# Set Theory: Union Of Sets

In these lessons, we will learn the union of sets and the complement of the union of sets.

Related Topics:

More Lessons on Sets
## Union of Sets

The

**union** of two sets

*A* and

*B* is the set of elements, which are in

*A*** or** in

*B*** or** in both. It is denoted by

*A* ∪

*B* and is read ‘

*A* union

*B*’

**Example : **

Given U* =* {1, 2, 3, 4, 5, 6, 7, 8, 10}

* X* = {1, 2, 6, 7} and *Y* = {1, 3, 4, 5, 8}

Find * X* ∪ *Y * and draw a Venn diagram to illustrate *X* ∪ *Y*.

**Solution: **

* X* ∪ *Y* = {1, 2, 3, 4, 5, 6, 7, 8} ← 1 is written only once.

**If ***X* ⊂ *Y* then *X* ∪ *Y* = *Y.* We will illustrate this relationship in the following example.

**Example:**

Given U* =* {1, 2, 3, 4, 5, 6, 7, 8, 10}

* X* = {1, 6, 9} and *Y* = {1, 3, 5, 6, 8, 9}

Find * X* ∪ *Y * and draw a Venn diagram to illustrate *X* ∪ *Y*.

**Solution: **

* X* ∪* Y* = {1, 3, 5, 6, 8, 9}

## Complement of the Union of Sets

The **complement of the set ***X*** ∪***Y* is the set of elements that are members of the universal set U but are not in* X* ∪*Y*. It is denoted by (*X* ∪ *Y* ) ’

**Example: **

Given: U* =* {1, 2, 3, 4, 5, 6, 7, 8, 9}

* X* = {1, 2, 6, 7} and *Y* = {1, 3, 4, 5, 8}

a) Draw a Venn diagram to illustrate (* X* ∪ *Y* ) ’

b) Find (* X* ∪ *Y * ) ’

**Solution: **

a) First, fill in the elements for *X* ∩ *Y* = {1}

Fill in the other elements for *X* and *Y* and for U

Shade the region outside *X* ∪ *Y* to indicate (*X* ∪ *Y* ) ’

b) We can see from the Venn diagram that

(*X* ∪ *Y* ) ’ = {9}

Or we find that *X* ∪ *Y* = {1, 2, 3, 4, 5, 6, 7, 8} and so

(*X* ∪ *Y* ) ’ = {9}

**Example: **

Given U = {*x* : 1 ≤ *x *≤10, *x* is an integer}, *A* = The set of odd numbers, *B* = The set of factors of 24 and *C* = {3, 10}.

a) Draw a Venn diagram to show the relationship.

b) Using the Venn diagram or otherwise, find:

i) (*A* ∪ *B* ) ’ ii) (*A* ∪ *C* ) ’ iii) (*A* ∪ *B* ∪ *C* ) ’

**Solution: **

* A = * {1, 3, 5, 7, 9}, * B* = {1, 2, 3, 4, 6, 8} and * C* = {3, 10}

a) First, fill in the elements for * A* ∩ *B* ∩*C* = {3}, *A* ∩ *B * {1, 3},

* A* ∩ *C * = {3}, *B* ∩ *C* = {3} and then the other elements.

b) We can see from the Venn diagram that

i) (*A* ∪ *B* ) ’ = {10}

ii) (*A* ∪ *C* ) ’* = * {2, 4, 6, 8}

iii) (*A* ∪ *B * ∪ *C* ) ’ = { }

## Videos

Sets: Union and Intersection.

A mathematics lesson on set operation of union.

Union of Sets.

The following video describes the Union and Intersection of Sets.

Union, Intersection and Complement.

Venn Diagrams: Shading Regions. This video shows how to shade the union, intersection and complement of two sets.