SEVENTH EDITION and EXPANDED SEVENTH EDITION

Chapter 2 Sets

2.1 Set Concepts

Set A collection of objects, which are called elements or members of the set. Listing the elements of a set inside a pair of braces, { }, is called roster form .

Well-defined Set A set which has no question about what elements should be included. Its elements can be clearly determined. No opinion is associated with the members.

Roster Form This is the form of the set where the elements are all listed, each separated by commas. Example: Set N is the set of all natural numbers less than or equal to 25. Solution: N = {1, 2, 3, 4, 5,…25} The 25 after the ellipsis indicates that the elements continue up to and including the number 25.

Set-Builder (or Set-Generator) Notation A formal statement that describes the members of a set is written between the braces. A variable may represent any one of the members of the set. Example: Write set B = {2, 4, 6, 8, 10} in set-builder notation. Solution:

Finite Set A set that contains no elements or the number of elements in the set is a natural number. Example: Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number.

Infinite Set An infinite set contains an indefinite (uncountable) number of elements. The set of natural numbers is an example of an infinite set because it continues to increase forever without stopping, making it impossible to count its members.

Equal Sets Equal sets have the exact same elements in them, regardless of their order. Symbol: A = B

Cardinal Number The number of elements in set A is its cardinal number. Symbol: n(A)

Equivalent Sets Equivalent sets have the same number of elements in them. Symbol: n(A) = n(B)

Empty (or Null) Set A null (or empty set ) contains absolutely NO elements. Symbol:

Universal Set The universal set contains all of the possible elements which could be discusses in a particular problem. Symbol: U

2.2 Subsets

Subsets A set is a subset of a given set if and only if all elements of the subset are also elements of the given set. Symbol: To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B.

Determining Subsets A B Í . Example: Determine whether set A is a subset of set B. A = { 3, 5, 6, 8 } B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Solution: All of the elements of set A are contained in set B, so A B Í .

Proper Subset All subsets are proper subsets except the subset containing all of the given elements. Symbol:

Determining Proper Subsets Example: Determine whether set A is a proper subset of set B. A = { dog, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, and sets A and B are not equal, therefore A B.

Determining Proper Subsets continued Example: Determine whether set A is a proper subset of set B. A = { dog, bird, fish, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, but sets A and B are equal, therefore A B.

Number of Distinct Subsets The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A. Example: Determine the number of distinct subsets for the given set { t , a , p , e }. List all the distinct subsets for the given set: { t , a , p , e }.

Number of Distinct Subsets continued Solution: Since there are 4 elements in the given set, the number of distinct subsets is 24 = 2 • 2 • 2 • 2 = 16 subsets. {t,a,p,e}, {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e}, {t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { }

Venn Diagrams and Set Operations 2.3 Venn Diagrams and Set Operations

Venn Diagrams A Venn diagram is a technique used for picturing set relationships. A rectangle usually represents the universal set, U. The items inside the rectangle are divided into subsets of U and are represented by circles.

Disjoint Sets Two sets which have no elements in common are said to be disjoint. The intersection of disjoint sets is the empty set. Disjoint sets A and B are drawn in this figure. There are no elements in common since there is no overlap- ping area of the two circles.

Overlapping Sets For sets A and B drawn in this figure, notice the overlapping area shared by the two circles. This section represents the elements are in the intersection of set A and set B.

Complement of a Set The set known as the complement contains all the elements of the universal set, which are not listed in the given subset. Symbol: A’

Intersection The intersection of two given sets contains only those elements common to those sets. Symbol:

Union The union of two given sets contains all of the elements for those sets. The union “unites” that is, it brings together everything into one set. Symbol:

Subsets When every element of B is also an element of A. Circle B is completely inside circle A.

Equal Sets When set A is equal to set B, all the elements of A are elements of B, and all the elements of B are elements of A. Both sets are drawn as one circle.

Venn Diagrams with Three Sets 2.4 Venn Diagrams with Three Sets

General Procedure for Constructing Venn Diagrams with Three Sets Find the elements that are common to all three sets and place in region V.

General Procedure for Constructing Venn Diagrams with Three Sets continued Find the elements for region II. Find the elements in . The elements in this set belong in regions II and V. Place the elements in the set that are not listed in region V in region II. The elements in regions IV and VI are found in a similar manner.

General Procedure for Constructing Venn Diagrams with Three Sets continued Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are found in a similar manner.

General Procedure for Constructing Venn Diagrams with Three Sets continued Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII. U I II III A V B IV VI VII C VIII

Example: Constructing a Venn diagram for Three Sets Construct a Venn diagram illustrating the following sets. U = {1, 2, 3, 4, 5, 6, 7, 8} A = { 1, 2, 5, 8} B = {2, 4, 5} C = {1, 3, 5, 8} Solution: Find the intersection of all three sets and place in region V, {5}.

Example: Constructing a Venn diagram for Three Sets continued Determine the intersection of sets A and B and place in region II. {2, 5} Element 5 has already been placed in region V, so 2 must be placed in region II. Now determine the numbers that go into region V. { 1, 2, 5, 8} Since 5 has been placed in region V, place 1 and 8 in region IV.

Example: Constructing a Venn diagram for Three Sets continued Now determine the numbers that go in region VI. {5} There are now new numbers to be placed in this region. Since all numbers in set A have been placed, there are no numbers in region I. The same procedures using set B completes region III. Using set C completes region VII.

Example: Constructing a Venn diagram for Three Sets continued The Venn diagram is then completed. U A B C V I III VII VI IV VIII II 2 4 5 1,8 3 6 7

De Morgan’s Laws A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms.

2.5 Applications of Sets

Example: Toothpaste Taste Test A drug company is considering manufacturing a new toothpaste. They are considering two flavors, regular and mint. In a sample of 120 people, it was found that 74 liked the regular, 62 liked the mint, and 35 liked both types. How many liked only the regular flavor? How many liked either one or the other or both? How many people did not like either flavor?

Solution Begin by setting up a Venn diagram with sets A (regular flavor) and B (mint flavor). Since some people liked both flavors, the sets will overlap and the number who liked both with be placed in region II. 35 people liked both flavors.

Solution continued Next, region I will refer to those who liked only the regular and region III will refer to those who liked only the mint. In order to get the number of people in each region, find the difference between all the people who liked each toothpaste and those who liked both. 74 – 35 = 39 62 – 35 = 27 U R egular Mint 35 bot h 39 regular only 27 mint

Solution continued “One or the other or both” represents the UNION of the two sets. Therefore, 39 + 27 + 35 = 101 people who liked one or the other or both.

Solution continued Take the total number of people in the entire sample and subtract the number who liked one or the other or both. 19 people did not like either flavor.

2.6 Infinite Sets

Infinite Sets An infinite set is a set that can be placed in a one-to-one correspondence with a proper subset of itself. These sets are “unbounded”.

Example: The Set of Multiples of Four Show that it is an infinite set. {4, 8, 12, 16, 20, …,4n, …} Solution: We establish one-to-one correspondence between the counting numbers and a proper subset of itself. Given set: {4, 8, 12, 16, 20, …, 4n, …} Proper subset: {4, 8, 12, 16, 20, …, 4n + 4, …} Therefore, the given set is infinite.

Countable Sets A set is countable if it is finite or if it can be placed in a one-to-one correspondence with the set of counting numbers. Any set that can be placed in a one-to-one correspondence with a set of counting numbers has cardinality aleph-null and is countable.