Chapter 2 The Basic Concepts of Set Theory © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 2: The Basic Concepts of Set Theory 2.1 Symbols and Terminology 2.2 Venn Diagrams and Subsets 2.3 Set Operations and Cartesian Products 2.4 Surveys and Cardinal Numbers 2.5 Infinite Sets and Their Cardinalities © 2008 Pearson Addison-Wesley. All rights reserved

Section 2-1 Chapter 1 Symbols and Terminology © 2008 Pearson Addison-Wesley. All rights reserved

Symbols and Terminology Designating Sets Sets of Numbers and Cardinality Finite and Infinite Sets Equality of Sets © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Designating Sets A set is a collection of objects. The objects belonging to the set are called the elements, or members of the set. Sets are designated using: 1) word description, the listing method, and set-builder notation. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Designating Sets Word description The set of even counting numbers less than 10 The listing method {2, 4, 6, 8} Set-builder notation {x|x is an even counting number less than 10} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Designating Sets Sets are commonly given names (capital letters). A = {1, 2, 3, 4} The set containing no elements is called the empty set (null set) and denoted by { } or To show 2 is an element of set A use the symbol © 2008 Pearson Addison-Wesley. All rights reserved

Example: Listing Elements of Sets Give a complete listing of all of the elements of the set {x|x is a natural number between 3 and 8} Solution {4, 5, 6, 7} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Sets of Numbers Natural (counting) {1, 2, 3, 4, …} Whole numbers {0, 1, 2, 3, 4, …} Integers {…,–3, –2, –1, 0, 1, 2, 3, …} Rational numbers May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333… Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat. Real numbers {x | x can be expressed as a decimal} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Cardinality The number of elements in a set is called the cardinal number, or cardinality of the set. The symbol n(A), read “n of A,” represents the cardinal number of set A. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Cardinality Find the cardinal number of each set. a) K = {a, l, g, e, b, r} b) M = {2} c) © 2008 Pearson Addison-Wesley. All rights reserved

Finite and Infinite Sets If the cardinal number of a set is a particular whole number, we call that set a finite set. Whenever a set is so large that its cardinal number is not found among the whole numbers, we call that set an infinite set. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Infinite Set The odd counting numbers are an infinite set. Word description The set of all odd counting numbers Listing method {1, 3, 5, 7, 9, …} Set-builder notation {x|x is an odd counting number} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Equality of Sets Set A is equal to set B provided the following two conditions are met: 1. Every element of A is an element of B, AND 2. Every element of B is an element of A. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Equality of Sets State whether the sets in each pair are equal. a) {a, b, c, d} and {a, c, d, b} b) {2, 4, 6} and {x|x is an even number} © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.1: Symbols and Terminology Which of the following is an example of set- builder notation? a) The counting numbers less than 5 b) {1, 2, 3, 4} c) {x | x is a counting number less than 5} © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.1: Symbols and Terminology 2. Are sets {6, 7, 8} and {7, 8, 6} equal? a) Yes b) No © 2008 Pearson Addison-Wesley. All rights reserved

Section 2-2 Chapter 1 Venn Diagrams and Subsets © 2008 Pearson Addison-Wesley. All rights reserved

Venn Diagrams and Subsets Complement of a Set Subsets of a Set Proper Subsets Counting Subsets © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Venn Diagrams In set theory, the universe of discourse is called the universal set, typically designated with the letter U. Venn Diagrams were developed by the logician John Venn (1834 – 1923). In these diagrams, the universal set is represented by a rectangle and other sets of interest within the universal set are depicted as circular regions. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Venn Diagrams The rectangle represents the universal set, U, while the portion bounded by the circle represents set A. A U © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Complement of a Set The colored region inside U and outside the circle is labeled A' (read “A prime”). This set, called the complement of A, contains all elements that are contained in U but not in A. A U © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Complement of a Set For any set A within the universal set U, the complement of A, written A', is the set of all elements of U that are not elements of A. That is © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Subsets of a Set Set A is a subset of set B if every element of A is also an element of B. In symbols this is written B A U © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Subsets Fill in the blank with to make a true statement. a) {a, b, c} ___ { a, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} © 2008 Pearson Addison-Wesley. All rights reserved

