set up to identify all related information. there is a one to one relationship between the primary key and any other information associated with it. database design and analysis is a high demand career with a median entry level salary of about 85, 000 per year, according to salary. com. writing equivalent subsets of an infinite set using natural numbers, multiples of 3 are given by the sequence. write this set using set builder notation by expressing each multiple of 3 using a formula in terms of a natural number,. in this example, is a multiple of 3 and natural number. the symbol is read as is a member or element of. because there is a one to one correspondence between the set of multiples of 3 and the natural numbers, the set of multiples of 3 is an equivalent subset of the natural your turn 1. 15 1. using natural numbers, multiples of 5 are given by the sequence.
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. write this set using set builder notation by associating each multiple of 5 in terms of a natural number, creating equivalent subsets of a finite set that are not equal a fast food restaurant offers a deal where you can select two options from the following set of four menu items for 6 : a chicken sandwich, a fish sandwich, a cheeseburger, or 10 chicken nuggets. javier and his friend michael are each purchasing lunch using this deal. create two equivalent, but not equal, subsets that javier and michael could choose to have for lunch. the possible two element subsets are : chicken sandwich, fish sandwich, chicken sandwich, cheeseburger, chicken sandwich, chicken nuggets, fish sandwich, cheeseburger, fish sandwich, chicken nuggets, and cheeseburger, chicken nuggets. one possible solution is that javier picked the set chicken sandwich, chicken nuggets, while michael chose the cheeseburger, chicken nuggets. because javier and michael both picked two items, but not exactly the same two items, these sets are equivalent, but not equal. your turn 1. 16 1. serena and venus williams walk into the same restaurant as javier and michael, but they order the same pair of items, resulting in equal sets of choices. if venus ordered a fish sandwich and chicken nuggets, what did serena 1 sets creating equivalent subsets of a finite set a high school volleyball team at a small school consists of the following players : angie, brenda, colleen, estella, maya, maria, penny, shantelle. create two possible equivalent starting line ups of six players that the coach could select for the next game. there are actually 28 possible ways that the coach could choose his starting line up. two such equivalent subsets are angie, brenda, maya, maria, penny, shantelle and angie, brenda, colleen, estella, maria, shantelle. each subset has six members, but they are not identical, so the two sets are equivalent but not equal. your turn 1. 17 1. consider the same group of volleyball players from above : angie, brenda, colleen, estella, maya, maria, penny, shantelle. the team needs to select a captain and an assistant captain from their members. list two possible equivalent subsets that they could select. check your understanding 8. every member of a of a set is also a member of the set. 9. explain what distinguishes a proper subset of a set from a subset of a set.
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. 10. the set is a proper subset of every set except itself. 11. is the following statement true or false? 12. if the cardinality of set, then set has a total of subsets. is to set 14. if every member of set is a member of set and every member of set is also a member set, then set to set section 1. 2 exercises for the following exercises, list all the proper subsets of each set. for the following exercises, determine the relationship between the two sets and write the relationship symbolically. for the following exercises, calculate the total number of subsets of each set. 1. 2 subsets for the following exercises, use the set of letters in the word largest as the set, 25. find a subset of that is equivalent, but not equal, to the set : 26. find a subset of that is equal to the set : 27. find a subset of that is equal to the set : 28. find a subset of that is equivalent, but not equal, to the set 29. find a subset of that is equivalent, but not equal, to the set : 30. find a subset of that is equal to the set : 31. find two three character subsets of set that are equivalent, but not equal, to each other. 32. find two three character subsets of set that are equal to each other. 33. find two five character subsets of set that are equal to each other. 34. find two five character subsets of set that are equivalent, but not equal, to each other. for the following exercises, use the set of integers as the set 35. find two equivalent subset of with a cardinality of 7. 36. find two equal subsets of with a cardinality of 4. 37. find a subset of that is equivalent, but not equal to, 38. find a subset of that is equivalent, but not equal to, 39. true or false. the set of natural numbers,, is equivalent to set 40. true or false. set is an equivalent subset of the set of rational numbers, 1. 3 understanding venn diagrams figure 1. 6 when assembling furniture, instructions with images are easier to follow, just like how set relationships are easier to understand when depicted graphically. credit : time to assemble more ikea furniture! by rod herrea flickr, cc after completing this section, you should be able to : utilize a universal set with two sets to interpret a venn diagram.
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. utilize a universal set with two sets to create a venn diagram. determine the complement of a set. 1 sets have you ever ordered a new dresser or bookcase that required assembly? when your package arrives you excitedly open it and spread out the pieces. then you check the assembly guide and verify that you have all the parts required to assemble your new dresser. now, the work begins. luckily for you, the assembly guide includes step by step instructions with images that show you how to put together your product. if you are really lucky, the manufacturer may even provide a url or qr code connecting you to an online video that demonstrates the complete assembly process. we can likely all agree that assembly instructions are much easier to follow when they include images or videos, rather than just written directions. the same goes for the relationships between sets. interpreting venn diagrams venn diagrams are the graphical tools or pictures that we use to visualize and understand relationships between sets. venn diagrams are named after the mathematician john venn, who first popularized their use in the 1880s. when we use a venn diagram to visualize the relationships between sets, the entire set of data under consideration is drawn as a rectangle, and subsets of this set are drawn as circles completely contained within the rectangle. the entire set of data under consideration is known as the universal set. consider the statement : all trees are plants. this statement expresses the relationship between the set of all plants and the set of all trees. because every tree is a plant, the set of trees is a subset of the set of plants. to represent this relationship using a venn diagram, the set of plants will be our universal set and the set of trees will be the subset. recall that this relationship is expressed symbolically as : to create a venn diagram, first we draw a rectangle and label the universal set then we draw a circle within the universal set and label it with the word trees. this section will introduce how to interpret and construct venn diagrams. in future sections, as we expand our knowledge of relationships between sets, we will also develop our knowledge and use of venn diagrams to explore how multiple sets can be combined to form new sets. interpreting the relationship between sets in a venn diagram write the relationship between the sets in the following venn diagram, in words and symbolically. the set of terriers is a subset of the universal set of dogs. in other words, the venn diagram depicts the relationship that all terriers are dogs.
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. this is expressed symbolically as your turn 1. 18 1. write the relationship between the sets in the following venn diagram, in words and symbolically. 1. 3 understanding venn diagrams so far, the only relationship we have been considering between two sets is the subset relationship, but sets can be related in other ways. lions and tigers are both different types of cats, but no lions are tigers, and no tigers are lions. because the set of all lions and the set of all tigers do not have any members in common, we call these two sets disjoint sets, or non overlapping sets. are disjoint sets if they do not share any elements in common. that is, if is a member of set is not a member of set is a member of set is not a member of set. to represent the relationship between the set of all cats and the sets of lions and tigers using a venn diagram, we draw the universal set of cats as a rectangle and then draw a circle for the set of lions and a separate circle for the set of tigers within the rectangle, ensuring that the two circles representing the set of lions and the set of tigers do not touch or overlap in any way. describing the relationship between sets describe the relationship between the sets in the following venn diagram. the set of triangles and the set of squares are two disjoint subsets of the universal set of two dimensional figures. the set of triangles does not share any elements in common with the set of squares. no triangles are squares and no squares are triangles, but both squares and triangles are 2d figures. your turn 1. 19 1. describe the relationship between the sets in the following venn diagram. 1 sets creating venn diagrams the main purpose of a venn diagram is to help you visualize the relationship between sets. as such, it is necessary to be able to draw venn diagrams from a written or symbolic description of the relationship between sets. to create a venn diagram : draw a rectangle to represent the universal set, and label it draw a circle within the rectangle to represent a subset of the universal set and label it with the set name. if there are multiple disjoint subsets of the universal set, their separate circles should not touch or overlap. drawing a venn diagram to represent the relationship between two sets draw a venn diagram to represent the relationship between each of the sets. all rectangles are parallelograms. all women are people.
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. the set of rectangles is a subset of the set of parallelograms. first, draw a rectangle to represent the universal set and label it with, then draw a circle completely within the rectangle, and label it with the name of the set it represents, in this example, both letters and names are used to represent the sets involved, but this is not necessary. you may use either letters or names alone, as long as the relationship is clearly depicted in the diagram, as shown below. 1. 3 understanding venn diagrams the universal set is the set of people, and the set of all women is a subset of the set of people. your turn 1. 20 1. draw a venn diagram to represent the relationship between each of the sets. all natural numbers are. draw a venn diagram to represent this relationship. drawing a venn diagram to represent the relationship between three sets all bicycles and all cars have wheels, but no bicycle is a car. draw a venn diagram to represent this relationship. step 1 : the set of bicycles and the set of cars are both subsets of the set of things with wheels. the universal set is the set of things with wheels, so we first draw a rectangle and label it with step 2 : because the set of bicycles and the set of cars do not share any elements in common, these two sets are disjoint and must be drawn as two circles that do not touch or overlap with the universal set. your turn 1. 21 1. airplanes and birds can fly, but no birds are airplanes. draw a venn diagram to represent this relationship. the complement of a set recall that if set is a proper subset of set, the universal set written symbolically as, then there is at least one element in set that is not in set. the set of all the elements in the universal set that are not in the subset called the complement of set. in set builder notation this is written symbolically as : 1 sets is used to represent the phrase, is a member of, and the symbol is used to represent the phrase, is not a member of. in the venn diagram below, the complement of set is the region that lies outside the circle and inside the rectangle. the universal set includes all of the elements in set and all of the elements in the complement of set and nothing else. consider the set of digit numbers.
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. let this be our universal set, now, let set consisting of all the prime numbers in set the complement of set the following venn diagram represents this relationship graphically. finding the complement of a set for both of the questions below, is a proper subset of given the universal set given the universal set the complement of set is the set of all elements in the universal set that are not in set the complement of set is the set of all dogs that are not beagles. all members of set are in the universal set because they are dogs, but they are not in set because they are not beagles. this relationship can be expressed in set build notation as follows : your turn 1. 22 for both of the questions below, is a proper subset of 1. given the universal set 2. given the universal set check your understanding 15. a venn diagram is a graphical representation of the between sets. 16. in a venn diagram, the set of all data under consideration, the set, is drawn as a rectangle. 17. two sets that do not share any elements in common are sets. 1. 3 understanding venn diagrams 18. the of a subset or the universal set,, is the set of all members of that are not in 19. the sets are subsets of the universal set. section 1. 3 exercises for the following exercises, interpret each venn diagram and describe the relationship between the sets, symbolically and in words. 1 sets for the following exercises, create a venn diagram to represent the relationships between the sets. 9. all birds have wings. 10. all cats are animals. 11. all almonds are nuts, and all pecans are nuts, but no almonds are pecans. 12. all rectangles are quadrilaterals, and all trapezoids are quadrilaterals, but no rectangles are trapezoids. insects and ants insects, but no ants are ladybugs. reptiles and snakes reptiles, but no lizards are snakes. are disjoint subsets of are disjoint subsets of is a subset of is a subset of music are sets with the following relationships : music are sets with the following relationships : for the following exercises, the universal set is the set of single digit numbers,. find the complement of each subset of for the following exercises, the universal set is bashful, doc, dopey, grumpy, happy, sleepy, sneezy.
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. find the complement of each subset of happy, bashful, grumpy doc, grumpy, happy, sleepy, bashful, sneezy, dopey for the following exercises, the universal set is. find the complement of each subset of for the following exercises, use the venn diagram to determine the members of the complement of set 1. 3 understanding venn diagrams 1. 4 set operations with two sets figure 1. 18 a large, multigenerational family contains an intersection and a union of sets. credit : family photo shoot bani syakur by mainur risyada flickr, cc by 2. 0 after completing this section, you should be able to : determine the intersection of two sets. determine the union of two sets. determine the cardinality of the union of two sets. apply the concepts of and and or to set operations. draw conclusions from venn diagrams with two sets. the movie yours, mine, and ours was originally released in 1968 and starred lucille ball and henry fonda. this movie, which is loosely based on a true story, is about the marriage of helen, a widow with eight children, and frank, a widower with ten children, who then have an additional child together. the movie is a comedy that plays on the interpersonal and organizational struggles of feeding, bathing, and clothing twenty people in one household. if we consider the set of helen s children and the set of frank s children, then the child they had together is the intersection of these two sets, and the collection of all their children combined is the union of these two sets. in this section, we will explore the operations of union and intersection as it relates to two sets. 1 sets the intersection of two sets the members that the two sets share in common are included in the intersection of two sets. to be in the intersection of two sets, an element must be in both the first set and the second set. in this way, the intersection of two sets is a logical and statement. symbolically, is written as : is written in set builder let us look at helen s and frank s children from the movie yours, mine, and ours. helen s children consist of the set and frank s children are included in the set is the set of children they had together.
