04/20/23 Section 2.1 1

Section 2.1Basic Set Concepts

Objectives

1. Use three methods to represent sets

2. Define and recognize the empty set

3. Use the symbols and .

4. Apply set notation to sets of natural numbers.

5. Determine a set’s cardinal number.

6. Recognize equivalent sets.

7. Distinguish between finite and infinite sets.

8. Recognize equal sets.

04/20/23 Section 2.1 2

Sets

• A collection of objects whose contents can be clearly determined.

• Elements or members are the objects in a set.

• A set must be well defined, meaning that its contents can be clearly determined.

• The order in which the elements of the set are listed is not important.

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Methods for Representing Sets

Capital letters are generally used to name sets.• Word description: Describing the members:

Set W is the set of the days of the week.

• Roster method: Listing the members:W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,

Sunday}

Commas are used to separate the elements of the set.

Braces are used to designate that the enclosed elements form a set.

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Example 1Representing a Set Using a Description

• Write a word description of the set:P = {Washington, Adams, Jefferson, Madison, Monroe}

• Solution: P is the set of the first five presidents of the United States.

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Example 2Representing a Set Using the Roster Method

• Write using the roster method:

Set C is the set of U.S. coins with a value of less than a dollar.

• Solution:

C = {penny, nickel, dime, quarter, half-dollar}

04/20/23 Section 2.1 6

Set-Builder Notation

– Before the vertical line is the variable x, which represents an element in general

– After the vertical line is the condition x must meet in order to be an element of the set.

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Example 3Converting from Set-Builder to Roster Notation

• Express set

A = {x | x is a month that begins with the letter M}

Using the roster method.• Solution:

There are two months, namely March and May.

Thus,

A = { March, May}

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The Empty Set• Also called the null set

• Set that contains no elements

• Represented by { } or Ø

• The empty set is NOT represented by { Ø }. This notation represents a set containing the element Ø.

• These are examples of empty sets:– Set of all numbers less than 4 and greater than 10– {x | x is a fawn that speaks}

04/20/23 Section 2.1 9

Example 4Recognizing the Empty Set

• Which of the following is the empty set?

a. {0}

No. This is a set containing one element.

b. 0

No. This is a number, not a set

c. { x | x is a number less than 4 or greater than 10 }

No. This set contains all numbers that are either less than 4, such as 3, or greater than 10, such as 11.

d. { x | x is a square with three sides}

Yes. There are no squares with three sides.

04/20/23 Section 2.1 10

Notations for Set Membership

is used to indicate that an object is an element of a set. The symbol is used to replace the words “is an element of.”

is used to indicate that an object is not an element of a set. The symbol is used to replace the words “is not an element of.”

04/20/23 Section 2.1 11

Example 5Using the symbols and

Determine whether each statement is true or false:

a. r {a,b,c,…,z}

True

b. 7 {1,2,3,4,5}

True

c. {a} {a,b}

False. {a} is a set and the set {a} is not an element of the set {a,b}.

04/20/23 Section 2.1 12

Example 6Sets of Natural Numbers

= {1,2,3,4,5,…}• Ellipsis, the three dots after the 5 indicate that there is

no final element and that the listing goes on forever.Express each of the following sets using the roster method

a. Set A is the set of natural numbers less than 5.A = {1,2,3,4}

b. Set B is the set of natural numbers greater than or equal to 25.

B = {25, 26, 27, 28,…}c. E = { x| x and x is even}.

E = {2, 4, 6, 8,…}

04/20/23 Section 2.1 13

Inequality Notation and SetsInequality Symbol Set Builder Roster and Meaning Notation Method

04/20/23 Section 2.1 14

Example 7Representing Sets of Natural Numbers

Express each of the following sets using the roster method:

a. { x | x and x ≤ 100}

Solution: {1, 2, 3, 4,…,100}

b. { x | x and 70 ≤ x <100 }

Solution: {70, 71, 72, 73, …, 99}

04/20/23 Section 2.1 15

Example 8Cardinality of Sets

• The cardinal number of set A, represented by n(A), is the number of distinct elements in set A. – The symbol n(A) is read “n of A.”– Repeating elements in a set neither adds new

elements to the set nor changes its cardinality.Find the cardinal number of each set:a. A = { 7, 9, 11, 13 }

n(A) = 4b. B = { 0 }

n(B) = 1c. C = { 13, 14, 15,…,22, 23}

n(C)=11

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Equivalent Sets

• Set A is equivalent to set B if set A and set B contain the same number of elements. For equivalent sets, n(A) = n(B).

