Answer
Prove: (A − (A ∩ B)) ∩ (B − (A ∩ B))
Proof:
(A − (A ∩ B)) ∩ (B − (A ∩ B)) = (A ⋂ (A ∩ B)c) ∩ (B ⋂ (A ∩ B)c) (by definition of set difference)
= (A ⋂ (Ac ⋃ Bc)) ∩ (B ⋂ (Ac ⋃ Bc)) (by DeMorgan’s law)
= ((A ⋂ Ac) ⋃ (A ⋂ Bc)) ∩ ((B ⋂ Ac) ⋃ (B ⋂ Bc)) (by Distributive law)
= (∅ ⋃ (A ⋂ Bc)) ∩ ((B ⋂ Ac) ⋃ ∅) (by Complement law)
= (A ⋂ Bc) ∩ (B ⋂ Ac) (by Identity law)
= (A ⋂ Ac) ∩ (B ⋂ Bc) (by Associative law)
= ∅ ⋂ ∅ (by Complement law)
= ∅ (by Idempotent law)
Work Step by Step
(A − (A ∩ B)) ∩ (B − (A ∩ B)) = (A ⋂ (A ∩ B)c) ∩ (B ⋂ (A ∩ B)c) (by definition of set difference)
= (A ⋂ (Ac ⋃ Bc)) ∩ (B ⋂ (Ac ⋃ Bc)) (by DeMorgan’s law)
= ((A ⋂ Ac) ⋃ (A ⋂ Bc)) ∩ ((B ⋂ Ac) ⋃ (B ⋂ Bc)) (by Distributive law)
= (∅ ⋃ (A ⋂ Bc)) ∩ ((B ⋂ Ac) ⋃ ∅) (by Complement law)
= (A ⋂ Bc) ∩ (B ⋂ Ac) (by Identity law)
= (A ⋂ Ac) ∩ (B ⋂ Bc) (by Associative law)
= ∅ ⋂ ∅ (by Complement law)
= ∅ (by Idempotent law)
