# Union of sets

## What is Union of Set?

If A & B are two given set, then the union of set will contain:
⟹ Unique elements present in set A
⟹ Unique elements present in set B
⟹ Common elements between A & B

For Example;
Given below are two sets A & B;

Then Union of set A & B will contain;
⟹ Unique elements in set A (i.e. 4, 13 & 17 )
⟹ Unique elements in set B (i.e. 15, 21 & 24 )
⟹ Common elements of A & B (i.e. 9 )

Hence, for A union B we get the following elements;

Conclusion
Basically in union operation we combine all the elements of the given sets.

## Representing Union of Set

The union between two set is represented by symbol ” ∪ “.

Hence, the union between set A & B can be expressed as:

Note that we can also do union operation in three or more sets.

Let A, B & C are the three sets, then the union is expressed as;

### Representing Union through Venn Diagram

We already know that Venn diagram is a representation of set in graphical form.

Here the sets are represented in form of circles and rectangle represents the universal set.

Now consider two sets A & B with following elements;
A = { 9, 15, 17, 20 }
B = { 10, 13, 17, 21 }

Below is the Venn diagram of union of sets A & B;

The union of set A & B consists of following elements:

⟹ Unique element in set A (i.e 9, 25, 20 )

⟹ Unique element in set B (i.e. 10, 13, 21 )

⟹ Common element in set A & B (i.e. 17)

Hence, in above Venn diagram, the union of set A & B is represented by:

⟹ Pink part of circle
⟹ Green part of circle
⟹ Middle portion representing number 17

## Union of three sets

The union operation is not limited to only two sets.

You can easily do union of three or more sets using the above mentioned process.

Let there are three sets A, B & C.
The union of above three sets will contain:

⟹ Unique elements in set A, B & C
⟹ Common elements in set AB, BC, CA and ABC

Note:
In simple language, for calculating union write down all the elements in the given sets but do not repeat the elements common among the given sets.

Example
Given below are set A, B & C.
A = { 2, 5, 7 }
B = { 1, 3 }
C = { 2, 7 }

Find A ∪ B ∪ C.

Solution
To find union of all sets, write down all the elements present in set A, B & C but do not repeat the common elements present among the given sets.

A ∪ B ∪ C = { 1, 2, 3, 5, 7 }

### Representing union of three sets in Venn diagram

If three sets are given A, B & C, then the union of the given set is represented as:

Here the enclosed pink area in the above image represents the union of three sets A, B and C.

## Solved Questions of Union of sets

(01) Given below are three sets A, B & C

A = {1, 2, 3, 4}
B = { 3, 4, 5, 6}
C = { 5, 6, 7, 8 }

Find:
(i) A ∪ B
(ii) B ∪ C
(iii) C ∪ A
(iv) A ∪ B ∪ C

Solution
(i) A ∪ B

To find A ∪ B in less time, write down all the elements of A & B without repeating any common element.

A ∪ B = { 1, 2, 3, 4, 5, 6 }

(ii) B ∪ C

Writing down all the elements of B & C without repeating common elements.

B ∪ C = { 3, 4, 5, 6, 7, 8 }

(iii) C ∪ A = {1, 2, 3, 4, 5, ,6 7, 8 }

(iv) A ∪ B ∪ C

There are two methods to find union of three sets.

First Method
(i) Write unique elements of all three sets
(ii) Now write common elements present in AB, BC, CA & ABC

Shortcut method
Write down all the elements present in set A, B & C without repeating common elements.

A ∪ B ∪ C = { 1, 2, 3, 4, 5, 6, 7, 8}

(02) Given below are following sets A & B.
A = { x : x is greater than 3 and less than 9 }
B = { 5, 7, 12, 15, 17}

Find A ∪ B

Solution
Set A can be written as;
A = { 4, 5, 6, 7, 8 }

Hence, we got two sets in roster form;
A = { 4, 5, 6, 7, 8 }
B = { 5, 7, 12, 15, 17}

A ∪ B = {4, 5, 6, 7, 8, 12, 15, 17 }

(03) Given below are two sets A & B

Find A ∪ B

Solution
To get union of two sets simply write all the elements without repeating the common ones.

(04) Given below are two sets A & B.

Find A ∪ B and represent it using Venn diagram.

Solution
Given below are the elements of A union B.

Given below is the representation of union in Venn diagram.

Example 05
Find the union of below sets A & B

Solution
Let us first express the sets in Roster form.

A = { x : x is even number & x > 2 }

Given above is the set name A.
Set A contains elements x such that;
⟹ x is an even number, and;
⟹ x is greater than 2.

So the roster form of set A is;
A = { 4, 6, 8, 10, 12, 14 . . . . .}

Given above is set B.
B = { x : x is odd number & x > 3 }

Set B contains element x such that;
⟹ x is an odd number, and;
⟹ x is greater than 3.

The roster form of set B can be written as:
B = { 5, 7, 9, 11, 13, . . . .}

Taking the union of set A & B we get;
A ∪ B = { 4, 5, 6, 7, 8, 9, 10, 11, 12. . . . }

The above set is the solution.