Answer:
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