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., because joseph is in both set finding the intersection of set the intersection of sets include the elements that set have in common : 3, 5, and 7. your turn 1. 23 notice that if sets are disjoint sets, then they do not share any elements in common, and empty set, as shown in the venn diagram below. determining the intersection of disjoint sets are disjoint, they do not share any elements in common. so, the intersection of set the empty set. your turn 1. 24 notice that if set is a subset of set is equal to set, as shown in the venn diagram below. 1. 4 set operations with two sets finding the intersection of a set and a subset is a subset of set is equal to set the set of odd natural your turn 1. 25 the union of two sets like the union of two families in marriage, the union of two sets includes all the members of the first set and all the members of the second set. to be in the union of two sets, an element must be in the first set, the second set, or both. in this way, the union of two sets is a logical inclusive or statement. symbolically, is written as : is written in set builder notation as : let us consider the sets of helen s and frank s children from the movie yours, mine, and ours again. helen s children is and frank s children is set. the union of these two sets is the collection of all nineteen of their children, notice, joseph is in both set, but he is only one child, so, he is only listed once in the union. finding the union of sets is the set formed by including all the unique elements in set, or both sets the first five elements of the union are the five unique elements in set. even though 3, 5, and 7 are also members of set, these elements are only listed one time. lastly, set includes the unique element 2, so 2 is also included as part of the union of sets your turn 1. 26 when observing the union of sets, notice that both set are subsets of can be represented in several different ways depending on the members that they have in common. if are disjoint sets, then would be represented with two disjoint circles within the universal set, as shown in the 1 sets venn diagram below.
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. share some, but not all, members in common, then the venn diagram is drawn as two separate circles if every member of set is also a member of set is a subset of set would be equal to set to draw the venn diagram, the circle representing set should be completely enclosed in the circle containing set finding the union of sets are disjoint, the union is simply the set containing all the elements in both set your turn 1. 27 finding the union of sets when one set is a subset of the other is a subset of set is equal to set 1. 4 set operations with two sets your turn 1. 28 set operation practice sets challenge is an application available on both android and iphone smartphones that allows you to practice and gain familiarity with the operations of set union, intersection, complement, and difference. figure 1. 24 google play store image of sets challenge game. credit : screenshot from google play the sets challenge application game uses some notation that differs from the notation covered in the text. the complement of set in this text is written symbolically as but the sets challenge game uses represent the complement operation. in the text we do not cover set difference between two sets, represented in the game as game this operation removes from set all the elements in for example, if set are subsets of the universal set there is a project at the end of the chapter to research the set determining the cardinality of two sets the cardinality of the union of two sets is the total number of elements in the set. symbolically the cardinality of. if two sets are disjoint, the cardinality of is the sum of the cardinality of 1 sets and the cardinality of set. if the two sets intersect, then is a subset of both set means that if we add the cardinality of set, we will have added the number of elements in twice, so we must then subtract it once as shown in the formula that follows.
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. the cardinality of is found by adding the number of elements in set to the number of elements in set, then subtracting the number of elements in the intersection of set are disjoint, then and the formula is still valid, but simplifies to determining the cardinality of the union of two sets the number of elements in set is 10, the number of elements in set is 20, and the number of elements in is 4. find the number of elements in using the formula for determining the cardinality of the union of two sets, we can say your turn 1. 29 determining the cardinality of the union of two disjoint sets are disjoint sets and the cardinality of set is 37 and the cardinality of set is 43, find the cardinality of to find the cardinality of, apply the formula, is the empty set, therefore your turn 1. 30 applying concepts of and and or to set operations to become a licensed driver, you must pass some form of written test and a road test, along with several other requirements depending on your age. to keep this example simple, let us focus on the road test and the written test. if you pass the written test but fail the road test, you will not receive your license. if you fail the written test, you will not be allowed to take the road test and you will not receive a license to drive. to receive a driver s license, you must pass the written test and the road test. for an and statement to be true, both conditions that make up the statement must be true. similarly, the intersection of two sets is the set of elements that are in both set. to be a, an element must be in set and also must be in set. the intersection of two sets corresponds to a logical and statement. the union of two sets is a logical inclusive or statement. say you are at a birthday party and the host offers leah, 1. 4 set operations with two sets lenny, maya, and you some cake or ice cream for dessert. leah asks for cake, lenny accepts both cake and ice cream, maya turns down both, and you choose only ice cream. leah, lenny, and you are all having dessert. the or statement is true if at least one of the components is true. maya is the only one who did not have cake or ice cream ; therefore, she did not have dessert and the or statement is false.
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. to be in the union of two sets, an element must be in set or both set applying the and or or operation find the set consisting of elements in : because only the elements 0 and 12 are members of both set because the set is the collection of all elements in set because the set is the collection of all elements in set, or both. parentheses are evaluated first : because the only member that both set share in common is 8. so, now we need to find the word translates to the union operation, the problem becomes which is equal to your turn 1. 31 find the set consisting of elements in : determine and apply the appropriate set operations to solve the problem don woods is serving cake and ice cream at his juneteenth celebration. the party has a total of 54 guests in attendance. suppose 30 guests requested cake, 20 guests asked for ice cream, and 12 guests did not have either cake or ice cream. how many guests had cake or ice cream? how many guests had cake and ice cream? the total number of people at the party is 54, and 12 people did not have cake or ice cream. recall that the total number of elements in the universal set is always equal to the number of elements in a subset plus the number of elements in the complement of the set,, or equivalently, a total of 42 people at the party had cake or ice cream. to determine the number of people who had both cake and ice cream, we need to find the intersection of the set of people who had cake and the set of people who had ice cream. from question 1, the number of people who had 1 sets cake or ice cream is 42. this is the union of the two sets. the formula for the union of two sets is use the information given in the problem and substitute the known values into the formula to solve for the number of people in the intersection : adding 30 and 20, the equation simplifies to your turn 1. 32 ravi and priya are serving soup and salad along with the main course at their wedding reception. the reception will have a total of 150 guests in attendance. a total of 92 soups and 85 salads were ordered, while 23 guests did not order any soup or salad. 1. how many guests had soup or salad or both? 2. how many guests had both soup and a salad?
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? the real inventor of the venn diagram john venn, in his writings, references works by both john boole and augustus de morgan, who referred to the circle diagrams commonly used to present logical relationships as euler s circles. leonhard euler s works were published over 100 years prior to venn s, and euler may have been influenced by the works of gottfried leibniz. so, why does john venn get all the credit for these graphical depictions? venn was the first to formalize the use of these diagrams in his book symbolic logic, published in 1881. further, he made significant improvements in their design, including shading to highlight the region of interest. the mathematician c. l. dodgson, also known as lewis carroll, built upon venn s work by adding an enclosing universal set. invention is not necessarily coming up with an initial idea. it is about seeing the potential of an idea and applying it to a new situation. margaret e. baron. a note on the historical development of logic diagrams : leibniz, euler and venn. the mathematical gazette, vol. 53, no. 384, 1969, pp. 113 125. jstor, www. jstor. org stable 3614533. accessed 15 july deborah bennett. drawing logical conclusions. math horizons, vol. 22, no. 3, 2015, pp. 12 15. jstor, www. jstor. org stable 10. 4169 mathhorizons. 22. 3. 12. accessed 15 july 2021. drawing conclusions from a venn diagram with two sets all venn diagrams will display the relationships between the sets, such as subset, intersecting, and or disjoint. in addition to displaying the relationship between the two sets, there are two main additional details that venn diagrams can include : the individual members of the sets or the cardinality of each disjoint subset of the universal set. a venn diagram with two subsets will partition the universal set into 3 or 4 sections depending on whether they are disjoint or intersecting sets. recall that the complement of set is the set of all elements in the universal set that are not in set figure 1. 25 side by side venn diagrams with disjoint and intersecting sets, respectively.
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. 1. 4 set operations with two sets using a venn diagram to draw conclusions about set membership is the collection of all elements in set are disjoint sets, there are no elements that are in both the empty set, the complement of set is the set of all elements in the universal set that are not in set the cardinality, or number of elements in set your turn 1. 33 venn diagram with two intersecting sets and members. using a venn diagram to draw conclusions about set cardinality figure 1. 27 venn diagram with two intersecting sets and number of elements in each section indicated. 1 sets the number of elements in is the number of elements in the number of elements in is the number of elements in the number of elements in set is the sum of all the numbers enclosed in the circle representing set your turn 1. 34 venn diagram with two disjoint sets and number of elements in each section. check your understanding 20. the of two sets is the set of all elements that they share in common. 21. the of two sets is the collection of all elements that are in set, or both set 22. the union of two sets is represented symbolically as. 23. the intersection of two sets is represented symbolically as. 24. if set is a subset of set is equal to set. 25. if set is a subset of set is equal to set. 26. if set are disjoint sets, then is the set. 27. the cardinality of, is found using the formula :. section 1. 4 exercises for the following exercises, determine the union or intersection of the sets as indicated. for the following exercises, use the sets provided to apply the and or or operation as indicated to find the resulting 1. 4 set operations with two sets 13. find the set consisting of elements in 14. find the set consisting of elements in 15. find the set consisting of elements in 16. find the set consisting of elements in 17. find the set consisting of elements in 18. find the set consisting of elements in 19. find the set consisting of the elements in 20. find the set consisting of the elements in 21. find the set consisting of the elements in 22. find the set consisting of the elements in 23. find the set consisting of elements in 24. find the set consisting of elements in for the following exercises, use the venn diagram provided to answer the following questions about the sets.
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. for the following exercises, use the venn diagram provided to answer the following questions about the sets. for the following exercises, use the venn diagram provided to answer the following questions about the sets. 1 sets for the following exercises, determine the cardinality of the union of set 43. if set 44. if set, find the number of elements in 45. if set, find the number of elements in 46. if set for the following exercises, use the venn diagram to determine the cardinality of 1. 4 set operations with two sets 1. 5 set operations with three sets figure 1. 28 companies like google collect data on how you use their services, but the data requires analysis to really mean something. credit : man holding smartphone and searches through google by nenad stojkovic flickr, cc by 2. 0 after completing this section, you should be able to : interpret venn diagrams with three sets. create venn diagrams with three sets. apply set operations to three sets. prove equality of sets using venn diagrams. have you ever searched for something on the internet and then soon after started seeing multiple advertisements for that item while browsing other web pages? large corporations have built their business on data collection and analysis. as we start working with larger data sets, the analysis becomes more complex. in this section, we will extend our knowledge of set relationships by including a third set. a venn diagram with two intersecting sets breaks up the universal set into four regions ; simply adding one additional set will increase the number of regions to eight, doubling the complexity of the problem. venn diagrams with three sets below is a venn diagram with two intersecting sets, which breaks the universal set up into four distinct regions. next, we see a venn diagram with three intersecting sets, which breaks up the universal set into eight distinct 1 sets shading venn diagrams venn diagram is an android application that allows you to visualize how the sets are related in a venn diagram by entering expressions and displaying the resulting venn diagram of the set shaded in gray. figure 1. 31 google play store image of venn diagram app. credit : screenshot from google play the venn diagram application uses some notation that differs from the notation covered in this text. the complement of set in this text is written symbolically as, but the venn diagram app uses 1. 5 set operations with three sets represent the complement operation.
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. the set difference operation,, is available in the venn diagram app, although this operation is not covered in it is recommended that you explore this application to expand your knowledge of venn diagrams prior to continuing with the next example. in the next example, we will explore the three main blood factors, a, b and rh. the following background information about blood types will help explain the relationships between the sets of blood factors. if an individual has blood factor a or b, those will be included in their blood type. the rh factor is indicated with a. for example, if a person has all three blood factors, then their blood type would be. in the venn diagram, they would be in the intersection of all if a person did not have any of these three blood factors, then their blood type would be and they would be in the set which is the region outside all three circles. interpreting a venn diagram with three sets use the venn diagram below, which shows the blood types of 100 people who donated blood at a local clinic, to answer the following questions. how many people with a type a blood factor donated blood? julio has blood type if he needs to have surgery that requires a blood transfusion, he can accept blood from anyone who does not have a type a blood factor. how many people donated blood that julio can accept? how many people who donated blood do not have the how many people had type a and type b blood? the number of people who donated blood with a type a blood factor will include the sum of all the values included in the a circle. it will be the union of sets in part 1, it was determined that the number of donors with a type a blood factor is 46. to determine the number of people who did not have a type a blood factor, use the following property, union is equal to, which means thus, 54 people donated blood that julio can this would be everyone outside the circle, or everyone with a negative rh factor, to have both blood type a and blood type b, a person would need to be in the intersection of sets. the two circles overlap in the regions labeled add up the number of people in these two regions to get the this can be written symbolically as 1 sets your turn 1. 35 use the same venn diagram in the example above to answer the following questions. 1. how many people donated blood with a type b blood factor? 2. how many people who donated blood did not have a type b blood factor?
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? 3. how many people who donated blood had a type b blood factor or were rh? most people know their main blood type of a, b, ab, or o and whether they are, but did you know that the international society of blood transfusion recognizes twenty eight additional blood types that have important implications for organ transplants and successful pregnancy? for more information, check out this article : creating venn diagrams with three sets in general, when creating venn diagrams from data involving three subsets of a universal set, the strategy is to work from the inside out. start with the intersection of the three sets, then address the regions that involve the intersection of two sets. next, complete the regions that involve a single set, and finally address the region in the universal set that does not intersect with any of the three sets. this method can be extended to any number of sets. the key is to start with the region involving the most overlap, working your way from the center out. creating a venn diagram with three sets a teacher surveyed her class of 43 students to find out how they prepared for their last test. she found that 24 students made flash cards, 14 studied their notes, and 27 completed the review assignment. of the entire class of 43 students, 12 completed the review and made flash cards, nine completed the review and studied their notes, and seven made flash cards and studied their notes, while only five students completed all three of these tasks. the remaining students did not do any of these tasks. create a venn diagram with subsets labeled : notes, flash cards, and review to represent how the students prepared for the test. step 1 : first, draw a venn diagram with three intersecting circles to represent the three intersecting sets : notes, flash cards, and review. label the universal set with the cardinality of the class. step 2 : next, in the region where all three sets intersect, enter the number of students who completed all three tasks. 1. 5 set operations with three sets step 3 : next, calculate the value and label the three sections where just two sets overlap. review and flash card overlap. a total of 12 students completed the review and made flash cards, but five of these twelve students did all three tasks, so we need to subtract :. this is the value for the region where the flash card set intersects with the review set. review and notes overlap.