These are equivalent sets:

The line with arrowheads, , indicate that each element of set A can be paired with exactly one element of set B and each element of set B can be paired with exactly one element of set A.

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One-To-One Correspondences and Equivalent Sets

• If set A and set B can be placed in a one-to-one correspondence, then A is equivalent to B: n(A) = n(B).

• If set A and set B cannot be placed in a one-to-one correspondence, then A is not equivalent to B:

n(A) ≠n(B).

04/20/23 Section 2.1 18

Example 9Determining if Sets are Equivalent

• This Table shows the celebrities who hosted NBC’s Saturday Night Live most frequently and the number of times each starred on the show.

A = the set of the five most frequent hosts.

B = the set of the number of times each host starred on the show.

• Are the sets equivalent?

Most Frequent Host of

Saturday Night Live

Celebrity Number of Shows Hosted

Steve Martin 14

Alec Baldwin 12

John Goodman 12

Buck Henry 10

Chevy Chase 9

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Example 9 continued

• Method 1: Trying to set up a One-to-One Correspondence.• Solution:The lines with the arrowheads indicate that the correspondence between the sets in not one-to-one. The elements Baldwin and Goodman from set A are both paired with the element 12 from set B. These sets are not equivalent.

04/20/23 Section 2.1 20

Example 9 continued

• Method 2: Counting Elements• Solution:Set A contains five distinct elements: n(A) = 5. Set B contains four distinct elements: n(B) = 4. Because the sets do not contain the same number of elements, they are not equivalent.

04/20/23 Section 2.1 21

Finite and Infinite Sets,Equal Sets

• Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number.

• Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number 0א read “aleph-null”.

• Equal sets: Set A is equal to set B if set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize the equality of sets A and B using the statement A = B.

If two sets are equal, then they must be equivalent!

04/20/23 Section 2.1 22

Example 10Determining Whether Sets are Equal

Determine whether each statement is true or false:

a. { 4, 8, 9 } = { 8, 9, 4 }

True

b. { 1, 3, 5 } = {0, 1, 3, 5 }

False

04/20/23 Section 2.2 23

Section 2.2Subsets

Objectives1. Recognize subsets and use the notation .

2. Recognize proper subsets and use the notation .

3. Determine the number of subsets of a set

4. Apply concepts of subsets and equivalent sets to infinite sets.

04/20/23 Section 2.2 24

Subsets

• Set A is a subset of set B, expressed as

A B,

if every element in set A is also an element in set B.

• The notation means that A is not a subset of B. A is not a subset of set B if there is at least one element of set A that is not an element of set B.

• Every set is a subset of itself.

04/20/23 Section 2.2 25

Example 1Subsets

• Applying the subset definition to the set of people age

25 -29 in

this table:

Percentage of Tattooed Americans, By Age GroupAge Group

Percent Tattooed

18-24 13%

25-29 36%

30-39 28%40-49 14%50-64 10%65+ 7%

04/20/23 Section 2.2 26

Example 1 Continued

• Given:

A = {1, 2, 3} B = {1, 2 }

Is A a subset of B?

No. A B

Is B a subset of A?

Yes. B A

04/20/23 Section 2.2 27

Example 2Proper Subsets

• Set A is a proper subset of set B, expressed as A B, if set A is a subset of set B and sets A and B are not equal ( A ≠ B).

Write , , or both in the blank to form a true statement.A = { x | x is a person and x lives in San Francisco}

B = { x | x is a person and x lives in California}A ____B

Solution: A , B

b. A = { 2, 4, 6, 8}B = { 2, 8, 4, 6}A ____B

Solution: A B

04/20/23 Section 2.2 28

Subsets and the Empty Set

• The Empty Set as a Subset

1. For any set B, Ø B.

2. For any set B other than the empty set, Ø B.

04/20/23 Section 2.2 29

The Number of Subsets of a Given Set

• As we increase the number of elements in the set by one, the number of subsets doubles.

