# Complement of a Set

Before studying about the Complement of a set, let us understand what are sets?

## Sets Definition

A well-defined collection of objects or elements is known as a set. Any set consisting of all the objects or elements related to a particular context is defined as a universal set. It is represented by $U$.
For any set A which is a subset of the universal set $U$, the complement of the set $A$ consists of those elements which are the members or elements of the universal set $U$ but not of the set $A$. The complement of any set $A$ is denoted by $A'$.

## Complement of a Set Definition

If $U$ is a universal set and $A$ be any subset of $U$ then the complement of $A$ is the set of all members of the universal set $U$ which are not the elements of $A$.

$A'$ = {${x ~:~ x~ ∈ ~U ~and ~x ~∉ ~A}$}

Alternatively it can be said that the difference of the universal set $U$ and the subset $A$ gives us the complement of set $A$.

## Venn Diagram for the Complement of a set

The Venn diagram to represent the complement of a set A is given by:

### Complement of a Set Examples

To make it more clear consider a universal set $U$ of all natural numbers less than or equal to 20.

Let the set $A$ which is a subset of $U$ be defined as the set which consists of all the prime numbers.

Thus we can see that $A$ = {${2, 3, 5, 7, 11, 13, 17, 19}$}

Now the complement of this set A consists of all those elements which is present in the universal set but not in $A$. Therefore, $A'$ is given by:

$A'$={${1,4,6,8,9,10,12,14,15,16,18,20}$}

Example: Let $U$ be the universal set which consists of all the integers greater than 5 but less than or equal to 25. Let $A$ and $B$ be the subsets of $U$ defined as:

$A$ = {${x~:x~ ∈~U ~and~ x~ is~ a~ perfect~ square}$}

$B$ = ${7, 9, 16, 18, 24}$

Find the complement of sets A and B and the intersection of both the complemented sets.

Solution: The universal set is defined as:

$U$ = {${6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}$}

Also, $A$ = {${9,16,25}$} and

$B$ = {${7,9,16,18,24}$}

The complement of set A is defined as:

$A'$ = {${x~:~x~∈~U~ and ~x~∉~A}$}

Therefore, $A'$ = {${6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24}$}

Similarly the complement of set B can be given by:

$B'$ = {${6,8,10,11,12,13,14,15,17,19,20,21,22,23,25}$}

The intersection of both the complemented sets is given by $A’∩ B'$.

$\Rightarrow A’∩ B'$= {${6,7,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25}$}

We can see from the above discussions that if a set $A$ is a subset of the universal set $U$ then the complement of set $A$ i.e. $A'$ is also a subset of $U$..

Now we are clear on the concept of the complement of sets. There is a lot more to learn. Enrich your knowledge and reach new horizons of success by downloading BYJU’S – The Learning App to know more or visit our website.