Show that for any sets A and B,
A = ( A ∩ B ) ∪ ( A – B ) and A ∪ ( B – A ) = ( A ∪ B )

Explanation:

To prove: A = (A ∩ B) υ (A – B)

Let x ∈ A

Case I

x ∈ A ∩ B

Then, x ∈ (A ∩ B) ⊂ (A υ B) υ (A – B)

Case II

x ∉ (A ∩ B)

Then, x ∉ A or x ∉ B

x ∉ (A – B) ⊂ (A υ B) υ (A – B)

A ⊂ (A ∩ B) υ (A – B) .....(1)

It is clear that

A ∩ B ⊂ A and (A – B) Ì A

(A ∩ B) υ ( A – B) ⊂ A ....(2)

From (1) and (2), we obtain

A = (A ∩ B) υ (A – B)

To prove: A υ (B - A) = (A υ B)

Let x ∈ A υ (B – A)

x ∈ A or x ∈ (B – A)

x ∈ A or (x ∈ B and x ∉ A)

(x ∈ A or x ∈ B ) and (x ∈ A or x ∉ A)

x ∈ ( A υ B)

A υ (B – A) ⊂ (A υ B) ....(3)

Next, we show that (A υ B) ⊂ A υ (B – A).

Let y ∈ (A υ B)

y ∈ A or y ∈ B

(y ∈ A or y ∈ B) and (y ∈ A or y ∉ A)

y ∈ A or (y ∈ B and y ∉ A)

y ∈ A υ (B – A)

A υ B ⊂ A υ (B – A) ....(4)

Hence, from (3) and (4), we obtain

A υ (B - A) = (A υ B) 

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NCERT Solutions Class 11 Maths Chapter 1 Exercise ME Question 8

Show that for any sets A and B, A = (A ∩ B) υ (A – B) and A υ (B - A) = (A υ B)

Two sets are given. We have proved that A υ (B - A) = (A υ B)