Chapter 2
Section 1 – Sets
Definition – any group or collection of objects is
called a set.
 The objects that belong to sets are called
elements or members.
 Each object in a set must be unique; repetitions
are not allowed.
Basic Number Sets
Natural Numbers
 Consists of the positive numbers
 N = {1, 2, 3, …}
 An ellipse, …, is used to indicate the members
continue as suggested by those listed.
Basic Number Sets
Whole Numbers
 Consists of the positive numbers and zero.
 W = {0, 1, 2, 3, …}
Basic Number Sets
Integers
 Consists of the positive and negative numbers.
 I = {…,-3,-2,-1,0, 1, 2, 3, …}
Basic Number Sets
Rational Numbers
 Q = {all terminating or repeating decimals}
 Terminating decimal: ⅞=.875
 Repeating decimal:
7
= .7
9
Basic Number Sets
Irrational Numbers
 I = {all nonterminating or repeating decimals}
2 = 1.414213562
π = 3.141592654
Basic Number Sets
Real Numbers
 R= {all rational and irrational numbers}
7
16
π = 3.141592654
Roster Method
Sets can be written by placing the elements between
braces.
Example:
the set of states that border the Pacific Ocean
Pacific = {Washington, Hawaii, Oregon,
Alaska, California}
Roster Method
The solution set of 2x+5=19 is {7}
2x + 5 =
2x + 5 − 5 =
2x =
1
⋅ 2x =
2
x=
19
19 − 5
14
1
⋅ 14
2
7
Roster Method


The set of all integers less than or equal to 2 is
{…, -2, -1, 0, 1, 2}
The set of all integers less than 2 is {…, -2, -1, 0, 1}
Well Defined Set
Definition
A set is well defined if it is possible to whether
or not an item is a member of a set or not.


The set of good movies released during 2005 is
not well defined.
The set containing the Oscar winning best movie
for 2005 is a well defined set.
Element of a Set
Pacific = {Washington, Hawaii, Oregon,
Alaska, California}
Washington is an element of the set Pacific can
be written as
Washington ∈ Pacific
Indiana is not an element of the set Pacific can be
written as
Indiana ∉ Pacific
Empty Set
Definition
The empty set, or null set, is the set that
contains no elements and is written Ø.
Example
The set of mountains in the state of Indiana.
Set-Builder Notation
Set builder notation uses a rule to define which
elements are members of a set

p
Q =  x | x = where p, q ∈ I , q ≠
q


0

Cardinality
Definition
the cardinal number of a set S is the number of
elements in the set. It is written n(S).