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. a total of 9 students completed the review and studied their notes, but again, five of these nine students completed all three tasks. so, we subtract :. this is the value for the region where the review set intersects with the notes set. flash card and notes overlap. a total of 7 students made flash cards and studied their notes ; subtracting the five students that did all three tasks from this number leaves 2 students who only studied their notes and made flash cards. add these values to the venn diagram. step 4 : now, repeat this process to find the number of students who only completed one of these three tasks. a total of 24 students completed flash cards, but we have already accounted for of these. thus, students who just made flash cards. a total of 14 students studied their notes, but we have already accounted for of these. thus, students only studied their notes. a total of 27 students completed the review assignment, but we have already accounted for students only completed the review assignment. add these values to the venn diagram. 1 sets step 5 : finally, compute how many students did not do any of these three tasks. to do this, we add together each value that we have already calculated for the separate and intersecting sections of our three sets :. because there 43 students in the class, and, this means only one student did not complete any of these tasks to prepare for the test. record this value somewhere in the rectangle, but outside of all the circles, to complete the venn diagram. your turn 1. 36 1. a group of 50 people attending a conference who preordered their lunch were able to select their choice of soup, salad, or sandwich. a total of 17 people selected soup, 29 people selected salad and 35 people selected a sandwich. of these orders, 11 attendees selected soup and salad, 10 attendees selected soup and a sandwich, and 18 selected a salad and a sandwich, while eight people selected a soup, a salad, and a sandwich. create a venn diagram with subsets labeled soup, salad, and sandwich, and label the cardinality of each section of the venn diagram as indicated by the data. applying set operations to three sets set operations are applied between two sets at a time. parentheses indicate which operation should be performed first. as with numbers, the inner most parentheses are applied first. next, find the complement of any sets, then perform any union or intersections that remain.
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. 1. 5 set operations with three sets applying set operations to three sets perform the set operations as indicated on the following sets : the elements common to both because the only elements that are in both sets are 0 and 6. the collection of all elements in set because the intersection of these two sets is the set of elements that are common to both sets. the complement of set is the set of elements in the universal set that are not in set your turn 1. 37 using the same sets from example 1. 37, perform the set operations indicated. notice that the answers to the your turn are the same as those in the example. this is not a coincidence. the following equivalences hold true for sets : these are the associative property for set intersection and set union. these are the commutative property for set intersection and set union. these are the distributive property for sets over union and intersection, respectively. proving equality of sets using venn diagrams to prove set equality using venn diagrams, the strategy is to draw a venn diagram to represent each side of the equality, then look at the resulting diagrams to see if the regions under consideration are identical. augustus de morgan was an english mathematician known for his contributions to set theory and logic. de morgan s law for set complement over union states that. in the next example, we will use venn diagrams to prove de morgan s law for set complement over union is true. but before we begin, let us confirm de morgan s law works for a specific example. while showing something is true for one specific example is not a proof, it will provide us with some reason to believe that it may be true for all cases. we will use these sets in the equation to begin, find the value of the set defined by each side of the equation. is the collection of all unique elements in set of a union b,, is the set of all elements in the universal set that are not in. so, the left side the equation is equal to the set step 2 : the right side of the equation is is the set of all members of the universal set that are not in set step 3 : finally, is the set of all elements that are in both the numbers 1 and 7 are common to both we have demonstrated that de morgan s law for set 1 sets complement over union works for this particular example. the venn diagram below depicts this relationship.
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. proving de morgan s law for set complement over union using a venn diagram de morgan s law for the complement of the union of two sets use a venn diagram to prove that de morgan s law is true. step 1 : first, draw a venn diagram representing the left side of the equality. the regions of interest are shaded to highlight the sets of interest. is shaded on the left, and is shaded on the right. step 2 : next, draw a venn diagram to represent the right side of the equation. is shaded and is shaded. because mix to form is also shaded. figure 1. 40 venn diagram of intersection of the complement of two sets. step 3 : verify the conclusion. because the shaded region in the venn diagram for matches the shaded region in the venn diagram for, the two sides of the equation are equal, and the statement is true. this completes the proof that de morgan s law is valid. your turn 1. 38 1. de morgan s law for the complement of the intersection of two sets use a venn diagram to prove that de morgan s law is true. check your understanding 28. when creating a venn diagram with two or more subsets, you should begin with the region involving the most, then work your way from the center outward. 29. to construct a venn diagram with three subsets, draw and label three circles that overlap in a common 1. 5 set operations with three sets region inside the rectangle of the universal set to represent each of the three subsets. 30. in a venn diagram with three sets, the area where all three sets, overlap is equal to the set 31. when performing set operations with three or more sets, the order of operations is inner most first, then find the of any sets, and finally perform any union or intersection operations that remain. 32. to prove set equality using venn diagrams, draw a venn diagram to represent each side of the and then compare the diagrams to determine if they match or not. if they match, the statement is, otherwise it is not. section 1. 5 exercises a gamers club at baily middle school consisting of 25 members was surveyed to find out who played board games, card games, or video games. use the results depicted in the venn diagram below to answer the following exercises. 1. how many gamers club members play all three types of games : board games, card games, and video games?
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? 2. how many gamers are in the set board 3. if javier is in the region with a total of three members, what type of games does he play? 4. how many gamers play video games? 5. how many gamers are in the set board 6. how many members of the gamers club do not play video games? 7. how many members of this club only play board games? 8. how many members of this club only play video games? 9. how many members of the gamers club play video and card games? 10. how many members of the gamers club are in the set a blood drive at city honors high school recently collected blood from 140 students, staff, and faculty. use the results depicted in the venn diagram below to answer the following exercises. 11. blood type is the universal acceptor. of the 140 people who donated at city honors, how many had blood 1 sets 12. blood type is the universal donor. anyone needing a blood transfusion can receive this blood type. how many people who donated blood during this drive had 13. how many people donated with a type a blood factor? 14. how many people donated with a type a and type b blood factor that is, they had type ab blood. 15. how many donors were 16. how many donors were not 17. opal has blood type. if she needs to have surgery that requires a blood transfusion, she can accept blood from anyone who does not have a type b blood factor. how many people donated blood during this drive at city honors that opal can accept? for the following exercises, create a three circle venn diagram to represent the relationship between the described 21. the number of elements in the universal set, are subsets of 22. the number of elements in the universal set, are subsets of 23. the number of elements in the universal set, are subsets of 24. the number of elements in the universal set, are subsets of 25. the universal set,, has a cardinality of 36. 26. the universal set,, has a cardinality of 63. 27. the universal set,, has a cardinality of 72. 28. the universal set,, has a cardinality of 81. 29. the anime drawing club at pratt institute conducted a survey of its 42 members and found that 23 of them sketched with pastels, 28 used charcoal, and 17 used colored pencils.
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. of these, 10 club members used all three mediums, 18 used charcoal and pastels, 11 used colored pencils and charcoal, and 12 used colored pencils and pastels. the remaining club members did not use any of these three mediums. 30. a new suv is selling with three optional packages : a sport package, a tow package, and an entertainment package. a dealership gathered the following data for all 31 of these vehicles sold during the month of july. a total of 18 suvs included the entertainment package, 11 included the tow package, and 16 included the sport package. of these, five suvs included all three packages, seven were sold with both the tow package and sport package, 11 were sold with the entertainment and sport package, and eight were sold with the tow package and entertainment package. the remaining suvs sold did not include any of these optional packages. for the following exercises, perform the set operations as indicated on the following sets : for the following exercises, perform the set operations as indicated on the following sets : 1. 5 set operations with three sets for the following exercises, use venn diagrams to prove the following properties of sets : 43. commutative property for the union of two sets : 44. commutative property for the intersection of two sets : 45. associative property for the intersection of three sets : 46. associative property for the union of three sets : 47. distributive property for set intersection over set union : 48. distributive property for set union over set intersection : 1 sets 1. 1 basic set concepts cardinality of a set 1. 3 understanding venn diagrams complement of a set 1. 4 set operations with two sets intersection of two sets union of two sets 1. 1 basic set concepts identify a set as being a well defined collection of objects and differentiate between collections that are not well defined and collections that are sets. represent sets using both the roster or listing method and set builder notation which includes a description of the members of a set. in set theory, the following symbols are universally used : the set of natural numbers, which is the set of all positive counting numbers. the set of integers, which is the set of all the positive and negative counting numbers and the number zero. the set of rational numbers or fractions. distinguish between finite sets, infinite sets, and the empty set to determine the size or cardinality of a set.
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. distinguish between equal sets which have exactly the same members and equivalent sets that may have different members but must have the same cardinality or size. 1 chapter summary every member of a subset of a set is also a member of the set containing it. a proper subset of a set does not contain all the members of the set containing it. there is a least one member of that is not a member of set the number subsets of a finite set members is equal to 2 raised to the the empty set is a subset of every set and must be included when listing all the subsets of a set. understand how to create and distinguish between equivalent subsets of finite and infinite sets that are not equal to the original set. 1. 3 understanding venn diagrams a venn diagram is a graphical representation of the relationship between sets. in a venn diagram, the universal set, is the largest set under consideration and is drawn as a rectangle. all subsets of the universal set are drawn as circles within this rectangle. the complement of set includes all the members of the universal set that are not in set. a set and its complement are disjoint sets, they do not share any elements in common. to find the complement of set remove all the elements of set from the universal set, the set that includes only the remaining elements is the complement of set determine the complement of a set using venn diagrams, the roster method and set builder notation. 1. 4 set operations with two sets the intersection of two sets, is the set of all elements that they have in common. any member of must be is both set the union of two sets,, is the collection of all members that are in either in set or both sets two sets that share at least one element in common, so that they are not disjoint are represented in a venn diagram using two circles that overlap. the region of the overlap is the set the regions that include everything in the circle representing set or the circle representing set overlap is the set apply knowledge of set union and intersection to determine cardinality and membership using venn diagrams, the roster method and set builder notation. 1. 5 set operations with three sets a venn diagram with two overlapping sets breaks the universal set up into four distinct regions. when a third overlapping set is added the venn diagram is broken up into eight distinct regions. analyze, interpret, and create venn diagrams involving three overlapping sets. including the blood factors : a, b and rh to find unions and intersections.
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. to find cardinality of both unions and intersections. when performing set operations with three or more sets, the order of operations is inner most parentheses first, then fine the complement of any sets, then perform any union or intersection operations that remain. to prove set equality using venn diagrams the strategy is to draw a venn diagram to represent each side of the equality or equation, then look at the resulting diagrams to see if the regions under consideration are identical. if they regions are identical the equation represents a true statement, otherwise it is not true. 1. 1 basic set concepts 1. 4 set operations with two sets the number of subsets of a finite set is equal to 2 raised to the power of is the number of elements 1 chapter summary : number of subsets of set 1. 4 set operations with two sets the cardinality of cardinality of infinite sets in set theory, it has been shown that the set of irrational numbers has a cardinality greater than the set of natural numbers. that is, the set of irrational numbers is so large that it is uncountably infinite. perform a search with the phrase, who first proved that the real numbers are uncountable? who first proved that the real numbers are uncountable? what was the significance of this proof to the development of set theory and by extension other fields of recent discoveries in the field of set theory include the solution to a 70 year old problem previously thought to be what does it mean for two infinite sets to have the same size? the real numbers are sometimes referred to as what? summarize your understanding of the problem known as the continuum hypothesis. malliaris and shelah s proof of this 70 year old problem is opening up investigation in what two fields of summarize your understanding of infinity. define what it means to be infinite. explain the difference between countable and uncountable sets. research the difference between a discrete set and a continuous set, then summarize your findings. in arithmetic, the operation of addition is represented by the plus sign,, but multiplication is represented in multiple and parentheses, such as 5 3. several set operations also are written in different forms based on the preferences of the mathematician and often their publisher. search for set complement on the internet and list at least three ways to represent the complement of a set. both the set challenge and venn diagram smartphone apps highlighted in the tech check sections have an operation for set difference.
