# Difference of Sets using Venn Diagram

How to find the difference of sets using Venn diagram?

The difference of two subsets A and B is a subset of U, denoted by A – B and is defined by

A – B = {x : x ∈ A and x ∉ B}.

Let A and B be two sets. The difference of A and B, written as A - B, is the set of all those elements of A which do not belongs to B.

Thus A – B = {x : x ∈ A and x ∉ B} or A – B = {x ∈ A : x ∉ B}.

Clearly, x ∈ A – B

⇒ x ∈ A and x ∉ B

In the adjoining figure the shaded part represents A – B.

Similarly, the difference B – A is the set of all those elements of B that do not belongs to A.

Thus, B – A = {x : x ∈ A and x ∉ B} or A – B = {x ∈ B : x ∉ A}.

In the adjoining figure the shaded part represents B – A.

In particular, A – B = ∅ if A ⊂ B and A – B = A if A ∩ B = ∅.

The subset of A – B is also called the complement of B relative to A.

The difference A – B can be expressed in terms of the complement as A – b = A ∩ B’.

Properties of difference of sets:

1. A – (B ∩ C) = (A – B) ∪ (A – C)

2. A – (B ∪ C) = (A – B) ∩ (A – C)

Solved example to find the difference of sets using Venn diagram:

1. If A = {2, 3, 4, 5, 6, 7} and B = {3, 5, 7, 9, 11, 13}, then find (i) A – B and (ii) B – A.

Solution:

According to the given statement; A = {2, 3, 4, 5, 6, 7} and B = {3, 5, 7, 9, 11, 13}

(i) A – B

= {2, 4, 6}

(ii) B – A

= {9, 11, 13}

2. Given three sets A, B and C such that: A = {x : x is a natural number between 10 and 16}, B = {set of even numbers between 8 and 20} and C = {7, 9, 11, 14, 18, 20}.

Find the difference of sets using Venn diagram:

(i) A – B

(ii) B – C

(iii) C – A

(iv) B – A

Solution:

According to the given statement

A = {11, 12, 13, 14, 15}

B = {10, 12, 14, 16, 18}

C = {7, 9, 11, 14, 18, 20}

(i) A – B

= {Those elements of set A which are not in set B}

= {11, 13, 15}

(ii) B – C

= {Those elements of set B which are not in set C}

= {10, 12, 16}

(iii) C – A

= {Those elements of set C which are not in set A}

= {7, 9, 18, 20}

(iv) B – A

= {Those elements of set B which are not in set A}

= {10, 16, 18}

Set Theory