n(Pacific)=5
n(Ø)=0
For z={0}, n(Z)=1
Cardinality
Problem
What is the cardinality of the set
F = { 3,7,11,15,19,23}
n( F ) = 6
Cardinality
Problem
What is the cardinality of the set
F = { − 5,− 3,0,1,6}
(a ) 0
(b) 5
(c) - 5
(d ) - 5
Cardinality
Problem
What is the cardinality of the set
F1 = {1,2,3,4,5,6}
6
F2 = { 4,8,12,16,20,24}
6
F3 = {1,5,9,13,17,20}
6
Cardinality
Problem
What is the cardinality of the set
F1 = { 3,8,13,18,23,28,33}
7
F2 = { 5,10,15,20,25,30,35}
7
F3 = {1,2,3,4,5,6,7}
7
Cardinality
Problem
What is the cardinality of the set
V1 = {12,19,26,33, ,250}
V2 = { 7,14,21,28,  ,245}
V3 = {1,2,3,4, ,35}
The cardinality is 35
Cardinality
Problem
What is the cardinality of the set
F = { 8,14,20,26,...,260}
(a ) 5
(b) 6
(c) 258
(d ) 43
Finite and Infinite Sets
Definition
If the cardinality of a set is finite if the number
of elements in the set is a whole number.
Otherwise, the set is infinite.
n( R ) = ∞
Equal Sets
Definition
Set A is equal to set B, written as A=B, if and
only if A and B have the same elements.
P1 = {Washington, Hawaii, Oregon, Alaska, California}
P2 = {x | x is a state that borders the Pacific Ocean}
P1 = P2
Equivalent Sets
Definition
Set A is equivalent to set B, denoted by A~B, if
and only if A and B have the same number of
elements.
P1 = {Washington, Hawaii, Oregon, Alaska, California}
P3 = {Indiana, Illinois, Michigan, Kentucky, Ohio}
P1 ~ P3, P1 ≠ P3
Assignment
Page 57 –
1, 2, 3, 6, 11, 13, 14, 15,
21, 25, 33, 35, 36, 42, 43,
47, 49, 51, 53, 56, 60
Section 2 – Subsets
Definition
The set of all elements under consideration is
called the universal set. It is denoted by U.
Universal Set
For the set Pacific, the universal set is the set of
the United States.
{x | x∈
I and x > 7} I is the universal set
{x|x>7}, it is assumed that R is the
universal set, but when there could be
doubt, it should be specified.
Complement
Definition
the complement of a set A, denoted by A,′ is
the set of all elements of the universal set U
that are not elements of A.
Complement
Problem
This year you need to take Basic and Intermediate
Math, French I and French II, Anatomy, and
Psychology. In the Fall semester you take Basic
Math, French I, and Psychology. What will you
take in the Spring semester?
U = {Basic Math, Int Math, French I,
French II, Anatomy, Psychology}
E = {Basic Math, French I, Psychology}
E′ = {Int Math, French II, Anatomy}
Complement
Problem
U = {the United States}
E = {x | x is a state east of the Mississippi R}
E′ = {x | x is a state west of the Mississippi R}
Complement
Problem
U = {1,2,3,4,5,6,7,8,9}
E = {2,4,6,8}
E′ = {1,3,5,7,9}
Complement
Example
U′ = Ø
That is, the complement of the
universal set is the null set.
Ø′ = U
That is, the complement of the null set
is the universal set.
Subsets
Definition
A is a subset of B, denoted by A ⊆ B , if
and only if every element of A is also an
element of B.
Subsets
Q⊆ R
For any set A, Ø ⊆ A
For any set A, A ⊆ A
{1,3,5,7,9} ⊆ {1,2,3,4,5,6,7,8,9}
Venn Diagrams
Venn diagrams are used to display the
relationships between sets.
The rectangle represents
the universal set.
The circle represents
the set.
Proper Subset
Definition
⊂ B , if
Set A is a proper subset of B, denoted by A every element of A is an element of B, and there is
at least one element of B that is not an element of
A.
Using Venn diagrams
Proper Subset
Problem
N⊂ W
{1,3,5,7,9} ⊂ {1,2,3,4,5,6,7,8,9}
Bears ⊂ { x | x ∈ NFL}
Number of Subsets
Problem
List all the subsets of the set {1}
Ø, {1}
Number of Subsets
Problem
List all the subsets of the set {1,2}
Ø, {1},{2},{1,2}
Number of Subsets
In general, a set with n elements has 2n subsets.
{1,2,3,4,5} has 25 subsets, or 32
{January, February, …, December} has 212
subsets, or 4,096.
Assignment
Page 66 –
1, 3, 5, 6, 10, 13, 15, 25, 30,
32, 41, 47, 54
Section 3 – Set Operations
