1. Section 2.1 Basic Set Concepts
Objectives
Use three methods to represent sets
Define and recognize the empty set
Use the symbols  and .
Apply set notation to sets of natural numbers.
Determine a set’s cardinal number.
Recognize equivalent sets.
Distinguish between finite and infinite sets.
Recognize equal sets.
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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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3. 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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4. Example 1 Representing 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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5. Example 2 Representing 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}
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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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7. Example 3 Converting 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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8. 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}
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9. Example 4 Recognizing the Empty Set
Which of the following is the empty set?
{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.
{ x | x is a square with three sides}
Yes. There are no squares with three sides.
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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.”
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11. Example 5 Using the symbols  and 
Determine whether each statement is true or false:
r  {a,b,c,…,z}
True
7  {1,2,3,4,5}
c. {a}  {a,b}
False. {a} is a set and the set {a} is not an element of the set {a,b}.
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12. Example 6 Sets 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
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,…}
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13. Inequality Notation and Sets
Inequality Symbol Set Builder Roster
and Meaning Notation Method
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14. Example 7 Representing Sets of Natural Numbers
Express each of the following sets using the roster method:
{ x | x   and x ≤ 100}
Solution: {1, 2, 3, 4,…,100}
b. { x | x  and 70 ≤ x <100 }
Solution: {70, 71, 72, 73, …, 99}
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15. Example 8 Cardinality 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 = { 7, 9, 11, 13 }
n(A) = 4
b. B = { 0 }
n(B) = 1
c. C = { 13, 14, 15,…,22, 23}
n(C)=11
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16. 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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17. 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).
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18. Example 9 Determining 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
Buck Henry 10
Chevy Chase 9
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19. 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.
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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.
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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 א0 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!
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22. Example 10 Determining Whether Sets are Equal
Determine whether each statement is true or false:
{ 4, 8, 9 } = { 8, 9, 4 }
True
b. { 1, 3, 5 } = {0, 1, 3, 5 }
False
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23. Section 2.2 Subsets Objectives
Recognize subsets and use the notation .
Recognize proper subsets and use the notation .
Determine the number of subsets of a set
Apply concepts of subsets and equivalent sets to infinite sets.
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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.
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25. Example 1 Subsets
Percentage of Tattooed Americans, By Age Group
Age Group
Percent Tattooed
18-24
13%
25-29
36%
30-39
28%
40-49
14%
50-64
10%
65+ 7%
Applying the subset definition to the set of people age in
this table:
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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
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27. Example 2 Proper 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
A = { 2, 4, 6, 8}
B = { 2, 8, 4, 6}
Solution: A  B
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28. Subsets and the Empty Set
The Empty Set as a Subset
For any set B, Ø  B.
For any set B other than the empty set, Ø  B.
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29. The Number of Subsets of a Given Set
Number of Elements
List of All Subsets
Number of Subsets
{ } 1
{a}
{a},{ } 2
{a,b}
{a,b},{a}, {b},{ } 4
{a,b,c} 3
{a,b,c},{a,b}, {a,c},{ b,c },
{a},{b},{c}, { } 8
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.
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30. Example 3 Finding the Number of Subsets and Proper Subsets
Find the number of subsets and the number of proper subsets.
{a, b, c, d, e }
There are 5 elements so there are 25 = 32 subsets and 25-1 = 31 proper subsets.
{ 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.
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31. The Number of Subsets of Infinite Sets
There are א0 natural numbers.
It has 2 א0 subsets.
It has 2 א0 – 1 proper subsets
2 א0 > א0
Denote 2 א0 by א1
א1 > א0
א0 is the “smallest” transfinite cardinal number in an infinite hierarchy of different infinities.
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32. Cardinal Numbers of Infinite Sets
Georg Cantor (1845 – 1918) studied the mathematics of infinity and assigned the transfinite cardinal number א0 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.
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Section 2.2

33. Section 2.3 Venn diagrams and Set Operations
Objectives
Understand the meaning of a universal set.
Understand the basic ideas of a Venn diagram.
Use Venn diagrams to visualize relationships between two sets.
Find the complement of a set
Find the intersection of two sets.
Find the union of two sets.
Perform operations with sets.
Determine sets involving set operations from a Venn diagram.
Understand the meaning of and and or.
Use the formula for n (A U B).
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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.
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Section 2.3