• 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.

Set Number of Elements

List of All Subsets Number of Subsets

{ } 0 { } 1

{a} 1 {a},{ } 2

{a,b} 2 {a,b},{a}, {b},{ } 4

{a,b,c}3

{a,b,c},{a,b}, {a,c},{ b,c },

{a},{b},{c}, { }8

04/20/23 Section 2.2 30

Example 3Finding the Number of Subsets and Proper Subsets

Find the number of subsets and the number of proper subsets.

a. {a, b, c, d, e } There are 5 elements so there are 25 = 32 subsets and 25-1

= 31 proper subsets.

b. { x | x and 9 ≤ x ≤ 15 }In roster form, we see that there are 7 elements:

{ 9, 10, 11, 12, 13, 14, 15 }There are 27 = 128 subsets and 27-1 = 127 proper subsets.

04/20/23 Section 2.2 31

The Number of Subsets of Infinite Sets

• There are 0א natural numbers.

– It has 2 א0 subsets.

– It has 2 א0 – 1 proper subsets

– 0א > 0א 2

– Denote 2 א0 by 1א

– 0א > 1א

• 0א is the “smallest” transfinite cardinal number in an infinite hierarchy of different infinities.

04/20/23 Section 2.2 32

Cardinal Numbers of Infinite Sets

Georg Cantor (1845 – 1918) studied the mathematics of infinity and assigned the transfinite cardinal number 0א to the set of natural numbers. He used one-to-one

correspondences to establish some surprising equivalences between the set of natural numbers and its proper subsets.

04/20/23 Section 2.3 33

Section 2.3Venn diagrams and Set Operations

Objectives1. Understand the meaning of a universal set.2. Understand the basic ideas of a Venn diagram.3. Use Venn diagrams to visualize relationships between two

sets.4. Find the complement of a set5. Find the intersection of two sets.6. Find the union of two sets.7. Perform operations with sets.8. Determine sets involving set operations from a Venn

diagram.9. Understand the meaning of and and or.10. Use the formula for n (A U B).

04/20/23 Section 2.3 34

Universal Sets and Venn Diagrams

• The universal set is a general set that

contains all elements under discussion.

• John Venn (1843 – 1923) created Venn

diagrams to show the visual relationship among sets.

• Universal set is represented by a rectangle

• Subsets within the universal set are depicted by circles, or sometimes ovals or other shapes.

04/20/23 Section 2.3 35

Example 1Determining Sets From a Venn Diagram

• Use the Venn diagram to determine each of the following sets:

a. U

U = { O , ∆ , $, M, 5 }

b. A

A = { O , ∆ }

c. The set of elements in U that are not in A.

{$, M, 5 }

04/20/23 Section 2.3 36

Representing Two Sets in a Venn Diagram

Disjoint Sets: Two sets that have Equal Sets: If A = B then AB

no elements in common. and B A.

Proper Subsets: All elements of Sets with Some Common Elements

set A are elements of set B. Some means “at least one”. The

representing the sets must overlap.

04/20/23 Section 2.3 37

Example 2Determining sets from a Venn Diagram

Solutions:

a. U = { a, b, c, d, e, f, g }

b. B = {d, e }

c. {a, b, c }

d. {a, b, c, f, g }

e. {d}

• Use the Venn Diagram to determine:

a. U

b. B

c. The set of elements in A but not B

d. The set of elements in U that are not in B

e. The set of elements in both A and B.

04/20/23 Section 2.3 38

The Complement of a Set

• The complement of set A, symbolized by A’ is the set of all elements in the universal set that are not in A. This idea can be expressed in set-builder notation as follows:

A’ = {x | x U and x A}

• The shaded region represents the complement of set A. This region lies outside the circle.

04/20/23 Section 2.3 39

Example 3Finding a Set’s Complement

• Let U = { 1, 2, 3, 4, 5, 5, 6, 8, 9} and A = {1, 3, 4, 7 }. Find A’.

