Set Theory: Combined Operations

Related Pages
Set Notation
Intersection of 2 Sets
Union of Sets
Venn Diagrams And Subsets
More Lessons on Sets

Combined operations involve the intersection, union and complement of sets. Perform the operations within parenthesis first. Other operations are performed from left to right.

Example:
Given that U = {x : 1 ≤ x ≤ 10, x is an integer},

G = {x : x is a prime number},

H = {x : x is an even number},

P = {1, 2, 3, 4, 5}.

List the elements of:

a) G ∩ H ∪ P

b) (G ∩ P) ’ ∪ H

c) H ’ ∩ (G ∪* P* )

d) (P ∪ H ∪ G) ’ ∩ (G ∩ H)

Solution:
G = {2, 3, 5, 7}, H = {2, 4, 6, 8, 10}

a) G ∩ H ∪ P = {2} ∪ P ← G ∩ H = {2}

= {1, 2, 3, 4, 5}

b) (G ∩ P) ’ ∪ H = {1, 4, 6, 7, 8, 9, 10} ∪ H

= {1, 2, 4, 6, 7, 8, 9, 10}

c) H ’ ∩ (G ∪ P ) = H ’ ∩ {1, 2, 3, 4, 5, 7}

= {1, 3, 5, 7}

d) (P ∪ H ∪ G) ’ ∩ (G ∩ H) = {9} ∩ (G ∩ H)

= {9} ∩ {2} = { }

Set operations : Intersection, union and complement
Example:
Universe {1,2,3,…8,9,10}
A = {2,4,6,8,10}
B = {6,7,8,9,10}
Find
A ∩ B
A ∪ B
A'
A ∩ B'3

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Bringing the set operations together
Example:
A = {3,7,-5,0,13}
B = {0,17,3,Blue,☆}
C = {Pink,☆,3,17}
Find A\(A∩(B\C)')∪(B∩C)

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Set Operations

1. Find the intersection of two sets
   The intersection of sets A and B, written A ∩ B, is the set of elements common to both set A and set B. The definition can be expressed in set-builder notation as follows:
   A ∩ B = {x|x ∈ A and x ∈ B}

Examples:
Find each of the following intersections:
a. {7,8,9,10,11} ∩ {6,8,10,12}
b. {1,3,5,7,9} ∩ {2,4,6,8}
c. {1,3,5,7,9} ∩ ∅

2. Find the union of two sets
   The union of sets A and B, written A ∪ B, is the set of elements that are members of set A or set B or both sets. The definition can be expressed in set-builder notation as follows:
   A ∪ B = {x|x ∈ A or x ∈ B}

Examples:
Find each of the following unions:
a. {7,8,9,10,11} ∪ {6,8,10,12}
b. {1,3,5,7,9} ∪ {2,4,6,8}
c. {1,3,5,7,9} ∪ ∅

3. Perform operations with sets

Examples:
Given:
U = {1,2,3,4,5,6,7,8,9,10}
A = {1,2,3,7,9}
B = {3,7,8,10}
find
a. (A ∪ B)'
b. A' ∩ B'

4. Determine sets involving set operations from a Venn Diagram
   Use the diagram to determine each of the following sets:
   a. A ∪ B
   b. (A ∪ B)'
   c. A' ∩ B
   d. A ∪ B'

5. Understand the meaning of “and” and “or”

6. Use the formula for n(A ∪ B)

Example:
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 are willing to do both.
How many students were willing to donate blood or serve breakfast?

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