Question

Show that for any sets A and B, 

A = (A ∩ B) ∩ (A – B)

Answer 1

We know that (A ∩ B) ⊂ A and (A – B) ⊂ A 

⇒(A ∩ B) ∩ (A – B)⊂ A….(1) 

Let and x ϵ (A ∩ B) ∩ (A – B) 

⇒ x ϵ (A ∩ B) and x ϵ (A–B) 

⇒ x ϵ A and x ϵ B and x ϵ A and x ∉ B 

⇒ x ϵ A and x ϵ A [∵ x ϵ B and x ∉ B are not possible simultaneously] 

→ x ϵ A 

∴ (A ∩ B) ∩ (A – B)⊂ A…(2) 

From (1) and (2), we get 

A = (A ∩ B) ∩ (A – B)