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. list at least two ways to represent set difference and provide a verbal description of how to calculate the difference between two sets when researching possible venn diagram applications, the greek letter delta, appeared as a symbol for a set operator. list at least one other symbol used for this same operation. search for list of possible set operations and their symbols. find and select two symbols that were not presented in this chapter. the real number system the set of real numbers and their properties are studied in elementary school today, but how did the number system evolve? the idea of natural numbers or counting numbers surfaced prior to written words, as evidenced by tally marks in cave writing. create a timeline for significant contributions to the real number system. use the following phrase to search online for information on the origins of the number zero : history of the number zero. then, record significant dates for the invention and common use of the number zero on your timeline. find out who is credited for discovering that the is irrational and add this information to your timeline. hint : search for, who was the first to discover irrational numbers? research georg cantor s contribution to the representation of real numbers as a continuum and add this to your research ernst zermelo s contribution to the real number system and add this to your timeline. 1 chapter summary basic set concepts 1. a is a well defined collection of distinct objects. 2. a collection of well defined objects without any members in it is called the. 3. write the set consisting of the last five letters of the english alphabet using the roster method. 4. write the set consisting of the numbers 1 through 20 inclusive using the roster method and an ellipsis. 5. write the set of all zebras that do not have stripes in symbolic form. 6. write the set of negative integers using the roster method and an ellipsis. 7. use set builder notation to write the set of all even integers. 8. write the set of all letters in the word mississippi and label it with a capital 9. determine whether the following collection describes a well defined set : a group of these five types of apples : granny smith, red delicious, mcintosh, fuji, and jazz. 10. determine whether the following collection describes a well defined set : a group of five large dogs.
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. 11. determine the cardinality of the set 12. determine whether the following set is a finite set or an infinite set : 13. determine whether sets are equal, equivalent, or neither : 14. determine if sets are equal, equivalent, or neither : 15. determine if sets are equal, equivalent, or neither : 16. if every member of set is also a member of set, then set is a of set 17. determine whether set is a subset, proper subset, or neither a subset nor proper subset of set 18. determine whether set is a subset, proper subset, or neither a subset nor proper subset of set 19. determine whether set is a subset, proper subset, or neither a subset nor proper subset of set 20. list all the subsets of the set 21. list all the subsets of the set 22. calculate the total number of subsets of the set scooby, velma, daphne, shaggy, fred. 23. calculate the total number of subsets of the set top hat, thimble, iron, shoe, battleship, cannon. 24. find a subset of the set that is equivalent, but not equal, to 25. find a subset of the set that is equal to 26. find two equivalent finite subsets of the set of natural numbers,, with a cardinality of 4. 27. find two equal finite subsets of the set of natural numbers,, with a cardinality of 3. understanding venn diagrams 28. use the venn diagram below to describe the relationship between the sets, symbolically and in words : 29. use the venn diagram below to describe the relationship between the sets, symbolically and in words : 1 chapter summary 30. draw a venn diagram to represent the relationship between the described sets : falcons 31. draw a venn diagram to represent the relationship between the described sets : natural numbers 32. the universal set is the set. find the complement of the set 33. the universal set is the set.
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. find the complement of the set 34. use the venn diagram below to determine the members of the set 35. use the venn diagram below to determine the members of the set set operations with two sets determine the union and intersection of the sets indicated : 36. what is 37. what is 38. write the set containing the elements in sets 39. write the set containing all the elements is both sets 42. find the cardinality of 44. use the venn diagram below to find 45. use the venn diagram below to find 1 chapter summary set operations with three sets use the venn diagram below to answer the following questions. 48. a food truck owner surveyed a group of 50 customers about their preferences for hamburger condiments. after tallying the responses, the owner found that 24 customers preferred ketchup, 11 preferred mayonnaise, and 31 preferred mustard. of these customers, eight preferred ketchup and mayonnaise, one preferred mayonnaise and mustard, and 13 preferred ketchup and mustard. no customer preferred all three. the remaining customers did not select any of these three condiments. draw a venn diagram to represent this data. 50. use venn diagrams to prove that if 1. determine whether the following collection describes a well defined set : a group of small tomatoes. classify each of the following sets as either finite or infinite. use the sets provided to answer the following questions : 9. determine if set is equivalent to, equal to, or neither equal nor equivalent to set. justify your answer. use the venn diagram below to answer the following questions. 17. draw a venn diagram to represent the relationship between the two sets : all flowers are plants. for the following questions, use the venn diagram showing the blood types of all donors at a recent mobile blood 1 chapter summary 18. find the number of donors who were ; that is, find 19. find the number of donors who were 20. use venn diagrams to prove that if 1 chapter summary 1 chapter summary figure 2. 1 logic is key to a well reasoned argument, in both math and law.
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. credit : modification of work lady justicia holding sword and scale bronze figurine with judge hammer on wooden table by jernej furman flickr, cc by 2. 0 2. 1 statements and quantifiers 2. 2 compound statements 2. 3 constructing truth tables 2. 4 truth tables for the conditional and biconditional 2. 5 equivalent statements 2. 6 de morgan s laws 2. 7 logical arguments what is logic? logic is the study of reasoning, and it has applications in many fields, including philosophy, law, psychology, digital electronics, and computer science. in law, constructing a well reasoned, logical argument is extremely important. the main goal of arguments made by lawyers is to convince a judge and jury that their arguments are valid and well supported by the facts of the case, so the case should be ruled in their favor. think about thurgood marshall arguing for desegregation in front of the u. s. supreme court during the brown v. board of education of topeka lawsuit in 1954, or ruth bader ginsburg arguing for equality in social security benefits for both men and women under the law during the mid 1970s. both these great minds were known for the preparation and thoroughness of their logical legal arguments, which resulted in victories that advanced the causes they fought for. thurgood marshall and ruth bader ginsburg would later become well respected justices on the u. s. supreme court themselves. in this chapter, we will explore how to construct well reasoned logical arguments using varying structures. your ability to form and comprehend logical arguments is a valuable tool in many areas of life, whether you re planning a dinner date, negotiating the purchase of a new car, or persuading your boss that you deserve a raise. 2 introduction 2. 1 statements and quantifiers figure 2. 2 construction of a logical argument, like that of a house, requires you to begin with the right parts. credit : modification of work barn raising by robert stinnett flickr, cc by 2. 0 after completing this section, you should be able to : identify logical statements. represent statements in symbolic form. negate statements in words. negate statements symbolically. translate negations between words and symbols. express statements with quantifiers of all, some, and none. negate statements containing quantifiers of all, some, and none. have you ever built a club house, tree house, or fort with your friends?
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? if so, you and your friends likely started by gathering some tools and supplies to work with, such as hammers, saws, screwdrivers, wood, nails, and screws. hopefully, at least one member of your group had some knowledge of how to use the tools correctly and helped to direct the construction project. after all, if your house isn t built on a strong foundation, it will be weak and could possibly fall apart during the next big storm. this same foundation is important in logic. in this section, we will begin with the parts that make up all logical arguments. the building block of any logical argument is a logical statement, or simply a statement. a logical statement has the form of a complete sentence, and it must make a claim that can be identified as being true or false. when making arguments, sometimes people make false claims. when evaluating the strength or validity of a logical argument, you must also consider the truth values, or the identification of true or false, of all the statements used to support the argument. while a false statement is still considered a logical statement, a strong logical argument starts with true statements. 2 logic identifying logical statements figure 2. 3 not all roses are red. credit : assorted pink yellow white red roses macro by proflowers flickr, cc by 2. 0 an example of logical statement with a false truth value is, all roses are red. it is a logical statement because it has the form of a complete sentence and makes a claim that can be determined to be either true or false. it is a false statement because not all roses are red : some roses are red, but there are also roses that are pink, yellow, and white. requests, questions, or directives may be complete sentences, but they are not logical statements because they cannot be determined to be true or false. for example, suppose someone said to you, please, sit down over there. this request does not make a claim and it cannot be identified as true or false ; therefore, it is not a logical statement. identifying logical statements determine whether each of the following sentences represents a logical statement. if it is a logical statement, determine whether it is true or false. tiger woods won the master s golf championship at least five times. please, sit down over there. all cats dislike dogs. this is a logical statement because it is a complete sentence that makes a claim that can be identified as being true or false.
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as of 2021, this statement is true: tiger woods won the master s in 1997, 2001, 2002, 2005 and 2019. this is not a logical statement because, although it is a complete sentence, this request does not make a claim that can be identified as being either true or false. this is a logical statement because it is a complete sentence that makes a claim that can be identified as being true or false. this statement is false because some cats do like some dogs. your turn 2.1 determine whether each of the following sentences represents a logical statement. if it is a logical statement, determine whether it is true or false. 1. the buffalo bills defeated the new york giants in super bowl xxv. 2. michael jackson s album thriller was released in 1982. 3. would you like some coffee or tea? representing statements in symbolic form when analyzing logical arguments that are made of multiple logical statements, symbolic form is used to reduce the amount of writing involved. symbolic form also helps visualize the relationship between the statements in a more 2.1 statements and quantifiers concise way in order to determine the strength or validity of an argument. each logical statement is represented symbolically as a single lowercase letter, usually starting with the letter to begin, you will practice how to write a single logical statement in symbolic form. this skill will become more useful as you work with compound statements in later sections. representing statements using symbolic form write each of the following logical statements in symbolic form. barry bonds holds the major league baseball record for total career home runs. some mammals live in the ocean. ruth bader ginsburg served on the u.s. supreme court from 1993 to 2020. : barry bonds holds the major league baseball record for total career home runs. the statement is labeled with a once the statement is labeled, use as a replacement for the full written statement to write and analyze the : some mammals live in the ocean. the letter is used to distinguish this statement from the statement in question 1, but any lower case letter may be used. : ruth bader ginsburg served on the u.s. supreme court from 1993 to 2020. when multiple statements are present in later sections, you will want to be sure to use a different letter for each separate logical statement. your turn 2.2 write each of the following logical statements in symbolic form. 1. the movie gandhi won the academy award for best picture in 1982. 2. soccer is the most popular sport in the world. 3. all oranges are citrus fruits.
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. mathematics is not the only language to use symbols to represent thoughts or ideas. the chinese and japanese languages use symbols known as hanzi and kanji, respectively, to represent words and phrases. at one point, the american musician prince famously changed his name to a symbol representing love. consider the false statement introduced earlier, all roses are red. if someone said to you, all roses are red, you might respond with, some roses are not red. you could then strengthen your argument by providing additional statements, such as, there are also white roses, yellow roses, and pink roses, to name a few. the negation of a logical statement has the opposite truth value of the original statement. if the original statement is false, its negation is true, and if the original statement is true, its negation is false. most logical statements can be negated by simply adding or removing the word not. for example, consider the statement, emma stone has green eyes. the negation of this statement would be, emma stone does not have green eyes. the table below gives some 2 logic gordon ramsey is a chef. gordon ramsey is not a chef. tony the tiger does not have spots. tony the tiger has spots. the way you phrase your argument can impact its success. if someone presents you with a false statement, the ability to rebut that statement with its negation will provide you with the tools necessary to emphasize the correctness of your negating logical statements write the negation of each logical statement in words. michael phelps was an olympic swimmer. tom is a cat. jerry is not a mouse. add the word not to negate the affirmative statement : michael phelps was not an olympic swimmer. add the word not to negate the affirmative statement : tom is not a cat. remove the word not to negate the negative statement : jerry is a mouse. your turn 2. 3 write the negation of each logical statement in words. 1. ted cruz was not born in texas. 2. adele has a beautiful singing voice. 3. leaves convert carbon dioxide to oxygen during the process of photosynthesis. negating logical statements symbolically the symbol for negation, or not, in logic is the tilde,. so, not is represented as. to negate a statement symbolically, remove or add a tilde. the negation of not not. symbolically, this equation is negating logical statements symbolically write the negation of each logical statement symbolically. : michael phelps was an olympic swimmer.
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. : tom is not a cat. : jerry is not a mouse. to negate an affirmative logical statement symbolically, add a tilde : because the symbol for this statement is, its negation is the symbol for this statement is, so to negate it we simply remove the, because the answer is. your turn 2. 4 write the negation of each logical statement symbolically. 2. 1 statements and quantifiers : ted cruz was not born in texas. : adele has a beautiful voice. : leaves convert carbon dioxide to oxygen during the process of photosynthesis. translating negations between words and symbols in order to analyze logical arguments, it is important to be able to translate between the symbolic and written forms of translating negations between words and symbols given the statements : : elmo is a red muppet. : ketchup is not a vegetable. write the symbolic form of the statement, elmo is not a red muppet. translate the statement elmo is not a red muppet is the negation of elmo is a red muppet, which is represented as. thus, we would write elmo is not a red muppet symbolically as is the symbol representing the statement, ketchup is not a vegetable, is equivalent to the statement, ketchup is a vegetable. your turn 2. 5 given the statements : : woody and buzz lightyear are best friends. : wonder woman is not stronger than captain marvel. 1. write the symbolic form of the statement, wonder woman is stronger than captain marvel. 2. translate the statement expressing statements with quantifiers of all, some, or none a quantifier is a term that expresses a numerical relationship between two sets or categories. for example, all squares are also rectangles, but only some rectangles are squares, and no squares are circles. in this example, all, some, and none are quantifiers. in a logical argument, the logical statements made to support the argument are called premises, and the judgment made based on the premises is called the conclusion. logical arguments that begin with specific premises and attempt to draw more general conclusions are called inductive arguments. consider, for example, a parent walking with their three year old child. the child sees a cardinal fly by and points it out. as they continue on their walk, the child notices a robin land on top of a tree and a duck flying across to land on a pond.