Definition
The intersection of sets A and B, denoted
by A ∩ B , is the set of elements common
to both A and B.
Intersection
Using set-builder notation, the intersection of
sets A and B is written as
A ∩ B = { x | x ∈ A and x ∈ B}
Intersection
Example
U={1,2,3,4,5,6,7,8,9,10,11,12}
T2={2,4,6,8,10,12}
T3={3,6,9,12}
T 2 ∩ T 3 = { 6,12}
Intersection
Example
U={1,2,3,4,5,6,7,8,9,10,11,12}
T2={2,4,6,8,10,12}
T3={3,6,9,12}
T 2′ = {1,3,5,7,9,11}
T 3′ = {1,2,4,5,7,8,10,11}
T 2′ ∩ T 3′ = {1,5,7, ,11}
Disjoint Sets
Definition
Two sets are disjoint if their intersection is the
empty set.
Disjoint Sets
Example
U={1,2,3,4,5,6,7,8,9,10,11,12}
Odd={1,3,5,7,9,11}
Even={2,4,6,8,10,12}
Odd ∩ Even = Ø
therefore, disjoint
Union
Definition
The union of sets A and B, denoted by A ∪ B ,
is the set of elements common to both A, or
to B, or both.
Union
Using set-builder notation, the union of sets A
and B is written as
A ∪ B = { x | x ∈ A or x ∈ B}
Union
Example
U = {1,2,3,4,5,6,7,8,9,10,11,12}
P = {1,2,3,5,7,11}
T 2 = { 2,4,6,8,10,12}
P ∪ T 2 = {1,2,3,4,5,6,7,8,10,11,12}
Union
Example
U = {1,2,3,4,5,6,7,8,9,10,11,12}
P = {1,2,3,5,7,11}
T 2 = { 2,4,6,8,10,12}
T 2′ = {1,3,5,7,9,11}
T 3 = { 3,6,9,12}
T 3′ = {1,2,4,5,7,8,10,11}
T 2′ ∪ T 3′ = {1,2,3,4,5,7,8,9,10,11}
P ∩ ( T 2′ ∪ T 3′ ) = {1,2,3,5,7,11}
De Morgan’s Laws
Example
U = {1,2,3,4,5,6,7,8,9,10,11,12}
T 2 = { 2,4,6,8,10,12}
T 2′ = {1,3,5,7,9,11}
T 3 = { 3,6,9,12}
T 3′ = {1,2,4,5,7,8,10,11}
( T 2 ∪ T 3) ′ = {1,5,7,11}
T 2′ ∩ T 3′ = {1,5,7,11}
T 2 ∪ T 3 = { 2,3,4,6,8,9,10,12}
( T 2 ∪ T 3)
′
= T 2′ ∩ T 3′
De Morgan’s Laws
Example
U = {1,2,3,4,5,6,7,8,9,10,11,12}
T 2 = { 2,4,6,8,10,12}
T 2′ = {1,3,5,7,9,11}
T 3 = { 3,6,9,12}
T 3′ = {1,2,4,5,7,8,10,11}
( T 2 ∩ T 3) ′ = {1,2,3,4,5,7,8,9,10,11}
T 2′ ∪ T 3′ = {1,2,3,4,5,7,8,9,10,11}
T 2 ∩ T 3 = { 6,12}
( T 2 ∩ T 3)
′
= T 2′ ∪ T 3′
Properties of Sets
Commutative Law
A∩ B = B ∩ A
A∪ B = B ∪ A
Properties of Sets
Associative Law
( A ∩ B) ∩ C = A ∩ ( B ∩ C )
( A ∪ B) ∪ C = A ∪ ( B ∪ C )
Properties of Sets
Distributive Law
A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C )
A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C )
Assignment
Page 78 – 1, 2, 3, 4, 5, 6, 12,
13, 18, 25, 29, 33,
39, 47, 48, 49
Section 4 – Applications of Sets
Example
There are 100 state senators discussing tuition for
the state universities. 70 favor raising tuition, 50
favor raising taxes to support education, and 30
favor raising tuition and raising taxes.
Applications of Sets
Let T = set of senators who want to raise tuition
X = set of senators who want to raise taxes
T
40
X
30
20
10
Applications of Sets
Example
In a survey of
75 college
students, it
was found
Number
23
18
14
10
9
8
5
Magazine
Read Time
Read Newsweek
Read US News
Read Time & Newsweek
Read Time & USNWR
Read Newsweek &
USNWR
Read all three
Applications of Sets
Time
Newsweek
9
5
5
5
3
4
2
U
US News
42
Applications of Sets
Example
Old McDonald
surveyed his
farm with the
following
results:
Quantity
9
2
26
37
18
6
5
7
Description
Fat red roosters
Fat red hens
Fat roosters
Fat chickens
Thin brown roosters
Thin red roosters
Thin red hens
Thin brown hens
Applications of Sets
Fat
Male
9
17
18
9
6
2
5
U
Red
7
The Inclusion – Exclusion
Principle
Let’s return to the example about the voting of
the state senators on tuition.
T
40
X
30
20
10
The Inclusion – Exclusion
Principle
Describe that portion of the diagram containing 40.
T
40
X
30
20
10
n( T ∩ X ′ ) = 40
The Inclusion – Exclusion
Principle
Describe that portion of the diagram containing 30.
T
40
X
30
20
10
n( T ∩ X ) = 30
The Inclusion – Exclusion
Principle
Describe that portion of the diagram containing 20.
T
40
X
30
20
10
n( T ′ ∩ X ) = 20
The Inclusion – Exclusion
Principle
Describe that portion of the diagram containing 10.
T
40
X
30
20
10
′
n ( T ∪ X )  = 10


