Math 166H-copyright Joe Kahlig, 12C
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Section 1.1: Introduction to Sets
Definition: A set is a well defined collection of objects. The objects in a set are called the elements
of the set.
roster notation
set-builder notation
Definition: The empty set is the set that doesn’t have any elements, denoted by φ or {}. The
universal set is the set that contains all of the elements for a problem, denoted by U .
Definition: Subset: J ⊆ K if for all x ∈ J then x ∈ K
Definition: Proper Subset: J ⊂ K if for all x ∈ J then x ∈ K and J 6= K
Example: If J is any set, is φ ⊆ J?
Example: Give all the subsets for these sets.
A) A = {1, 2}
B) B = {a, b, c}
Example: How many subsets does the set K have? How many proper subsets doe K have?
K = {a, b, c, d, e, f, g, h}
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Math 166H-copyright Joe Kahlig, 12C
Set Quiz
Circle the correct answer.
A = {0, 1, 2, ....., 9}
B = {1, 2, 3, 5, 8}
E = {1, 2, 5}
F = {φ, 7, 8}
C = {2, 3, 5, 7, 8}
D = {1, {2, 5}}
G = {1, 2, {1, 2}}
True
False
2, 7 ∈ A
True
False
1⊆D
True
False
φ∈A
True
False
C⊆B
True
False
{2, 5} ∈ D
True
False
φ∈F
True
False
{1, 2, 3} ∈ A
True
False
{2, 5} ⊆ D
True
False
φ=0
True
False
5∈D
True
False
{1, 2} ⊆ G
True
False
φ = {φ}
Set Operations
• Intersection: A ∩ B = {x|x ∈ A and x ∈ B}
• Union: A ∪ B = {x|x ∈ A or x ∈ B}
• Compliment: AC = A′ = {x|x ∈ U and x 6∈ A}
Example: Use these sets to answer the following.
U = {0, 1, 2, ..., 9}
A = {0, 2, 4, 6, 8}
C = {0, 3, 4, 5, 7}
D = {4, 5, 6, 7, 8, 9}
A) B ∩ D =
B) E ∩ F ∩ A =
B = {1, 3, 5, 7, 9}
E = {0, 6, 7, 9}
F = {1, 3, 8, 9}
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Math 166H-copyright Joe Kahlig, 12C
U = {0, 1, 2, ..., 9}
A = {0, 2, 4, 6, 8}
C = {0, 3, 4, 5, 7}
D = {4, 5, 6, 7, 8, 9}
B = {1, 3, 5, 7, 9}
E = {0, 6, 7, 9}
C) B ∪ E =
D) A ∪ B =
E) F c =
F) (D ∪ E)c ∩ (C ∪ F ) =
Example: Use the given sets to express the set operations in words.
Let U denote all students at Texas A& M
C = {x ∈ U |x lives in College Station }
A) B C ∩ S
B) C C ∪ S
B = {x ∈ U |x lives in Bryan}
S = {x ∈ U |x is a sophomore}
F = {1, 3, 8, 9}
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Math 166H-copyright Joe Kahlig, 12C
Venn Diagrams
A Venn Diagram is a pictorial way of representing a set.
Example: Shade the parts of a Venn diagram that represents these set operations.
A) A ∩ B
B) (A ∪ B) ∩ C
C) A ∩ (B ∪ C c )
Example: Write an expression both with set operations and with words that describes the shaded
region of the Venn Diagram.
Use the given sets to express the set operations in words.
Let U denote all students at Texas A& M
B = {x ∈ U |x plays basketball }
T
T = {x ∈ U |x plays tennis}
G = {x ∈ U |x plays golf}
B
G
Definition: Sets A and B are said to be disjoint if
T
B
G