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

Question

Philoid · Accepted Answer

Let x ϵ A

Then either x ϵ (A–B) or x ϵ (A ∩ B)

⇒ x ϵ (A–B) ∪ (A ∩ B)

∴ A ⊂ (A – B) ∪ (A ∩ B)….(1)

Consverly,

Let x ϵ (A–B) ∪ (A ∩ B)

⇒ x ϵ (A–B) or x ϵ (A ∩ B)

⇒ x ϵ A and x ∉ B or x ϵ B

⇒ x ϵ A

∴ (A–B) ∪ (A ∩ B) ⊂ A……….(2)

∴ From (1) and (2), We get

(A–B) ∪ (A ∩ B) =A

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