# Representing set difference using Venn diagram

In this chapter we will learn to represent set difference operation using Venn diagram.

Let’s first review the basics of set difference operation.

## What is Difference of set ?

Let there are two sets A & B.

The difference of set A – B will result in elimination of common elements of set A & B from set A.

For example;
Let A = { 3, 6, 9, 12, 15 }
B = { 8, 9, 10, 11, 12 }

Note that between set A & B, the numbers 9 & 12 are common elements.

So in A – B operation, the common elements 9 & 12 will be removed from set A.

Hence, A – B = {3, 6, 15}

### Representation of difference of sets

The difference between two sets are represented by symbol ” – “

If there are two sets A & B then there are two possibilities of set difference;

(a) A – B
In this operation, the common elements will be removed from set A.

(b) B – A
In this operation, the common elements will be removed from set B.

## Using Venn diagram in set difference

To learn basics of Venn diagram in set theory, click the red link.

The set difference operation can be presented graphically with the help of Venn diagram.

### Representing A – B in Venn diagram

Consider the two sets A & B, the difference of set A – B is shown by below image.

In the above image;

⟹ rectangular box represent universal set

⟹ Circle A & B represent set A & B respectively

⟹ The area in blue color represent the set difference A – B.

Important Points:
(a) In operation A – B, we have removed common elements of A & B from set A.

(b) The set A – B only contain unique elements of A.

### Representing B – A in Venn diagram

The Venn diagram representation of B – A is shown below;

In the above image;

⟹ The green area represent the difference of set B – A

⟹ Here we are subtracting common elements of set A & B from set B.

## Examples of Set difference with Venn diagram

(01) Given below are sets A & B. Represent A – B and B – A using Venn diagram.

A = { 1, 3, 5, 7, 9, 11 }
B = { 2, 3, 5, 7, 8 }

Solution
In set A & B, the common elements are 3, 5, & 7

Solving for A – B
In this operation we will remove common elements A & B from set A.

A – B = { 1, 9, 11 }

Venn diagram representation of A – B.

The area colored in grey represents A – B.

Solving for B – A

Here we will remove common elements of A & B from set B.

B – A = { 2 , 8 }

Venn diagram representation;

The green area represent B – A

(02) Represent P – Q in Venn diagram.

P = { Monday, Tuesday, Wednesday }
Q ={ Wednesday, Thursday, Friday }

Solution
In both set P & Q, the term ” Wednesday ” is the common element.

Solving for P – Q

In operation P – Q, we will eliminate common elements of P & Q from set P.

P – Q = { Monday, Tuesday }

(03) Given below are set A, B & C.

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

Find the following operation;
(a) A – B
(b) B – A
(c) B – C
(d) C – A

Solution

(a) Solving for A – B

A – B = { 2, 3 }

Yellow area represent operation A – B.

(b) Solving for B – A

B – A = { 5, 6 }

The pink area represent set B – A.

(c) Solving for B – C

B – C = { 5, 6 }

The grey area represent set B – C.

(d) Solving for C – A

Here number 2 & 4 are common between set C & A

C – A = { 7 }

The green area represent the set C – A.