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. the child recognizes that cardinals, robins, and ducks are all birds, then excitedly declares, all birds fly! the child has just made an inductive argument. they noticed that three different specific types of birds all fly, then synthesized this information to draw the more general conclusion that all birds can fly. in this case, the child s conclusion is false. the specific premises of the child s argument can be paraphrased by the following statements : premise : cardinals are birds that fly. premise : robins are birds that fly. premise : ducks are birds that fly. the general conclusion is : all birds fly! all inductive arguments should include at least three specific premises to establish a pattern that supports the general conclusion. to counter the conclusion of an inductive argument, it is necessary to provide a counter example. the parent can tell the child about penguins or emus to explain why that conclusion is false. 2 logic on the other hand, it is usually impossible to prove that an inductive argument is true. so, inductive arguments are considered either strong or weak. deciding whether an inductive argument is strong or weak is highly subjective and often determined by the background knowledge of the person making the judgment. most hypotheses put forth by scientists using what is called the scientific method to conduct experiments are based on inductive reasoning. in the following example, we will use quantifiers to express the conclusion of a few inductive arguments. drawing general conclusions to inductive arguments using quantifiers for each series of premises, draw a logical conclusion to the argument that includes one of the following quantifiers : all, some, or none. squares and rectangles have four sides. a square is a parallelogram, and a rectangle is a parallelogram. what conclusion can be drawn from these premises? of these, 1 and 2, 6 and 7, and 23 and 24 are consecutive integers ; 3, 13, and 47 are odd numbers. what conclusion can be drawn from these premises? sea urchins live in the ocean, and they do not breathe air. sharks live in the ocean, and they do not breathe air. eels live in the ocean, and they do not breath air. what conclusion can be drawn from these premises? the conclusion you would likely come to here is some four sided figures are parallelograms.
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. however, it would be incorrect to say that all four sided figures are parallelograms because there are some four sided figures, such as trapezoids, that are not parallelograms. this is a false conclusion. from these premises, you may draw the conclusion all sums of two consecutive counting numbers result in an odd number. most inductive arguments cannot be proven true, but several mathematical properties can be. if we let represent our first counting number, then would be the next counting number and is divisible by two, it is an even number, and if you add one to any even number the result is always an odd number. thus, the conclusion is true! based on the premises provided, with no additional knowledge about whales or dolphins, you might conclude no creatures that live in the ocean breathe air. even though this conclusion is false, it still follows from the known premises and thus is a logical conclusion based on the evidence presented. alternatively, you could conclude some creatures that live in the ocean do not breathe air. the quantifier you choose to write your conclusion with may vary from another person s based on how persuasive the argument is. there may be multiple acceptable answers. your turn 2. 6 for each series of premises, draw a logical conclusion to the argument that includes one of the following quantifiers : all, some, or none.. of these, 1 and 2 are consecutive integers, 5 and 6 are consecutive integers, and 14 and 15 are consecutive integers. also, their sums, 3, 11, and 29 are all prime numbers. prime numbers are positive integers greater than one that are only divisible by one and the number itself. what conclusion can you draw from these premises? 2. a robin is a bird that lays blue eggs. a chicken is a bird that typically lays white and brown eggs. an ostrich is a bird that lays exceptionally large eggs. if a bird lays eggs, then they do not give live birth to their young. what conclusion can you draw from these premises? 3. all parallelograms have four sides. all rectangles are parallelograms. all squares are rectangles. what additional conclusion can you make about squares from these premises? it is not possible to prove definitively that an inductive argument is true or false in most cases. negating statements containing quantifiers recall that the negation of a statement will have the opposite truth value of the original statement.
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. there are four basic forms that logical statements with quantifiers take on. 2. 1 statements and quantifiers the negation of logical statements that use the quantifiers all, some, or none is a little more complicated than just adding or removing the word not. for example, consider the logical statement, all oranges are citrus fruits. this statement expresses as a subset relationship. the set of oranges is a subset of the set of citrus fruit. this means that there are no oranges that are outside the set of citrus fruit. the negation of this statement would have to break the subset relationship. to do this, you could say, at least one orange is not a citrus fruit. or, more concisely, some oranges are not citrus fruit. it is tempting to say no oranges are citrus fruit, but that would be incorrect. such a statement would go beyond breaking the subset relationship, to stating that the two sets have nothing in common. the negation of is a subset of would be to state is not a subset of, as depicted by the venn diagram in figure 2. 4. the statement, all oranges are citrus fruit, is true, so its negation, some oranges are not citrus fruit, is false. now, consider the statement, no apples are oranges. this statement indicates that the set of apples is disjointed from the set of oranges. the negation must state that the two are not disjoint sets, or that the two sets have a least one member in common. their intersection is not empty. the negation of the statement, is the empty set, is the statement that is not empty, as depicted in the venn diagram in figure 2. 5. the negation of the true statement no apples are oranges, is the false statement, some apples are oranges. table 2. 2 summarizes the four different forms of logical statements involving quantifiers and the forms of their associated negations, as well as the meanings of the relationships between the two categories or sets 2 logic logical statements with quantifiers negation of logical statements w quantifiers is a subset of all zebras have stripes. true is not a subset of some zebras do not have stripes. false is not empty, some fish are sharks. true no fish are sharks. false no trees are evergreens. false is not empty, some trees are evergreens.
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. true is not a subset of some horses are not mustangs. true is a subset of all horses are mustangs. false we covered sets in great detail in chapter 1. to review, is a subset of means that every member of set is also a member of set. the intersection of two sets is the set of all elements that they share in common. if is the empty set, then sets do not have any elements in common. the two sets do not overlap. they are disjoint. negating statements containing quantifiers all, some, or none given the statements : : all leopards have spots. : some apples are red. : no lemons are sweet. write each of the following symbolic statements in words. the statement all leopards have spots is and has the form all. according to table 2. 2, the negation will have the form some. the negation of is the statement, some leopards do not have spots. the statement some apples are red has the form some. this indicates that the categories overlap or intersect. according to table 2. 2, the negation will have the form, no, indicating that do not intersect. this results in the opposite truth value of the original statement, so the negation of some apples are red is the statement : no apples are red. is the statement : no lemons are sweet, is asserting that the set of lemons does not intersect with the set of sweet things. the negation of,, must make the opposite claim. it must indicate that the set of lemons intersects with the set of sweet things. this means at least one lemon must be sweet. the statement, some lemons are sweet is. the negation of the statement, no, is the statement, some, as indicated in 2. 1 statements and quantifiers your turn 2. 7 given the statements : : some apples are not sweet. : no triangles are squares. : some vegetables are green. write each of the following symbolic statements in words. check your understanding 1. a is a complete sentence that makes a claim that may be either true or false. 2. the of a logical statement has the opposite truth value of the original statement. represents the logical statement, marigolds are yellow flowers, then represents the statement, marigolds are not yellow flowers. 4. the statement has the same truth value as the statement. 5. the logical statements used to support the conclusion of an argument are called.
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. 6. arguments attempt to draw a general conclusion from specific premises. 7. all, some, and none are examples of, words that assign a numerical relationship between two or more groups. 8. the negation of the statement, all giraffes are tall, is. section 2. 1 exercises for the following exercises, determine whether the sentence represents a logical statement. if it is a logical statement, determine whether it is true or false. 1. a loan used to finance a house is called a mortgage. 2. all odd numbers are divisible by 2. 3. please, bring me that notebook. 4. robot, what s your function? 5. in english, a conjunction is a word that connects two phrases or parts of a sentence together. 8. what is 7 plus 3? for the following exercises, write each statement in symbolic form. 9. grammy award winning singer, lady gaga, was not born in houston, texas. 10. bruno mars performed during the super bowl halftime show twice. 11. coco chanel said, the most courageous act is still to think for yourself. aloud. 12. bruce wayne is not superman. for the following exercises, write the negation of each statement in words. 13. bozo is not a clown. 14. ash is pikachu s trainer and friend. 15. vanilla is the most popular flavor of ice cream. 16. smaug is a fire breathing dragon. 17. elephant and piggy are not best friends. 18. some dogs like cats. 19. some donuts are not round. 20. all cars have wheels. 21. no circles are squares. 22. nature s first green is not gold. 23. the ancient greek philosopher plato said, the greatest wealth is to live content with little. 2 logic 24. all trees produce nuts. for the following exercises, write the negation of each statement symbolically and in words. : their hair is red. : my favorite superhero does not wear a cape. : all wolves howl at the moon. : nobody messes with texas. : i do not love new york. : some cats are not tigers. : no squares are not parallelograms. : the president does not like broccoli. for the following exercises, write each of the following symbolic statements in words.
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. : kermit is a green frog ; translate : mick jagger is not the lead singer for the rolling stones ; translate 35. given : : all dogs go to heaven ; translate : some pizza does not come with pepperoni on it ; translate : no pizza comes with pineapple on it ; translate 38. given : : not all roses are red ; translate : thelonious monk is not a famous jazz pianist ; translate : not all violets are blue ; translate for the following exercises, draw a logical conclusion from the premises that includes one of the following quantifiers : all, some, or none. 41. the ford motor company builds cars in michigan. general motors builds cars in michigan. chrysler builds cars in michigan. what conclusion can be drawn from these premises? 42. michelangelo buonarroti was a great renaissance artist from italy. raphael sanzio was a great renaissance artist from italy. sandro botticelli was a great renaissance artist from italy. what conclusion can you draw from 43. four is an even number and it is divisible by 2. six is an even number and it is divisible by 2. eight is an even number and it is divisible by 2. what conclusion can you draw from these premises? 44. three is an odd number and it is not divisible by 2. seven is an odd number and it is not divisible by 2. twenty one is an odd number and it is not divisible by 2. what conclusion can you draw from these premises? 45. the odd number 5 is not divisible by 3. the odd number 7 is not divisible by 3. the odd number 29 is not divisible by 3. what conclusion can you draw from these premises? 46. penguins are flightless birds. emus are flightless birds. ostriches are flightless birds. what conclusion can you draw from these premises? 47. plants need water to survive. animals need water to survive. bacteria need water to survive. what conclusion can you draw from these premises? 48. a chocolate chip cookie is not sour. an oatmeal cookie is not sour. an oreo cookie is not sour. what conclusion can you draw from these premises? 2. 1 statements and quantifiers 2. 2 compound statements figure 2. 6 a person seeking their driver s license must pass two tests. a compound statement can be used to explain performance on both tests at once.
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. credit : modification of work drivers license teen driver by state farm flickr, cc by after completing this section, you should be able to : translate compound statements into symbolic form. translate compound statements in symbolic form with parentheses into words. apply the dominance of connectives. suppose your friend is trying to get a license to drive. in most places, they will need to pass some form of written test proving their knowledge of the laws and rules for driving safely. after passing the written test, your friend must also pass a road test to demonstrate that they can perform the physical task of driving safely within the rules of the law. consider the statement : my friend passed the written test, but they did not pass the road test. this is an example of a compound statement, a statement formed by using a connective to join two independent clauses or logical statements. the statement, my friend passed the written test, is an independent clause because it is a complete thought or idea that can stand on its own. the second independent clause in this compound statement is, my friend did not pass the road test. the word but is the connective used to join these two statements together, forming a compound statement. so, did your friend acquire their driving license?. this section introduces common logical connectives and their symbols, and allows you to practice translating compound statements between words and symbols. it also explores the order of operations, or dominance of connectives, when a single compound statement uses multiple connectives. common logical connectives understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying the topics presented in this chapter. the chapter will discuss each connective introduced here in more detail. the joining of two logical statements with the word and or but forms a compound statement called a conjunction. in logic, for a conjunction to be true, all the independent logical statements that make it up must be true. the symbol for a. consider the compound statement, derek jeter played professional baseball for the new york yankees, and he was a shortstop. if represents the statement, derrick jeter played professional baseball for the new york yankees, and if represents the statement, derrick jeter was a short stop, then the conjunction will be written the joining of two logical statements with the word or forms a compound statement called a disjunction.
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. unless otherwise specified, a disjunction is an inclusive or statement, which means the compound statement formed by joining two independent clauses with the word or will be true if a least one of the clauses is true. consider the compound statement, the office manager ordered cake for for an employee s birthday or they ordered ice cream. this is a disjunction because it combines the independent clause, the office manager ordered cake for an employee s birthday, with the independent clause, the office manager ordered ice cream, using the connective, or. this disjunction is true if 2 logic the office manager ordered only cake, only ice cream, or they ordered both cake and ice cream. inclusive or means you can have one, or the other, or both! joining two logical statements with the word implies, or using the phrase if first statement, then second statement, is called a conditional or implication. the clause associated with the if statement is also called the hypothesis or antecedent, while the clause following the then statement or the word implies is called the conclusion or consequent. the conditional statement is like a one way contract or promise. the only time the conditional statement is false, is if the hypothesis is true and the conclusion is false. consider the following conditional statement, if pedro does his homework, then he can play video games. the hypothesis antecedent is the statement following the word if, which is pedro does did his homework. the conclusion consequent is the statement following the word then, which is pedro can play his video games. joining two logical statements with the connective phrase if and only if is called a biconditional. the connective phrase if and only if is represented by the symbol, in the biconditional statement, is called the hypothesis or is called the conclusion or consequent. for a biconditional statement to be true, the truth values of must match. they must both be true, or both be false. the table below lists the most common connectives used in logic, along with their symbolic forms, and the common names used to describe each connective. disjunction, inclusive or, then implies if and only if these connectives, along with their names, symbols, and properties, will be used throughout this chapter and should be memorized. associate connectives with symbols and names for each of the following connectives, write its name and associated symbol.
