# Difference of Sets

## What is Difference of sets?

If A & B are given two sets, then the difference of set A & B will results in:

⟹ removal of common element between A & B from set A

⟹ only the unique element of set A will remain after subtraction.

For Example
Given below are sets A & B; Do the subtraction A – B.

In the operation the main set A is subtracted with set B.
To find the subtraction we have to remove the common elements of A & B from set A.

Note that elements 7 & 15 are the common elements between A & B.

The subtraction of sets A – B will;

⟹ remove common elements 7 & 15 from set A

⟹ only the unique elements of set A will be left.

Hence, the elements in set A – B are;

Conclusion
In difference of sets, the common elements are removed from the main set.

## Representing Set Difference

The difference between two sets in represented by symbol ” – “.

Hence, the difference of two sets A & B is represented as:

It means that set A is subtracted with set B.
In this operation we have to subtract the common elements of A & B from main set A.

Similarly the subtraction of set B with A is expressed as;

In this operation we have to remove common elements of A & B from main set B.

### Representing set difference through Venn diagram

Venn diagram is useful for graphical representation of different sets.

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

Consider two sets A & B with following elements;

A = { 1, 5, 9, 15, 17 }
B = { 5, 10, 15, 20 }

Given below is the Venn diagram for subtraction of set A – B.

The area covered in green represents the set A – B.

Note that the green area only contain the element which is unique to set A.

The common elements ( i.e. 5 & 15 ) have been removed from the difference operation.

Hence, A – B = { 1, 9, 17 }

## Solved Questions on Set Difference

(01) Given below are two sets A & B

Find A – B and B – A

Solution

Finding A – B

Here we have to subtract common elements of A & B from set A.

Note that elements 13, 15 & 17 are the common elements in set A & B.

A – B = { 19, 22, 25 }

Finding B – A
Subtract common elements of B & A from set B.

Again elements 13, 15 & 17 are the common elements in A & B.

B – A = { 5, 8, 20 }

(02) Given below are two sets A, B and C

Find;
( i ) A – B
( ii ) B – A
( iii ) C – A
( iv ) A – C
( v ) B – C

Solution
Let us first write all the sets in Roster form.

A = { x : x is greater than 15 and less than 25 }

Roster form of set A is;
A = { 16, 17, 18, 19, 20, 21, 22, 23, 24 }

B = { x : x is greater than 10 & less than 17 }

Roster form of set B is;
B = { 11, 12, 13, 14, 15, 16}

Hence, all the three sets in Roster form are;
A = { 16, 17, 18, 19, 20, 21, 22, 23, 24 }
B = { 11, 12, 13, 14, 15, 16}
C = { 11, 14, 17, 19, 23}

( i ) A – B
Number 16 is the only common element between A & B.

Removing the common element between A & B from set A we get;
A – B = { 17, 18, 19, 20, 21, 22, 23, 24}

( ii ) B – A
Again 16 is the common element between A & B.

Removing 16 from set B we get;
B – A = { 11, 12, 13, 14, 15}

( iii ) C – A
Number 17, 19 & 23 are the common element between set A & C.

Removing 17, 19 & 23 from set C we get;
C – A = { 11, 14 }

( iv ) A – C
Again number 17, 19 & 23 are the common elements between set A & C.

Removing common elements from set A.

A – C = { 16, 18, 20, 21, 22, 24 }

( v ) B – C
Number 11 & 14 are the common elements between set A & C.

Remove common elements from set B & C.

B – C = { 12, 13, 15, 16 }

(03) Given below are two sets A & B.
A = { Orange, Banana, Apple, Cherry }
B = { Mango, Banana, Strawberry, Guava, Cherry }

Find A – B and B – A

Solution
(i) A – B
Here Banana and Cherry are the common elements between set A & B.

Remove the common elements from set A , we get;
A – B = { Orange, Apple }

(ii) B – A
Remove the above mentioned common elements from set B, we get;

B – A = { Mango, Strawberry, Guava }

(04) Given below are two sets A & B

Find A – B

Solution
Let us first write the set A in Roster form

A = { x : x lies between 6 & 10 }

Set A can also be written as;
A = { 7, 8, 9 }

Hence, the two given sets in roster form are;
A = { 7, 8, 9 }
B = { 𝜙 }

Note that there is no common element between set A & B.
So no element will be removed from set A.

Hence, A – B = { 7, 8, 9 }