Set Equality (Alternative Definition) Suppose that A and B are sets. Then A = B if © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Proper Subset of a Set Set A is a proper subset of set B if In symbols, this is written © 2008 Pearson Addison-Wesley. All rights reserved

Example: Proper Subsets Decide whether or both could be placed in each blank to make a true statement. a) {a, b, c} ___ { a ,b, c, d} b) {1, 2, 3, 4} ___ {1, 2, 3, 4} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Counting Subsets One method of counting subsets involves using a tree diagram. The figure below shows the use of a tree diagram to find the subsets of {a, b}. Yes No {a, b} {a} {b} Yes No Yes No © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Number of Subsets The number of subsets of a set with n elements is 2n. The number of proper subsets of a set with n elements is 2n – 1. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Number of Subsets Find the number of subsets and the number of proper subsets of the set {m, a, t, h, y}. © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.2: Venn Diagrams and Subsets 1. Find the complement of {m} if U = {m, n}. a) {m} b) {n} c) U d) © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.2: Venn Diagrams and Subsets 2. Find the number of subsets of { $, #, @ }. a) 0 b) 3 c) 6 d) 8 © 2008 Pearson Addison-Wesley. All rights reserved

Section 2-3 Chapter 1 Set Operations and Cartesian Products © 2008 Pearson Addison-Wesley. All rights reserved

Set Operations and Cartesian Products Intersection of Sets Union of Sets Difference of Sets Ordered Pairs Cartesian Product of Sets Venn Diagrams De Morgan’s Laws © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Intersection of Sets The intersection of sets A and B, written is the set of elements common to both A and B, or © 2008 Pearson Addison-Wesley. All rights reserved

Example: Intersection of Sets Find each intersection. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Union of Sets The union of sets A and B, written is the set of elements belonging to either of the sets, or © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Union of Sets Find each union. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Difference of Sets The difference of sets A and B, written A – B, is the set of elements belonging to set A and not to set B, or © 2008 Pearson Addison-Wesley. All rights reserved

Example: Difference of Sets Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, h}, B = {c, e, g}, and C = {a, c, d, g, e}. Find each set. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Ordered Pairs In the ordered pair (a, b), a is called the first component and b is called the second component. In general Two ordered pairs are equal provided that their first components are equal and their second components are equal. © 2008 Pearson Addison-Wesley. All rights reserved

Cartesian Product of Sets The Cartesian product of sets A and B, written, is © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Cartesian Products Let A = {a, b}, B = {1, 2, 3} Find each set. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

Cardinal Number of a Cartesian Product If n(A) = a and n(B) = b, then © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Cardinal Numbers of Cartesian Products If n(A) = 12 and n(B) = 7, then find © 2008 Pearson Addison-Wesley. All rights reserved

Venn Diagrams of Set Operations B A B U U A A B U U © 2008 Pearson Addison-Wesley. All rights reserved

Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set © 2008 Pearson Addison-Wesley. All rights reserved

Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved De Morgan’s Laws For any sets A and B, © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.3: Set Operations and Cartesian Products 1. If U = {a, b, c} and A = {a} then find a) A b) {b, c} c) U d) © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.3: Set Operations and Cartesian Products 2. Which choice indicates the shaded region below? a) b) c) E G H U © 2008 Pearson Addison-Wesley. All rights reserved

Section 2-4 Chapter 1 Surveys and Cardinal Numbers © 2008 Pearson Addison-Wesley. All rights reserved