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. a compound statement formed with the connective word or is called a disjunction, and it is represented by the a compound statement formed with the connective word implies or phrase if, then is called a conditional statement or implication and is represented by the a compound statement formed with the connective words but or and is called a conjunction, and it is represented 2. 2 compound statements your turn 2. 8 for each connective write its name and associated symbol. 3. if and only if translating compound statements to symbolic form to translate a compound statement into symbolic form, we take the following steps. identify and label all independent affirmative logical statements with a lower case letter, such as,, or. identify and label any negative logical statements with a lowercase letter preceded by the negation symbol, such as replace the connective words with the symbols that represent them, such as consider the previous example of your friend trying to get their driver s license. your friend passed the written test, but they did not pass the road test. let represent the statement, my friend passed the written test. and, let the statement, my friend did not pass the road test. because the connective but is logically equivalent to the word and, the symbol for but is the same as the symbol for and ; replace but with the symbol the compound statement is symbolically written as :. my friend passed the written test, but they did not pass the road test. when translating english statements into symbolic form, do not assume that every connective and means you are dealing with a compound statement. the statement, the stripes on the u. s. flag are red and white, is a simple statement. the word white doesn t express a complete thought, so it is not an independent clause and does not get its own symbol. this statement should be represented by a single letter, such as : the stripes on the u. s. flag are red and white. translating compound statements into symbolic form represent the statement, it is a warm sunny day, and let represent the statement, the family will go to the beach. write the symbolic form of each of the following compound statements. if it is a warm sunny day, then the family will go to the beach. the family will go to the beach, and it is a warm sunny day. the family will not go to the beach if and only if it is not a warm sunny day. the family not go to the beach, or it is a warm sunny day. replace it is a warm sunny day with.
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. replace the family will go to the beach. with. next. next, because the connective is if, then place the conditional symbol, and. the compound statement is written replace the family will go to the beach with. replace it is a warm sunny day. with. next, because the connective is and, place the. the compound statement is written symbolically as : replace the family will not go to the beach. with. replace it is not a warm sunny day with. next, because the connective is or, if and only if, place the biconditional symbol,. the compound statement is written symbolically as : replace the family will not go to the beach with. replace it is a warm sunny day with. next, because the connective is or, place the. the compound statement is written symbolically as : your turn 2. 9 represent the statement, last night it snowed, and let represent the statement, today we will go skiing. write the symbolic form of each of the following compound statements : 1. today we will go skiing, but last night it did not snow. 2 logic 2. today we will go skiing if and only if it snowed last night. 3. last night is snowed or today we will not go skiing. 4. if it snowed last night, then today we will go skiing. translating compound statements in symbolic form with parentheses into words when parentheses are written in a logical argument, they group a compound statement together just like when calculating numerical or algebraic expressions. any statement in parentheses should be treated as a single component of the expression. if multiple parentheses are present, work with the inner most parentheses first. consider your friend s struggles to get their license to drive. let represent the statement, my friend passed the written represent the statement, my friend passed the road test, and let represent the statement, my friend received a driver s license. the statement can be translated into words as follows : the statement grouped together to form the hypothesis of the conditional statement and is the conclusion. the conditional statement has the form if therefore, the written form of this statement is : if my friend passed the written test and they passed the road test, then my friend received a driver s license. sometimes a compound statement within parentheses may need to be negated as a group. to accomplish this, add the phrase, it is not the case that before the translation of the phrase in parentheses.
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. for example, using,, and friend obtaining a license, let s translate the statement in this case, the hypothesis of the conditional statement is and the conclusion is to negate the hypothesis, add the phrase it is not the case before translating what is in parentheses. the translation of the hypothesis is the sentence, it is not the case that my friend passed the written test and they passed the road test, and the translation of the conclusion is, my friend did not receive a driver s license. so, a translation of the complete conditional statement, is : if it is not the case that my friend passed the written test and the road test, then my friend did not receive a driver s license. it is acceptable to interchange proper names with pronouns and remove repeated phrases to make the written statement more readable, as long the meaning of the logical statement is not changed. translating compound statements in symbolic form with parentheses into words represent the statement, my child finished their homework, let represent the statement, my child cleaned her represent the statement, my child played video games, and let represent the statement, my child streamed a movie. translate each of the following symbolic statements into words. replace with it is not the case, and with and. one possible translation is : it is not the case that my child finished their homework and cleaned their room. the hypothesis of the conditional statement is, my child finished their homework and cleaned their room. the conclusion of the conditional statement is, my child played video games or streamed a movie. one possible translation of the entire statement is : if my child finished their homework and cleaned their room, then they played video games or streamed a movie. the hypothesis of the biconditional statement is and is written in words as : it is not the case that my child played video games or streamed a movie. the conclusion of the biconditional statement is translates to : it is not the case that my child finished their homework and cleaned their room. because the translates to if and only if, one possible translation of the statement is : it is not the case that my child played video games or streamed a movie if and only if it is not the case that my child finished their homework and cleaned their room. 2. 2 compound statements your turn 2. 10 represent the statement, my roommates ordered pizza, let represent the statement, i ordered wings, and be the statement, our friends came over to watch the game. translate the following statements into words.
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. the dominance of connectives the order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. for example, if you were presented with the problem, and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. if you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. try inputting this expression into a scientific calculator. if you do, the calculator should return a value of 7, not 8. the order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. parentheses are used to indicate which operation addition or multiplication should be done first. adding parentheses can change and or clarify the order. the parentheses in the expression indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7 : as with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. this set of rules is called the dominance of connectives. when evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows : parentheses are the least dominant connective. so, any expression inside parentheses must be evaluated first. add as many parentheses as needed to any statement to specify the order to evaluate each connective. next, we evaluate negations. then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance. after evaluating all conjunctions and disjunctions, we evaluate conditionals. lastly, we evaluate the most dominant connective, the biconditional. if a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right. see figure 2. 7 for a visual breakdown of the dominance of connectives. let s revisit your friend s struggles to get their driver s license. let represent the statement, my friend passed the written test, let represent the statement, my friend passed the road test, and let represent the statement, my friend received a driver s license. let s use the dominance of connectives to determine how the compound statement should be evaluated. step 1 : there are no parentheses, which is least dominant of all connectives, so we can skip over that. step 2 : because negation is the next least dominant, we should evaluate first.
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. we could add parentheses to the statement to make it clearer which connecting needs to be evaluated first : is equivalent to 2 logic step 3 : next, we evaluate the conjunction, is equivalent to step 4 : finally, we evaluate the conditional, as this is the most dominant connective in this compound statement. when using spreadsheet applications, like microsoft excel or google sheets, have you noticed that core functions, such as sum, average, or rate, have the same name and syntax for use? this is not a coincidence. various standards organizations have developed requirements that software developers must implement to meet the conditions set by governments and large customers worldwide. the opendocument format oasis standard enables transferring data between different office productivity applications and was approved by the international standards organization iso and iec on may 6, 2006. prior to adopting these standards, a government entity, business, or individual could lose access to their own data simply because it was saved in a format no longer supported by a proprietary software product even data they were required by law to preserve, or data they simply wished to maintain access to. just as rules for applying the dominance of connectives help maintain understanding and consistency when writing and interpreting compound logical statements and arguments, the standards adopted for database software maintain global integrity and accessibility across platforms and user bases. applying the dominance of connectives for each of the following compound logical statements, add parentheses to indicate the order to evaluate the statement. recall that parentheses are evaluated innermost first. because negation is the least dominant connective, we evaluate it first : because conjunction and disjunction have the same dominance, we evaluate them left to right. so, we evaluate the conjunction next, as indicated by the additional set of parentheses : the only remaining connective is the disjunction, so it is evaluated last, as indicated by the third set of parentheses. the complete solution is : negation has the lowest dominance, so it is evaluated first : the remaining connectives are the conditional and the conjunction. because conjunction has a lower precedence than the conditional, it is evaluated next, as indicated by the second set of parentheses : the last step is to evaluate the conditional, as indicated by the third set of parentheses : this statement is known as de morgan s law for the negation of a disjunction. it is always true. section 2. 6 of this chapter will explore de morgan s laws in more detail. first, we evaluate the negations on the right side of the biconditional prior to the conjunction.
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. then, we evaluate the disjunction on the left side of the biconditional, followed by the negation of the disjunction on the left side. lastly, after completely evaluating each side of the biconditional, we evaluate the biconditional. it does not matter which side you begin with. the final solution is : your turn 2. 11 for each of the following compound logical statements, add parentheses to indicate the order in which to evaluate the statement. recall that parentheses are evaluated innermost first. 2. 2 compound statements work it out logic terms and properties matching game materials : for every group of four students, include 30 cards game, trading, or playing cards, 30 individual clear plastic gaming card sleeves, and 30 card size pieces of paper. prepare a list of 60 questions and answers ahead of time related to definitions and problems in statements and quantifiers and compound statements. provide each group of four students with 20 questions and their associated answers. instruct each group to select 15 of the 20 questions, and then, for each problem selected, create one card with the question and one card with the answer. instruct the groups to then place each problem and answer in a separate card sleeve. once they create 15 problem cards and 15 answer cards, have students shuffle the set of cards. to play the game, groups should exchange their card decks to ensure no team begins playing with the deck that they created. split each four person group into teams of two students. after shuffling the cards, one team lays cards face down on their desk in a five by six grid. the other team will go first. each player will have a turn to try matching the question with the correct answer by flipping two cards to the face up position. if a team makes a match, they get to flip another two cards ; if they do not get a match, they flip the cards face down and it is the other team s turn to flip over two cards. the game continues in this manner until teams match all question cards with their corresponding answer cards. the team with the most set of matching cards wins. in the first module of this chapter, we evaluated the truth value of negations. in the following modules, we will discuss how to evaluate conjunctions, disjunctions, conditionals, and biconditionals, and then evaluate compound logical statements using truth tables and our knowledge of the dominance of connectives.
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. check your understanding 9. a is a logical statement formed by combining two or more statements with connecting words, such as and, or, but, not, and if, then. 10. a is a word or symbol used to join two or more logical statements together to form a compound statement. 11. the most dominant connective is the. 12. are used to specify which logical connective should be evaluated first when evaluating a compound 13. both and have equal dominance and are evaluated from left to right when no parentheses are present in a compound logical statement. section 2. 2 exercises for the following exercises, translate each compound statement into symbolic form. : layla has two weeks for vacation, : marcus is layla s friend, : layla will travel to paris, france, and : layla and marcus will travel together to niagara falls, ontario. 1. if layla has two weeks for vacation, then she will travel to paris, france. 2. layla and marcus will travel together to niagara falls, ontario or layla will travel to paris, france. 3. if marcus is not layla s friend, then they will not travel to niagara falls, ontario together. 4. layla and marcus will travel to niagara falls, ontario together if and only if layla and marcus are friends. 5. if layla does not have two weeks for vacation and marcus is layla s friend, then marcus and layla will travel together to niagara falls, ontario. 6. if layla has two weeks for vacation and marcus is not her friend, then she will travel to paris, france. for the following exercises, translate each compound statement into symbolic form. : tom is a cat, : jerry is a mouse, : spike is a dog, : tom chases jerry, and : spike catches tom. 7. jerry is a mouse and tom is a cat. 8. if tom chases jerry, then spike will catch tom. 9. if spike does not catch tom, then tom did not chase jerry. 2 logic 10. tom is a cat and spike is a dog, or jerry is not a mouse. 11. it is not the case that tom is not a cat and jerry is not a mouse. 12. spike is not a dog and jerry is a mouse if and only if tom chases jerry, but spike does not catch tom. for the following exercises, translate the symbolic form of each compound statement into words. : tracy chapman plays guitar, : joan jett plays guitar, : all rock bands include guitarists, and : elton john plays the piano.
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. for the following exercises, translate the symbolic form of each compound statement into words. : the median is the middle number, : the mode is the most frequent number, : the mean is the average number, : the median, mean, and mode are equal, and : the data set is symmetric. for the following exercises, apply the proper dominance of connectives by adding parentheses to indicate the order in which the statement must be evaluated. 2. 2 compound statements 2. 3 constructing truth tables figure 2. 8 just like solving a puzzle, a computer programmer must consider all potential solutions in order to account for each possible outcome. credit : modification of work deadline xmas 2010 by allan henderson flickr, cc by 2. 0 after completing this section, you should be able to : interpret and apply negations, conjunctions, and disjunctions. construct a truth table using negations, conjunctions, and disjunctions. construct a truth table for a compound statement and interpret its validity. are you familiar with the choose your own adventure book series written by edward packard? these gamebooks allow the reader to become one of the characters and make decisions that affect what happens next, resulting in different sequences of events in the story and endings based on the choices made. writing a computer program is a little like what it must be like to write one of these books. the programmer must consider all the possible inputs that a user can put into the program and decide what will happen in each case, then write their program to account for each of these a truth table is a graphical tool used to analyze all the possible truth values of the component logical statements to determine the validity of a statement or argument along with all its possible outcomes. the rows of the table correspond to each possible outcome for the given logical statement identified at the top of each column. a single logical statement has two possible truth values, true or false. in truth tables, a capital t will represent true values, and a capital f will represent false values. in this section, you will use the knowledge built in statements and quantifiers and compound statements to analyze arguments and determine their truth value and validity. a logical argument is valid if its conclusion follows from its premises, regardless of whether those premises are true or false.
