# 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, 8, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23}

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.