Related Topics: More Lessons on Sets

**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?

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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 * ∩ *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} = { }

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'

Example:

A = {3,7,-5,0,13}

B = {0,17,3,Blue,☆}

C = {Pink,☆,3,17}

Find A\(A∩(B\C)')∪(B∩C)

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?

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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