# Set Operations - Union, Intersection, Complement

In these lessons we shall learn about union of sets.

A set is a well defined group of objects or symbols. The objects or symbols are called elements of the set. We will look at the following set operations: Union, Intersection and Complement.

The following figures give the set operations and Venn Diagrams for complement, subset, intersect and union. Scroll down the page for more examples and solutions. Intersection of Sets, Union of Sets and Venn Diagrams

This video gives an introduction into the intersection of sets and union of sets as it relates to Venn diagrams. It explains how to find the intersection of two sets as well as the union of two sets. This video contains plenty of examples and practice problems on intersection and union of sets.

Intersection of Sets

Example:
Set A = {1, 4, 6, 8}
Set B = {0, 2, 4, 8, 9}
U = {the digits}
Draw a Venn Diagram for A ∩ B

Complement of a Set
Learn what a complement of a set is.

Example:
Set A = {0, 1, 4, 5, 6, 7, 8}
U = {the digits}
Draw a Venn Diagram for A'

Boolean Set Operations
Intersection, union and complement set operations defined

Example:
Universe = {1,2,3,4,5,6,7,8,9,10}
A = {2,4,6,8,10}
B = {6,7,8,9,10}

Sets - Intersection, Union and Complement
A ∩ B pronounced as A intersection B are members that are common to both set A and set B.
A ∪ B pronounced as A union B are members that are in set A or set B or both.
A’ pronounced as A complement are members that are not in set A.

Example:
If U = {1,2,3,4,5,6,7,8,9,10}
A = {1,2,3,4,5,6}
B = {1,2,3,5,7}
C = {2,4} and D = {8,9}

Find
A ∩ B
A ∪ B
A’
A ∩ C
A ∪ C
B ∩ D
B ∪ D
B’
A ∩ B ∩ C
(A ∪ B)'

Set Operations and Venn Diagrams - Part 1 of 2
A Venn diagram is a visual diagram that shows the relationship of sets with one another.
The set of all elements being considered is called the universal set (U) and is represented by a rectangle.
The complement of A, A’, is the set of elements in U but not in A. A’ = {x|x ∈ U and x = ∉ A}
Set A and B are disjoint because they do not share any common elements.
B is a proper subset of A. This means B is a subset of A, but B ≠ A.
The intersection A and B is the set of elements in both set A and set B.
The union of A and B is the set of elements in set A or set B.
Intersection and Unions with the Empty Set
A ∩ ∅ = ∅
A ∪ ∅ = A

Set Operations and Venn Diagrams - Part 2 of 2

Examples:

1. Create a Venn diagram to show the relationship among the sets.
U is the set of whole numbers from 1 to 15.
A is the set of multiples of 3.
B is the set of primes.
C is the set of odd numbers

2. Given the following Venn diagram, determine each of the following sets.
a. A ∩ B.
b. A ∪ B
c. (A ∪ B)’
d. A’ ∩ B
e. A ∪ B'

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