 # Set Theory: Combined Operations

Related Topics: More Lessons on Sets

Combined operations involve the intersection, union and complement of sets. Perform the operations within brackets 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 HP

b) (GP) ’ ∪ H

c) H ’ ∩ (G P )

d) (P H G) ’ ∩ (GH)

Solution:

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

a) G HP = {2} ∪ P G H = {2}

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

b) (GP) ’ ∪ 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) ’ ∩ (GH) = {9} ∩ (GH)

= {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 &cap B'

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)
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. Performs operations with sets
Example:
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 of th campus blood drive survey indicated that 490 students were willing to donate bllod, 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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