Use the identities A = A ∩ S and S = B ∪ B and a distributive law to prove that:

1) A = (A ∩ B) ∪ (A ∩ B).

2) If B ⊂ A then A = B ∪ (A ∩ B).

Given that

A = A ∩ S - - (1)

S = B ∪ B - - (2)

To prove:

A = (A ∩ B) ∪ (A ∩ B)

(A ∩ B) ∪ (A ∩ B)

= [ (A∪A) ∩ (A∪B) ] ∩ [ (B∪A) ∩ (B∪B) ]

=[A ∩ (A∪B) ] ∩ [ (A∪B) ∩ S]

=A ∩ (A∪B)

=A

Hence proved.

2) If B ⊂ A then A = B ∪ (A ∩ B)

R. H. S = B ∪ (A ∩ B)

= (B ∪ A) ∩ (B∪B) - - (3)

As B is subset of A so

(B ∪ A) = A

From (2)

(B ∪ B) = S

(3) becomes

=A ∩ S

from (1)

A ∩ S = A

Hence proved