or n( T ′ ∩ X ′ ) = 10
The Inclusion – Exclusion
Principle
T
40
X
30
20
10
( 40 + 30) + ( 30 +
20 ) − 30 = 90
70 + 50 − 30 = 90
n( T ) + n( X ) − n ( T ∩ X ) = n ( T ∪ X )
The Inclusion – Exclusion
Principle
If we write our result as
n( T ∪ X ) = n( T ) + n ( X ) − n ( T ∩ X )
we have the Inclusion – Exclusion Principle
Note:
If T and X are disjoint, then
n( T ∪ X ) = n ( T ) + n ( X )
The Inclusion – Exclusion
Principle
Problem
A survey of 1500 commuters in New York City
showed that 1140 take the subway, 680 take
the bus, and 120 do not take either.
How many commuters take both the bus and
subway?
How many commuters take only the subway?
The Inclusion – Exclusion
Principle
Place 120 in its proper location in the diagram.
Subway
Bus
120
′
n ( S ∪ B )  = 120


The Inclusion – Exclusion
Principle
Since
( S ∪ B ) ∪ ( S ∪ B )′
=U
′

n ( S ∪ B ) + n ( S ∪ B )  = n ( U )


n( S ∪ B ) + 120 = 1500
n( S ∪ B ) = 1380
The Inclusion – Exclusion
Principle
By the Inclusion – Exclusion Principle
n( S ∪ B ) = n ( S ) + n ( B ) − n( S ∩ B )
1380 = 1140 + 680 − n( S ∩ B )
n( S ∩ B ) = 1140 + 680 − 1380
n( S ∩ B ) = 440
Therefore, the number of people that ride
both the subway and bus is 440.
The Inclusion – Exclusion
Principle
Now, to determine the number that ride the
subway only:
Subway
Bus
440
120
n( S ∩ B′ ) + n ( S ∩ B ) = n ( S )
The Inclusion – Exclusion
Principle
n( S ∩ B′ ) + n ( S ∩ B ) = n ( S )
n( S ∩ B′ ) + 440 = 1140
n( S ∩ B′ ) = 700
Therefore, the number that ride the subway
only is 700.
The Inclusion – Exclusion
Principle
Dwaine Tomlinson runs a basketball program in California. On the
first day of the season, 60 young men showed up and were
categorized by age level and preferred basketball position.
Guard
Forward
Center
Totals
Jr. High
9
6
4
19
Sr. High
12
5
9
26
College
5
8
2
15
Totals
26
19
15
60
The Inclusion – Exclusion
Principle
Guard
Forward
Center
Totals
Jr. High
9
6
4
19
Sr. High
12
5
9
26
College
5
8
2
15
Totals
26
19
15
60
n( J ∩ G ) = ?
n( J ∩ G ) = 9
The Inclusion – Exclusion
Principle
Guard
Forward
Center
Totals
Jr. High
9
6
4
19
Sr. High
12
5
9
26
College
5
8
2
15
Totals
26
19
15
60
n( N ∪ ( S ∩ F ) ) = ?
n( N ∪ ( S ∩ F ) ) = 4 + 9 + 2 + 5 = 20
The Inclusion – Exclusion
Principle
Guard
Forward
Center
Totals
Jr. High
9
6
4
19
Sr. High
12
5
9
26
College
5
8
2
15
Totals
26
19
15
60
n( S ′ ∩ ( G ∪ N ) ) = ?
n( S ′ ∩ ( G ∪ N ) ) = 9 + 4 + 5 + 2 = 20
The Inclusion – Exclusion
Principle
Guard
Forward
Center
Totals
Jr. High
9
6
4
19
Sr. High
12
5
9
26
College
5
8
2
15
Totals
26
19
15
60
n( N ′ ∩ ( S ′ ∩ C ′ ) ) = ?
n( N ′ ∩ ( S ′ ∩ C ′ ) ) = 9 + 6 = 15
Assignment
Page 89 – 1, 3, 5, 7, 9, 11,
13, 15, 17, 19, 21