Chapter 2
The Basic
Concepts of
Set Theory
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All rights reserved
Chapter 2: The Basic Concepts of
Set Theory
2.1
2.2
2.3
2.4
2.5
Symbols and Terminology
Venn Diagrams and Subsets
Set Operations and Cartesian Products
Surveys and Cardinal Numbers
Infinite Sets and Their Cardinalities
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2-3-2
Chapter 1
Section 2-3
Set Operations and Cartesian Products
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2-3-3
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
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2-3-4
Intersection of Sets
The intersection of sets A and B, written A B,
is the set of elements common to both A and
B, or
A B  {x | x  A and x  B}.
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Example: Intersection of Sets
Find each intersection.
a) {1,3,5, 7,9} {1, 2,3, 4,5, 6}
b) {2, 4, 6} 
Solution
a) {1, 3, 5}
b) 
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2-3-6
Union of Sets
The union of sets A and B, written A B,
is the set of elements belonging to either of
the sets, or
A B  {x | x  A and x  B}.
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Example: Union of Sets
Find each union.
a) {1,3,5, 7,9} {1, 2,3, 4,5, 6}
b) {2, 4, 6} 
Solution
a) {1, 2,3, 4,5, 6, 7,9}
b) {2, 4, 6}
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2-3-8
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
A  B  {x | x  A and x  B}.
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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) A  B
b)  B  A C
Solution
a) {a, b, h}
b) {g} {b, f, h} = {b, f, g, h}
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2-3-10
Ordered Pairs
In the ordered pair (a, b), a is called the
first component and b is called the second
component. In general (a, b)  (b, a ).
Two ordered pairs are equal provided that
their first components are equal and their
second components are equal.
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2-3-11
Cartesian Product of Sets
The Cartesian product of sets A and B,
written, A  B, is
A  B  {(a, b) | a  A and b  B}.
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Example: Finding Cartesian Products
Let A = {a, b}, B = {1, 2, 3}
Find each set.
a) A  B
b) B  B
Solution
a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3),
(3, 1), (3, 2), (3, 3)}
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2-3-13
Cardinal Number of a Cartesian Product
If n(A) = a and n(B) = b, then
n  A  B   n( B  A)
 n( A)  n( B)  n( B)  n( A)
 ab  ba
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Example: Finding Cardinal Numbers of
Cartesian Products
If n(A) = 12 and n(B) = 7, then find
n  A  B  and n  B  A .
Solution
n  A  B   n( B  A)
 n( A)  n( B)  n( B)  n( A)
 7 12  84
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2-3-15
Venn Diagrams of Set Operations
A B
A
A B
B
A
U
U
A
A
U
B
A B
A
A
B
U
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Example: Shading Venn Diagrams to
Represent Sets
Draw a Venn Diagram to represent the set
A B.
Solution
A
B
U
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2-3-17
Example: Shading Venn Diagrams to
Represent Sets
Draw a Venn Diagram to represent the set
 A
B C.
Solution
B
A
U
C
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2-3-18
De Morgan’s Laws
For any sets A and B,




A
B

A
B
and
A
B



  A B.
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2-3-19