• Solution:

Set A’ contains all the elements of set U that are not in set A.

Because set A contains the

elements 1,3,4,and 7, these

elements cannot be members of

set A’:

A’ = {2, 5, 6, 8, 9}

04/20/23 Section 2.3 40

The Intersection and Union of Sets

• The intersection of sets A and B, written A∩B, is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows:

A∩B = { x | x A and xB}• The union of sets A and B, written AUB is the set of

elements are in A or B or in both sets. This definition can be expressed in set-builder notation as follows:

AUB = { x | x A or xB}• For any set A:

– A∩Ø = Ø– AUØ = A

04/20/23 Section 2.3 41

Example 4Finding the Intersection of Two Sets

• Find each of the following intersections:

a. {7, 8, 9, 10, 11} ∩ {6, 8, 10, 12}

{8, 10}

a. {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8}

Ø

a. {1, 3, 5, 7, 9} ∩ Ø

Ø

04/20/23 Section 2.3 42

Example 5Finding the Union of Sets

• Find each of the following unions:

a. {7, 8, 9, 10, 11} U {6, 8, 10, 12}

b. {1, 3, 5, 7, 9} U {2, 4, 6, 8}

c. {1, 3, 5, 7, 9} U Ø

a. {6, 7, 8, 9, 10, 11, 12}

b. {1, 2, 3, 4, 5, 6, 7, 8, 9}

c. {1, 3, 5, 7, 9}

04/20/23 Section 2.3 43

Example 6Performing Set Operations

a. (A U B)’

• Solution:

A U B = {1, 3, 7, 8, 9, 10}

(A U B)’ = {2, 4, 5, 6}

b. A’ ∩ B’• Solution

A’ = {2, 4, 5, 6, 8, 10}

B’ = {1, 2, 4, 5, 6, 9}

A’ ∩ B’ = {2, 4, 5, 6 }

• Always perform any operations inside parenthesis first!Given:

U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}A = { 1, 3, 7, 9 }B = { 3, 7, 8, 10 }

• Find

04/20/23 Section 2.3 44

Example 7Determining Sets from a Venn Diagram

Set to Determine

Description of Set Regions in Venn Diagram

a. A B set of elements in A or B or Both I,II,III

b. (A B)’ set of elements in U that are not in A B IV

c. A B set of elements in both A and B II

d. (A B)’ set of elements in U that are not in A B I, III, IV

e. A’ B set of elements that are not in A and are in B III

f. A B’ set of elements that are in A or not in B or both I,II, IV

04/20/23 Section 2.3 45

Sets and Precise Use of Everyday English

• Set operations and Venn diagrams provide precise ways of organizing, classifying, and describing the vast array of sets and subsets we encounter every day.

• Or refers to the union of sets

• And refers to the intersection of sets

04/20/23 Section 2.3 46

Example 8The Cardinal Number of the Union of Two Finite

Sets

• Some of the results of the campus blood drive survey indicated that 490 students were willing to donate blood, 340 students were willing to help serve a free breakfast to blood donors, and 120 students were willing to do both.

How many students were willing to donate blood or serve breakfast?

04/20/23 Section 2.3 47

Example 8 continued

04/20/23 Section 2.4 48

Section 2.4

Objectives

1. Perform set operations with three sets.

2. Use Venn diagrams with three sets.

3. Use Venn diagrams to prove equality of sets.

04/20/23 Section 2.4 49

Example 1Set Operations with Three Sets

• Given

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {1, 2, 3, 4, 5}

B = {1, 2, 3, 6, 8}

C = {2, 3, 4, 6, 7}

• Find A ∩ (B U C’)

• Solution

Find C’

= {1, 5, 8, 9}

• Find (B U C’)

= {1, 2, 3, 6, 8} U {1, 5, 8, 9}

= {1, 2, 3, 5, 6, 8, 9}

• Find A ∩ (B U C’)

= {1, 2, 3, 5, 6, 8, 9} ∩ {1, 5, 8, 9}

= {1, 2, 3, 5}

04/20/23 Section 2.4 50

Venn diagrams with Three Sets

04/20/23 Section 2.4 51

Example 2Determining Sets from a Venn Diagram with Three

Intersecting SetsUse the Venn diagram to find:

a. A

b. A U B

c. B ∩ C

d. C’

e. A ∩ B ∩ C

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Example 4Proving the Equality of Sets

• Prove that (A ∩ B)’ = A’ U B’

• Solution– We can apply deductive reasoning using a Venn

diagram to prove this statement is true for all sets A and B.