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. you will then explore the truth tables for negation, conjunction, and disjunction, and use these truth tables to analyze compound logical statements containing these interpret and apply negations, conjunctions, and disjunctions the negation of a statement will have the opposite truth value of the original statement. when is false, and finding the truth value of a negation for each logical statement, determine the truth value of its negation. : all horses are mustangs. : new delhi is not the capital of india. 2 logic is true because 3 5 does equal 8 ; therefore, the negation of, is false. is false because there are other types of horses besides mustangs, such as clydesdales or arabians ; therefore, the negation of,, is true. is false because new delhi is the capital of india ; therefore, the negation of,, is true. your turn 2. 12 for each logical statement, determine the truth value of its negation. : some houses are built with bricks. : abuja is the capital of nigeria. a conjunction is a logical and statement. for a conjunction to be true, both statements that make up the conjunction must be true. if at least one of the statements is false, the and statement is false. finding the truth value of a conjunction determine the truth value of each conjunction. is true, and is false. because one statement is true, and the other statement is false, this makes the complete is false, so is true, and is true. therefore, both statements are true, making the complete conjunction true. is false, and is false. because both statements are false, the complete conjunction is false. your turn 2. 13 : yellow is a primary color, : blue is a primary color, and : green is a primary color, determine the truth value of each conjunction. the only time a conjunction is true is if both statements that make up the conjunction are true. a disjunction is a logical inclusive or statement, which means that a disjunction is true if one or both statements that form it are true. the only way a logical inclusive or statement is false is if both statements that form the disjunction are finding the truth value of a disjunction determine the truth value of each disjunction. 2. 3 constructing truth tables is true, and is false.
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. one statement is true, and one statement is false, which makes the complete disjunction is false, so is true, and is true. therefore, one statement is true, and the other statement is true, which makes the complete disjunction true. is false, and is false. when all of the component statements are false, the disjunction is false. your turn 2. 14 : yellow is a primary color, : blue is a primary color, and : green is a primary color, determine the truth value of each disjunction. in the next example, you will apply the dominance of connectives to find the truth values of compound statements containing negations, conjunctions, and disjunctions and use a table to record the results. when constructing a truth table to analyze an argument where you can determine the truth value of each component statement, the strategy is to create a table with two rows. the first row contains the symbols representing the components that make up the compound statement. the second row contains the truth values of each component below its symbol. include as many columns as necessary to represent the value of each compound statement. the last column includes the complete compound statement with its truth value below it. finding the truth value of compound statements construct a truth table to determine the truth value of each step 1 : the statement contains three basic logical statements,,, and, and three connectives, when we place parentheses in the statement to indicate the dominance of connectives, the statement becomes step 2 : after we have applied the dominance of connectives, we create a two row table that includes a column for each basic statement that makes up the compound statement, and an additional column for the contents of each parentheses. because we have three sets of parentheses, we include a column for the innermost parentheses, a the next set of parentheses, and in the last column for the third parentheses. step 3 : once the table is created, we determine the truth value of each statement starting from left to right. the truth values of,, and are true, false, and true, respectively, so we place t, f, and t in the second row of the table. step 4 : next, evaluate from the table : is false, and is also false, so is false, because a conjunction is only true if both of the statements that make it are true. place an f in the table below its heading.
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. step 5 : finally, using the table, we understand that is false and is true, so the complete statement is false or true, which is true because a disjunction is true whenever at least one of the statements that 2 logic make it is true. place a t in the last column of the table. the complete statement step 1 : applying the dominance of connectives to the original compound statement, we get step 2 : the table needs columns for step 3 : the truth values of,,, and are the same as in question 1. step 4 : next, is false or false, which is false, so we place an f below this statement in the table. this is the only time that a disjunction is false. step 5 : finally, are the conjunction of the statements, and, and so the expressions evaluate to false and true, which is false. recall that the only time an and statement is true is when both statements that form it are also true. the complete statement step 1 : applying the dominance of connectives to the original statement, we have : step 2 : so, the table needs the following columns : step 3 : the truth values of,, and are the same as in questions 1 and 2. step 4 : from the table it can be seen that is true and true, which is true. so the negation of because the negation of a statement always has the opposite truth value of the original statement. step 5 : finally, is the disjunction of with, and so we have false or false, which makes the complete statement false. your turn 2. 15 : yellow is a primary color, : blue is a primary color, and : green is a primary color, determine the truth value of each compound statement, by constructing a truth table. construct truth tables to analyze all possible outcomes recall from statements and questions that the negation of a statement will always have the opposite truth value of the original statement ; if a statement is false, then its negation is true, and if a statement is true, then its negation is false. to create a truth table for the negation of statement, add a column with a heading of, and for each row, input the truth value with the opposite value of the value listed in the column for, as depicted in the table below. 2. 3 constructing truth tables conjunctions and disjunctions are compound statements formed when two logical statements combine with the connectives and and or respectively.
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. how does that change the number of possible outcomes and thus determine the number of rows in our truth table? the multiplication principle, also known as the fundamental counting principle, states that the number of ways you can select an item from a group of items and another item from a group with items is equal to the product of. because each proposition has two possible outcomes, true or false, the multiplication principle states that there will be possible outcomes : tt, tf, ft, ff. the tree diagram and table in figure 2. 9 demonstrate the four possible outcomes for two propositions a conjunction is a logical and statement. for a conjunction to be true, both the statements that make up the conjunction must be true. if at least one of the statements is false, the and statement is false. a disjunction is a logical inclusive or statement. which means that a disjunction is true if one or both statements that make it are true. the only way a logical inclusive or statement is false is if both statements that make up the disjunction 2 logic constructing truth tables to analyze compound statements construct a truth table to analyze all possible outcomes for each of the following arguments. step 1 : because there are two basic statements, and, and each of these has two possible outcomes, we will have rows in our table to represent all possible outcomes : tt, tf, ft, and ff. the columns will include step 2 : every value in column will have the opposite truth value of the corresponding value in column : f, t, f, step 3 : to complete the last column, evaluate each element in column with the corresponding element in column using the connective or. step 1 : the columns will include. because there are two basic statements, and, the table will have four rows to account for all possible outcomes. step 2 : the column will be completed by evaluating the corresponding elements in columns respectively with the and connective. step 3 : the final column,, will be the negation of the step 1 : this statement has three basic statements,,, and. because each basic statement has two possible truth values, true or false, the multiplication principal indicates there are possible outcomes. so eight rows of outcomes are needed in the truth table to account for each possibility. half of the eight possibilities must be true for the first statement, and half must be false.
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. step 2 : so, the first column for statement, will have four t s followed by four f s. in the second column for statement, when is true, half the outcomes for must be true and the other half must be false, and the same pattern will repeat for when is false. so, column will have tt, ff, ff, ff. step 3 : the column for the third statement,, must alternate between t and f. once, the three basic propositions are listed, you will need a column for step 4 : the column for the negation of,, will have the opposite truth value of each value in column. step 5 : next, fill in the truth values for the column containing the statement the or statement is true if at least one of is true, otherwise it is false. 2. 3 constructing truth tables step 6 : finally, fill in the column containing the conjunction. to evaluate this statement, combine with the and connective. recall, that only time and is true is when both values are true, otherwise the statement is false. the complete truth table is : your turn 2. 16 construct a truth table to analyze all possible outcomes for each of the following arguments. determine the validity of a truth table for a compound statement a logical statement is valid if it is always true regardless of the truth values of its component parts. to test the validity of a compound statement, construct a truth table to analyze all possible outcomes. if the last column, representing the complete statement, contains only true values, the statement is valid. determining the validity of compound statements construct a truth table to determine the validity of each of the following statements. step 1 : because there are two statements, and, and each of these has two possible outcomes, there will be rows in our table to represent all possible outcomes : tt, tf, ft, and ff. step 2 : the columns, will include every value in column will have the opposite truth value of the corresponding value in column step 3 : to complete the last column, evaluate each element in column with the corresponding element in using the connective and. the last column contains at least one false, therefore the statement 2 logic step 1 : because the statement only contains one basic proposition, the truth table will only contain two may be either true or false. step 2 : the columns will include with the and connective, because the symbol represents a conjunction or logical and statement. true and false is false, and false and true is also false.
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. step 3 : the final column is the negation of each entry in the third column, both of which are false, so the negation of false is true. because all the truth values in the final column are true, the statement your turn 2. 17 construct a truth table to determine the validity of each of the following statements. check your understanding 14. a logical argument is if its conclusion follows from its premises. 15. a logical statement is valid if it is always. 16. a is a tool used to analyze all the possible outcomes for a logical statement. 17. the truth table for the conjunction,, has rows of truth values. 18. the truth table for the negation of a logical statement,, has rows of truth values. section 2. 3 exercises for the following exercises, find the truth value of each statement.. what is the truth value of : the sun revolves around the earth. what is the truth value of : the acceleration of gravity is m sec2. what is the truth value of? : dan brown is not the author of the book, the davinci code. what is the truth value of : broccoli is a vegetable. what is the truth value of for the following exercises, given, : five is an even number, and : seven is a prime number, find the truth value of each of the following statements. 2. 3 constructing truth tables for the following exercises, complete the truth table to determine the truth value of the proposition in the last column. for the following exercises, given all triangles have three sides, some rectangles are not square, and pentagon has eight sides, determine the truth value of each compound statement by constructing a truth table. for the following exercises, construct a truth table to analyze all the possible outcomes for the following arguments. for the following exercises, construct a truth table to determine the validity of each statement. 2 logic 2. 4 truth tables for the conditional and biconditional figure 2. 10 if then statements use logic to execute directions. credit : coding by carlos varela flickr, cc by 2. 0 after completing this section, you should be able to : use and apply the conditional to construct a truth table. use and apply the biconditional to construct a truth table. use truth tables to determine the validity of conditional and biconditional statements. computer languages use if then or if then else statements as decision statements : if the hypothesis is true, then do something.
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. or, if the hypothesis is true, then do something ; else do something else. for example, the following representation of computer code creates an if then else decision statement : check value of variable., then print hello, world! else print goodbye. in this imaginary program, the if then statement evaluates and acts on the value of the variable. for instance, if the program would consider the statement as true and hello, world! would appear on the computer screen. if, the program would consider the statement as false because 3 is greater than 1, and print goodbye on the screen. in this section, we will apply similar reasoning without the use of computer programs. people in mathematics the countess of lovelace, ada lovelace, is credited with writing the first computer program. she wrote an algorithm to work with charles babbage s analytical engine that could compute the bernoulli numbers in 1843. in doing so, she became the first person to write a program for a machine that would produce more than just a simple calculation. the computer programming language ada is named after her. reference : posamentier, alfred and spreitzer christian, chapter 34 ada lovelace : english 1815 1852 pp. 272 278, math makers : the lives and works of 50 famous mathematicians, prometheus books, 2019. 2. 4 truth tables for the conditional and biconditional use and apply the conditional to construct a truth table a conditional is a logical statement of the form if, then. the conditional statement in logic is a promise or contract. the only time the conditional, is false is when the contract or promise is broken. for example, consider the following scenario. a child s parent says, if you do your homework, then you can play your video games. the child really wants to play their video games, so they get started right away, finish within an hour, and then show their parent the completed homework. the parent thanks the child for doing a great job on their homework and allows them to play video games. both the parent and child are happy. the contract was satisfied ; true implies true is now, suppose the child does not start their homework right away, and then struggles to complete it. they eventually finish and show it to their parent. the parent again thanks the child for completing their homework, but then informs the child that it is too late in the evening to play video games, and that they must begin to get ready for bed. now, the child is really upset.
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. they held up their part of the contract, but they did not receive the promised reward. the contract was broken ; true implies false is false. so, what happens if the child does not do their homework? in this case, the hypothesis is false. no contract has been entered, therefore, no contract can be broken. if the conclusion is false, the child does not get to play video games and might not be happy, but this outcome is expected because the child did not complete their end of the bargain. they did not complete their homework. false implies false is true. the last option is not as intuitive. if the parent lets the child play video games, even if they did not do their homework, neither parent nor child are going to be upset. false implies true is true. the truth table for the conditional statement below summarizes these results. notice that the only time the conditional statement, is false is when the hypothesis,, is true and the conclusion,, is false. constructing truth tables for conditional statements assume both of the following statements are true : : my sibling washed the dishes, and : my parents paid them 5. 00. create a truth table to determine the truth value of each of the following conditional statements. 2 logic is true and is true, the statement is, if my sibling washed the dishes, then my parents paid them 5. 00. my sibling did wash the dishes, since is true, and the parents did pay the sibling 5. 00, so the contract was entered and completed. the conditional statement is true, as indicated by the truth table representing this case : t t t. translates to the statement, if my sibling washed the dishes, then my parents did not pay them 5. 00. is false. the sibling completed their end of the contract, but they did not get paid. the contract was broken by the parents. the conditional statement is false, as indicated by the truth table representing this case : t f f. translates to the statement, if my sibling did not wash the dishes, then my parents paid them 5. 00. is true. the sibling did not do the dishes. no contract was entered, so it could not be broken. the parents decided to pay them 5. 00 anyway. the conditional statement is true, as indicated by the truth table representing this case : f t t. your turn 2. 18 is true and : kevin vacuumed the living room, and : kevin s parents did not let him borrow the car.
