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

Question

Philoid · Accepted Answer

Let x ϵ A ∪ (B – A)

⇒ x ϵ A or x ϵ (B–A)

⇒ X ϵ A or x ϵ B or x ∉ A

⇒ x ϵ A or x ϵ B

⇒ x ϵ (A ∪ B)

∴ A ∪ (B – A) ⊂ (A ∪ B)…….(1)

Let and x ϵ (A ∪ B)

⇒ x ϵ A or x ϵ B

⇒ x ϵ A or x ϵ B and x ∉ A

⇒ x ϵ A or x ϵ B–A

⇒ x ϵ A ∪ (B–A)

∴ (A ∪ B) ⊂A ∪ (B – A)…….(2)

From (1) and(2), we get

A ∪ (B – A) = A ∪ B.

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