Surveys and Cardinal Numbers Cardinal Number Formula © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Surveys Problems involving sets of people (or other objects) sometimes require analyzing known information about certain subsets to obtain cardinal numbers of other subsets. The “known information” is often obtained by administering a survey. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Suppose that a group of 140 people were questioned about particular sports that they watch regularly and the following information was produced. 93 like football 40 like football and baseball 70 like baseball 25 like baseball and hockey 40 like hockey 28 like football and hockey 20 like all three a) How many people like only football? b) How many people don’t like any of the sports? © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing a Survey Construct a Venn diagram. Let F = football, B = baseball, and H = hockey. B F H © 2008 Pearson Addison-Wesley. All rights reserved

Cardinal Number Formula For any two sets A and B, © 2008 Pearson Addison-Wesley. All rights reserved

Example: Applying the Cardinal Number Formula Find n(A) if © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table On a given day, breakfast patrons were categorized according to age and preferred beverage. The results are summarized on the next slide. There will be questions to follow. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table Coffee (C) Juice (J) Tea (T) Totals 18-25 (Y) 15 22 18 55 26-33 (M) 30 25 77 Over 33 (O) 45 24 91 90 69 64 223 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Analyzing Data in a Table (C) (J) (T) Totals (Y) 15 22 18 55 (M) 30 25 77 (O) 45 24 91 90 69 64 223 Using the letters in the table, find the number of people in each of the following sets. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.4: Surveys and Cardinal Numbers Always Sometimes Never © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.4: Surveys and Cardinal Numbers 15 36 10 A 10 B 2 11 13 C © 2008 Pearson Addison-Wesley. All rights reserved

Section 2-5 Chapter 1 Infinite Sets and Their Cardinalities © 2008 Pearson Addison-Wesley. All rights reserved

Infinite Sets and Their Cardinalities One-to-One Correspondence and Equivalent Sets The Cardinal Number (Aleph-Null) Infinite Sets Sets That Are Not Countable © 2008 Pearson Addison-Wesley. All rights reserved

One-to-One Correspondence and Equivalent Sets A one-to-one correspondence between two sets is a pairing where each element of one set is paired with exactly one element of the second set and each element of the second set is paired with exactly one element of the first set. © 2008 Pearson Addison-Wesley. All rights reserved

Example: One-to-One Correspondence For sets {a, b, c, d} and {3, 7, 9, 11} a pairing to demonstrate one-to-one correspondence could be {a, b, c, d} {3, 7, 9, 11} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Equivalent Sets Two sets, A and B, which may be put in a one-to-one correspondence are said to be equivalent, written A ~ B. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved The Cardinal Number The basic set used in discussing infinite sets is the set of counting numbers, {1, 2, 3, …}. The set of counting numbers is said to have the infinite cardinal number (aleph-null). © 2008 Pearson Addison-Wesley. All rights reserved

Example: Showing That {2, 4, 6, 8,…} Has Cardinal Number To show that another set has cardinal number we show that it is equivalent to the set of counting numbers. {1, 2, 3, 4, …, n, …} {2, 4, 6, 8, …,2n, …} © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Infinite Sets A set is infinite if it can be placed in a one-to-one correspondence with a proper subset of itself. The whole numbers, integers, and rational numbers have cardinal number © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Countable Sets A set is countable if it is finite or if it has cardinal number © 2008 Pearson Addison-Wesley. All rights reserved

Sets That Are Not Countable The real numbers and irrational numbers are not countable and are said to have cardinal number c (for continuum). © 2008 Pearson Addison-Wesley. All rights reserved

Cardinal Numbers of Infinite Sets Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers c Real Numbers © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.5: Infinite Sets and Their Cardinalities The cardinal number of a finite set is always less than a) True b) False © 2008 Pearson Addison-Wesley. All rights reserved

Section 2.5: Infinite Sets and Their Cardinalities The set {1, 7, 11, 15, 19, …} is equivalent to a) the rational numbers. b) the irrational numbers. c) the reals. © 2008 Pearson Addison-Wesley. All rights reserved