# Complement of set

## What is Set Complement?

If A is the given set and U is the universal set then complement of set A will result in elements which is not in A.

Hence, set complement of A contain all the elements of Universal set except elements in A.

For Example
Given below are universal set (U) and set A.

The complement of set will contain all element of U except A.

Conclusion
The complement of any set contain all element except the one which is already present.

## Representing Set Complement

If A is the given set then there are three ways you can represent set complement.

⟹ Using symbol ” A’ “
We pronounce this as ” A dash “

⟹ Using complement power ” \mathtt{A^{c}}

⟹ Difference with Universal Set “U – A”

All the above expression represents the complement of set A.

### Representing set complement through Venn diagram

Venn diagram is useful for graphical representation of sets.

In Venn diagram, the sets are represented through circles and universal set is represented through rectangular box.

Given below is the universal set and set A.

U = { 7, 14, 21, 28, 35, 42, 49, 56 }
A = { 14, 28, 42 }

Given below is the Venn diagram for complement of set A.

The area covered in green represents the complement of set A.

Note that the green color contains elements of Universal set excluding the element of set A.

A’ = { 7, 21, 35, 49, 56 }

## Solved Questions on Set Complement

(01) Given below are universal set U, set A and Set B.
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
A = { 1, 2, 3, 4 }
B = { 2, 4, 6, 8}

Find;
(i) A’
(ii) B ‘
(iii) (A – B)’

Solution

(i) Find A’
To find A complement, write down all the element of U except A.

A’ = U – A

A’ = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } – { 1, 2, 3, 4 }

A’ = {5, 6, 7, 8, 9 }

(ii) Find B’
For B complement, get all the elements of U except elements of B

B’ = U – B

B’ = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } – { 2, 4, 6, 8 }

B’ = {1, 3, 5, 7, 9 }

(iii) (A – B)’

We have to find complement of A – B.

Let us first find A – B.
A – B = { 1, 2, 3, 4 } – {2, 4, 6, 8}

Here 2 & 4 are the common elements in A & B.
Eliminating elements 2 & 4 from set A we get.

A – B = { 1, 3 }

Now find the complement of A – B.

(A – B)’ = U – (A – B)

(A – B)’ = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } – {1, 3}

(A – B)’ = { 2, 4, 5, 6, 7, 8, 9 }

(02) Given below are Universal set U and Set A.

U = { Mango, Apple, Orange, Banana, Cherry, Guava }

A = { Mango, Cherry, Guava }

Find A’ and also draw Venn diagram.

Solution
To find A complement, we have to get all the elements which is not in set A.

A’ = U – A

A’ = { Apple, Orange, Banana }

The area colored in grey represent the set A’.

Note that the grey area contains all the elements not in set A.

(03) Given below are two sets A & B

Find the following sets;
(i) A’
(ii) B’
(iii) (A – B)’

Solution
Let us first write the sets in Roster form

A = { x : x is greater than 5 and less than 12 }
A = { 6, 7, 8, 9, 10, 11}

B = { x : x is prime number less than 13 }
B = { 2, 3, 5, 7, 11}

Universal Set U = { 1, 2, 3, 4, 5 . . . . . 15 }

(i) Complement of set A
For complement include all elements not in set A.

A’ = U – A
A’ = {1, 2, 3, 4, 5, 12, 13, 14, 15}

(ii) Complement of Set B
For B complement, include all elements not in set B.

B’ = U – B
B’ = { 1, 4, 6, 8, 9, 10, 12, 13, 14, 15 }

(iii) Complement of set A – B

First find the elements of A – B.
A – B can be found by removing the common element of A & B from set A.

A – B = { 6, 8, 9, 10}

Now find the complement of A – B by taking elements not present in set A-B.

(A – B)’ = {1, 2, 3, 4, 5, 7, 11, 12, 13, 14, 15 }

(04) Given below are Universal set and set A.

Find A’.

Solution
The set A is a null set which means that it doesn’t contain any element.

So the complement of set A contain all the element in universal set.

Hence, A’ = U = { x : x is a natural number starting from 1}

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