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. create a truth table to determine the truth value of each of the following conditional statements. determining validity of conditional statements construct a truth table to analyze all possible outcomes for each of the following statements then determine whether they are valid. applying the dominance of connectives, the statement is equivalent to so, the columns of the truth table will include because there are only two basic propositions, the table will have rows of truth values to account for all the possible outcomes. the statement is not valid because the last column is not all true. 2. 4 truth tables for the conditional and biconditional applying the dominance of connectives, the statement is equivalent to columns of the truth table will include because there are only two basic and, the table will have rows of truth values to account for all the possible outcomes. the statement is not valid because the last column is not all true. your turn 2. 19 construct a truth table to analyze all possible outcomes for each of the following statements, then determine whether they are valid. use and apply the biconditional to construct a truth table, is a two way contract ; it is equivalent to the statement is true whenever the truth value of the hypothesis matches the truth value of the conclusion, otherwise it is false. the truth table for the biconditional is summarized below. 2 logic constructing truth tables for biconditional statements assume both of the following statements are true : : the plumber fixed the leak, and : the homeowner paid the plumber 150. 00. create a truth table to determine the truth value of each of the following biconditional statements. is true and is true, the statement is the plumber fixed the leak if and only if the homeowner paid them 150. 00. because both are true, the leak was fixed and the plumber was paid, meaning both parties satisfied their end of the bargain. the biconditional statement is true, as indicated by the truth table representing this case : t t t. translates to the statement, the plumber fixed the leak if and only if the homeowner did not pay them 150. if the plumber fixed the leak and the homeowner did not pay them, the homeowner will have broken their end of the contract.
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. the biconditional statement is false, as indicated by the truth table representing this case : t f f. translates to the statement, the plumber did not fix the leak if and only if the homeowner did not pay them 150. in this case, neither party the plumber nor the homeowner entered into the contract. the leak was not repaired, and the plumber was not paid. no agreement was broken. the biconditional statement is true, as indicated by the truth table representing this case : f f t. your turn 2. 20 is true and the contractor fixed the broken window, and the homeowner paid the contractor 200. create a truth table to determine the truth value of each of the following biconditional statements. is true whenever the truth values of match, otherwise it is false. 2. 4 truth tables for the conditional and biconditional determining validity of biconditional statements construct a truth table to analyze all possible outcomes for each of the following statements, then determine whether they are valid. applying the dominance of connectives, the statement is equivalent to columns of the truth table will include because there are only two and, the table will have rows of truth values to account for all the possible outcomes. the statement is not valid because the last column is not all true. applying the dominance of connectives, the statement is equivalent to columns of the truth table will include because there are only two and, the table will have rows of truth values to account for all the possible outcomes. the statement is not valid because the last column is not all true. applying the dominance of connectives, the statement is equivalent to so, the columns of the truth table will include there are only two basic propositions, the table will have rows of truth values to account for all the possible outcomes. the statement is valid because the last column is all true. 2 logic applying the dominance of connectives, the statement is equivalent to so, the columns of the truth table will include,,, because there are three basic propositions,,, and, the table rows of truth values to account for all the possible outcomes. the statement is not valid because the last column is not all true. your turn 2. 21 construct a truth table to analyze all possible outcomes for each of the following statements, then determine whether they are valid. check your understanding 19. in logic, a conditional statement can be thought of as a.
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. 20. if the hypothesis,, of a conditional statement is true, the,, must also be true for the conditional statement to be true. 21. if the of a conditional statement is false, the conditional statement is true. 22. the symbolic form of the statement is 2. 4 truth tables for the conditional and biconditional 23. the statement is equivalent to the statement if and only if is whenever the truth value of matches the truth value of, otherwise it is false. section 2. 4 exercises for the following exercises, complete the truth table to determine the truth value of the proposition in the last column. 2 logic for the following exercises, assume these statements are true : faheem is a software engineer, ann is a project 2. 4 truth tables for the conditional and biconditional giacomo works with faheem, and the software application was completed on time. translate each of the following statements to symbols, then construct a truth table to determine its truth value. 11. if giacomo works with faheem, then faheem is not a software engineer. 12. if the software application was not completed on time, then ann is not a project manager. 13. the software application was completed on time if and only if giacomo worked with faheem. 14. ann is not a project manager if and only if faheem is a software engineer. 15. if the software application was completed on time, then ann is a project manager, but faheem is not a software 16. if giacomo works with faheem and ann is a project manager, then the software application was completed on 17. the software application was not completed on time if and only if faheem is a software engineer or giacomo did not work with faheem. 18. faheem is a software engineer or ann is not a project manager if and only if giacomo did not work with faheem and the software application was completed on time. 19. ann is a project manager implies faheem is a software engineer if and only if the software application was completed on time implies giacomo worked with faheem. 20. if giacomo did not work with faheem implies that the software application was not completed on time, then ann was not the project manager.
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. for the following exercises, construct a truth table to analyze all the possible outcomes and determine the validity of 2 logic 2. 5 equivalent statements figure 2. 11 how your logical argument is stated affects the response, just like how you speak when holding a conversation can affect how your words are received. credit : modification of work by goelshivi flickr, public domain after completing this section, you should be able to : determine whether two statements are logically equivalent using a truth table. compose the converse, inverse, and contrapositive of a conditional statement have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? the way you choose to express your ideas can be as, or even more, important than what you are saying. if your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly. logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience. in this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. developing this skill will provide the additional skills and knowledge needed to construct well reasoned, persuasive arguments that can be customized to address specific audiences. an alternate way to think about logical equivalence is that the truth values have to match. that is, whenever is also true, and whenever is also false. determine logical equivalence and, are logically equivalent when is a valid argument, or when the last column of the truth table consists of only true values. when a logical statement is always true, it is known as a tautology. to determine whether two statements are logically equivalent, construct a truth table for and determine whether it valid. if the last column is all true, the argument is a tautology, it is valid, and is logically equivalent to ; otherwise, not logically equivalent to. determining logical equivalence with a truth table create a truth table to determine whether the following compound statements are logically equivalent.
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. 2. 5 equivalent statements construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, because the last column it not all true, the biconditional is not valid and the statement is not logically equivalent to the statement construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, because the last column is true for every entry, the biconditional is valid and the statement equivalent to the statement your turn 2. 22 create a truth table to determine whether the following compound statements are logically equivalent. compose the converse, inverse, and contrapositive of a conditional statement the converse, inverse, and contrapositive are variations of the conditional statement, the converse is if, and it is formed by interchanging the hypothesis and the conclusion. the converse is logically equivalent to the inverse. the inverse is if, and it is formed by negating both the hypothesis and the conclusion. the inverse is logically equivalent to the converse. the contrapositive is if, and it is formed by interchanging and negating both the hypothesis and the conclusion. the contrapositive is logically equivalent to the conditional. the table below shows how these variations are presented symbolically. 2 logic writing the converse, inverse, and contrapositive of a conditional statement use the statements, : harry is a wizard and : hermione is a witch, to write the following statements : write the conditional statement,, in words. write the converse statement,, in words. write the inverse statement,, in words. write the contrapositive statement,, in words. the conditional statement takes the form, if, then, so the conditional statement is : if harry is a wizard, then hermione is a witch. remember the if then words are the connectives that form the conditional statement. the converse swaps or interchanges the hypothesis,, with the conclusion,. it has the form, if, then. so, the converse is : if hermione is a witch, then harry is a wizard. to construct the inverse of a statement, negate both the hypothesis and the conclusion. the inverse has the form,, so the inverse is : if harry is not a wizard, then hermione is not a witch. the contrapositive is formed by negating and interchanging both the hypothesis and conclusion.
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. it has the form, if, so the contrapositive statement is : if hermione is not a witch, then harry is not a wizard. your turn 2. 23 use the statements, : elvis presley wore capes and : some superheroes wear capes, to write the following 1. write the conditional statement,, in words. 2. write the converse statement,, in words. 3. write the inverse statement,, in words. 4. write the contrapositive statement,, in words. identifying the converse, inverse, and contrapositive use the conditional statement, if all dogs bark, then lassie likes to bark, to identify the following. write the hypothesis of the conditional statement and label it with a write the conclusion of the conditional statement and label it with a. identify the following statement as the converse, inverse, or contrapositive : if lassie likes to bark, then all dogs identify the following statement as the converse, inverse, or contrapositive : if lassie does not like to bark, then some dogs do not bark. which statement is logically equivalent to the conditional statement? 2. 5 equivalent statements the hypothesis is the phrase following the if. the answer is : all dogs bark. notice, the word if is not included as part of the hypothesis. the conclusion of a conditional statement is the phrase following the then. the word then is not included when stating the conclusion. the answer is : : lassie likes to bark. lassie likes to bark is and all dogs bark is. so, if lassie likes to bark, then all dogs bark, has the form if,, which is the form of the converse. lassie does not like to bark is and some dogs do not bark is. the statement, if lassie does not like to bark, then some dogs do not bark, has the form if, which is the form of the contrapositive. is logically equivalent to the conditional statement your turn 2. 24 use the conditional statement, if dora is an explorer, then boots is a monkey, to identify the following : 1. write the hypothesis of the conditional statement and label it with a 2. write the conclusion of the conditional statement and label it with a. 3. identify the following statement as the converse, inverse, or contrapositive : if dora is not an explorer, then boots is not a monkey.
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. 4. identify the following statement as the converse, inverse, or contrapositive : if boots is a monkey, then dora is an explorer. 5. which statement is logically equivalent to the inverse? determining the truth value of the converse, inverse, and contrapositive assume the conditional statement, if chadwick boseman was an actor, then chadwick boseman did not star in the movie black panther is false, and use it to answer the following questions. write the converse of the statement in words and determine its truth value. write the inverse of the statement in words and determine its truth value. write the contrapositive of the statement in words and determine its truth value. the only way the conditional statement can be false is if the hypothesis, : chadwick boseman was an actor, is true and the conclusion, : chadwick boseman did not star in the movie black panther, is false. the converse is and it is written in words as : if chadwick boseman did not star in the movie black panther, then chadwick boseman was an actor. this statement is true, because false true is true. the inverse has the form the written form is : if chadwick boseman was not an actor, then chadwick boseman starred in the movie black panther. because is true, and is false, and is true. this means the inverse is false true, which is true. alternatively, from question 1, the converse is true, and because the inverse is logically equivalent to the converse it must also be true. the contrapositive is logically equivalent to the conditional. because the conditional is false, the contrapositive is also false. this can be confirmed by looking at the truth values of the contrapositive statement. the contrapositive has the form is false and is true and is false. therefore, false, which is false. the written form of the contrapositive is if chadwick boseman starred in the movie black panther, then chadwick boseman was not an actor. your turn 2. 25 assume the conditional statement if my friend lives in san francisco, then my friend does not live in california is false, and use it to answer the following questions. 1. write the converse of the statement in words and determine its truth value. 2. write the inverse of the statement in words and determine its truth value. 2 logic 3. write the contrapositive of the statement in words and determine its truth value.
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. check your understanding 25. two statements are logically equivalent to each other if the biconditional statement, 26. the statement has the form, 27. the contrapositive is to the conditional statement, and has the form, if 28. the of the conditional statement has the form, if 29. the of the conditional statement is logically equivalent to the and has the form, if section 2. 5 exercises for the following exercises, determine whether the pair of compound statements are logically equivalent by constructing a truth table. for the following exercises, answer the following : write the conditional statement write the converse statement write the inverse statement write the contrapositive statement : six is afraid of seven and : seven ate nine. : hope is eternal and : despair is temporary. : tom brady is a quarterback and : tom brady does not play soccer. : shakira does not sing opera and : shakira sings popular music. : the shape does not have three sides and : the shape is not a triangle. : all birds can fly and : emus can fly. : penguins cannot fly and : some birds can fly. : some superheroes do not wear capes and : spiderman is a superhero. : no pok mon are little ponies and : bulbasaur is a pok mon. : roses are red, and violets are blue and : sugar is sweet, and you are sweet too. for the following exercises, use the conditional statement : if clark kent is superman, then lois lane is not a reporter, to answer the following questions. 21. write the hypothesis of the conditional statement, label it with a, and determine its truth value. 22. write the conclusion of the conditional statement, label it with a, and determine its truth value. 23. identify the following statement as the converse, inverse, or contrapositive, and determine its truth value : if clark kent is not superman, then lois lane is a reporter. 24. identify the following statement as the converse, inverse, or contrapositive, and determine its truth value : if lois lane is a reporter, then clark kent is not superman. 25. which form of the conditional is logically equivalent to the converse? for the following exercises, use the conditional statement : if the masked singer is not a music competition, then donnie wahlberg was a member of new kids on the block, to answer the following questions. 26. write the hypothesis of the conditional statement, label it with a, and determine its truth value.
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