35. Example 1 Determining Sets From a Venn Diagram
Use the Venn diagram to determine each of the following sets: U
U = { O , ∆ , $, M, 5 } A
A = { O , ∆ }
The set of elements in U that are not in A.
{$, M, 5 }
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36. Representing Two Sets in a Venn Diagram
Disjoint Sets: Two sets that have Equal Sets: If A = B then AB
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.
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37. Example 2 Determining sets from a Venn Diagram
Use the Venn Diagram to determine: U B
The set of elements in A but not B
The set of elements in U that are not in B
The set of elements in both A and B.
Solutions:
U = { a, b, c, d, e, f, g }
B = {d, e }
{a, b, c }
{a, b, c, f, g }
{d}
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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.
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39. Example 3 Finding 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}
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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 xB}
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 xB}
For any set A:
A∩Ø = Ø
AUØ = A
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41. Example 4 Finding the Intersection of Two Sets
Find each of the following intersections:
{7, 8, 9, 10, 11} ∩ {6, 8, 10, 12}
{8, 10}
{1, 3, 5, 7, 9} ∩ {2, 4, 6, 8} Ø
{1, 3, 5, 7, 9} ∩ Ø
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42. Example 5 Finding the Union of Sets
Find each of the following unions:
{7, 8, 9, 10, 11} U {6, 8, 10, 12}
{1, 3, 5, 7, 9} U {2, 4, 6, 8}
{1, 3, 5, 7, 9} U Ø
{6, 7, 8, 9, 10, 11, 12}
{1, 2, 3, 4, 5, 6, 7, 8, 9}
{1, 3, 5, 7, 9}
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43. Example 6 Performing Set Operations
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
(A U B)’
Solution:
A U B = {1, 3, 7, 8, 9, 10}
(A U B)’ = {2, 4, 5, 6}
A’ ∩ B’
Solution
A’ = {2, 4, 5, 6, 8, 10}
B’ = {1, 2, 4, 5, 6, 9}
A’ ∩ B’ = {2, 4, 5, 6 }
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44. Example 7 Determining 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
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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
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46. Example 8 The 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?
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Section 2.3

47. Example 8 continued
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48. Section 2.4 Objectives Perform set operations with three sets.
Use Venn diagrams with three sets.
Use Venn diagrams to prove equality of sets.
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49. Example 1 Set 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}
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50. Venn diagrams with Three Sets
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Section 2.4

51. Example 2 Determining Sets from a Venn Diagram with Three Intersecting Sets
Use the Venn diagram to find: A
A U B
B ∩ C C’
A ∩ B ∩ C
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52. Example 4 Proving 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.
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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.
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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.
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Section 2.4

55. Section 2.5 Survey Problems
Objectives
Use Venn Diagrams to visualize a survey’s results.
Use survey results to complete Venn diagrams and answer questions about the survey.
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Section 2.5

56. Example 1 Visualizing the Results of a Survey
The results of the survey are summarized in this figure.
How many students are willing
to donate blood?
Represented by regions I and II.
Thus, n(A) = = 490.
How many are willing to donate
blood but not serve breakfast?
Region 1 represents A ∩ B’ = 370
How many weren’t willing to do either?
A’ ∩ B’ is in region IV; those areas outside the circles = 290.
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Section 2.5

57. Solving Survey Problems
Use the survey’s description to define sets and draw a Venn diagram.
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.
Use the completed Venn diagram to answer the problem’s questions.
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Section 2.5

58. Example 2 Surveying 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
How many Mexicans agreed with the statement?
How many Mexicans disagreed with the statement?
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Section 2.5

59. Example 2 continued Solution: 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 area
outside the circle represents the set of
Mexicans. The group of people outside the A circle must
be the set of people disagreeing with the statement.
Determine the cardinality for each region in the Venn
diagram, starting with the innermost region. We are given
the following cardinalities:
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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:
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Section 2.5

61. Example 2 continued 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.
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.
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Section 2.5