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Section 2.2: Complements, Subsets, and Venn Diagrams
Practice HW from Mathematical Excursions Textbook (not to hand in)
p. 71 # 1-49 odd
In the section, we continue with our discussion of basic set concepts.
The Universal Set and Compliment of a Set
The universal set is the set U of all possible elements being considered for a particular
problem.
Example 1: Find an appropriate universal set for the elements Virginia, Texas, Virginia,
and Ohio
Solution:
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Definition: The compliment of set A is the set of all elements in the universal set U that are
not in the set A.
Notation for Compliment: Given a set A, the compliment is given by A .
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Example 2: Find the compliment of the following sets given that the universal set is
U  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} .
a. {0, 1, 4, 5, 6}
b. {x | x  6 and x  W }
c. { } or 
d. U
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Subsets
If every element of a set A is also an element of a set B, we say A is a subset of the set B.
Notation: The notation A  B denotes that the set A is a subset of the set B.
The notation A  B denotes that the set A is not a subset of the set B.
Facts
1. Every set is a subset of itself, that is, given a set A, then A  A .
2. The empty set  is a subset of every set, that is, for any set A,   A .
Example 3: For the following sets, insert  or  in the blank space between the sets to
make a true statement.
a. {1, 2, 3, 4, 5}
{1, 2, 3, 4, 5, 6, 7, 8}
b. {red, white, yellow}
{the colors in the American flag}
c. {baseball, football}
{all sports}
d. the set of rational numbers
the set of integers
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Proper Subsets
A set A is a proper subset of a set B, denoted by A  B , if every element of A is an
element of B, and A  B .
For example, the set {baseball, football} is a proper subset of the set {baseball, football, basketball }
and we write {baseball, football}  {baseball, football, basketball } .
However, the set {USA, Mexico, Canada} is not a subset of the set {Mexico, Canada, USA}
since the sets are equal. However, we can write
{USA, Mexico, Canada}  {Mexico, Canada, USA}
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Example 4: Given the sets U  {a, e, i, o, u} , W  {a, i, o, u} , X  {a, u} , and Y  {e} ,
determine whether the following is true or false.
a. X  W
b. X  W
c. Y   W
d. Y  U
e.   Y
Listing the Subsets of a Set
Fact: A subset containing n elements contains 2 n subsets.
Example 5: List the number and all of the subsets of the set {dog, cat}
Solution:
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Example 6: List the number and all of the subsets of the set {1, 3, 5, 7}
Solution:
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