– If both sets represent the same regions in this general diagram then this proves that they are equal.

04/20/23 Section 2.4 53

Example 4 continued

Prove that (A ∩ B)’ = A’ U B’

Solution: Begin with the regions represented by (A ∩ B)’.

Next, find the regions

Represented by A’U B’.

Set Regions

A I, II

B II, III

(A ∩ B) II

(A ∩ B)’ I, III, IV

Set Regions

A’ III, IV

B’ I,IV

A’ U B’ I, III,IV

Since both (A ∩ B)’ and A’ U B’ are represented by the same regions, the result proves that they are equal.

04/20/23 Section 2.4 54

De Morgan’s Laws

• (A ∩ B)’ = A’ U B’

The complement of the intersection of the two sets is the union of the complements of those sets.

• (A’ U B)’ = A’ ∩ B’

The complement of the union of two sets is the intersection of the complements of those sets.

04/20/23 Section 2.5 55

Section 2.5Survey Problems

Objectives

1. Use Venn Diagrams to visualize a survey’s results.

2. Use survey results to complete Venn diagrams and answer questions about the survey.

04/20/23 Section 2.5 56

Example 1Visualizing the Results of a Survey

The results of the survey are summarized in this figure.

a. How many students are willing

to donate blood?

Represented by regions I and II.

Thus, n(A) = 370 + 120 = 490.

b. How many are willing to donate

blood but not serve breakfast?

Region 1 represents A ∩ B’ = 370

c. How many weren’t willing to do either?

A’ ∩ B’ is in region IV; those areas outside the circles = 290.

04/20/23 Section 2.5 57

Solving Survey Problems

1. Use the survey’s description to define sets and draw a Venn diagram.

2. Use the survey’s results to determine the cardinality for each region in the Venn diagram. Start with the intersection of the sets, the innermost region, and work outward.

3. Use the completed Venn diagram to answer the problem’s questions.

04/20/23 Section 2.5 58

Example 2Surveying People’s Attitudes

A survey is taken that asks 2000 randomly selected U.S. and Mexican adults the following question: Do you agree or disagree that the primary cause of poverty is societal injustice?

The results of the survey showed that:

1060 people agreed with the statement

400 Americans agreed with the statement.

If half the adults surveyed were Americans

a. How many Mexicans agreed with the statement?

b. How many Mexicans disagreed with the statement?

04/20/23 Section 2.5 59

Example 2 continued

Solution:1. Define the sets and draw a Venn diagram. Set U.S. is the set of Americans surveyed. Set A (labeled “Agree”) is the set of people surveyed who agreed with the statement. The areaoutside the circle represents the set ofMexicans. The group of people outside the A circle mustbe the set of people disagreeing with the statement.

2. Determine the cardinality for each region in the Venn

diagram, starting with the innermost region. We are given

the following cardinalities:

04/20/23 Section 2.5 60

Example 2 continued• There were 2000 people surveyed: n(U) = 2000.

• Half the people surveyed were Americans: n(U.S.) = 1000.

• The number of people who agreed with the statement was 1060: n(A) = 1060.

• There were 400 Americans who agreed with the statement: n(U.S. ∩ A) = 400.

• Starting with region II and moving outwards to regions I and III:

04/20/23 Section 2.5 61

Example 2 continued

a. The Mexicans who agreed with the statement are shown in region III. This means that 660 Mexicans agreed that societal injustice is the primary cause of poverty.

b. The Mexicans who disagreed with the statement corresponds to region IV. This means that 340 Mexicans disagreed that societal injustice is the primary cause of poverty.