1.

For any two sets A and B, prove that A ∪ (B – A) = A ∪ B

Answer»

Let x ϵ A ∪ (B –A) ⇒ x ϵ A or x ϵ (B – A) 

⇒ x ϵ A or x ϵ B and x ∉ A 

⇒ x ϵ B 

⇒ x ϵ (A ∪ B) [∵ B ⊂ (A ∪ B)] 

This is true for all x ϵ A ∪ (B–A) 

∴ A∪(B–A)⊂(A∪B)……(1) 

Conversely, 

Let x ϵ (A ∪ B) ⇒ x ϵ A or x ϵ B 

⇒ x ϵ A or x ϵ (B–A) [∵ B ⊂ (A ∪ B)] 

⇒ x ϵ A ∪ (B–A) 

∴ (A∪B)⊂ A∪(B–A)……(2) 

From 1 and 2 we get…

 A ∪ (B – A) = A ∪ B



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