Prove the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, 𝒫(A ∩ B) = 𝒫(A) ∩ 𝒫(B).

Question

Question 1: Proof that 𝒫(A ∩ B) ⊆ 𝒫(A) ∩ 𝒫(B).

Question 2: Proof that 𝒫(A) ∩ 𝒫(B) ⊆ 𝒫(A ∩ B).

Answer 1

Let X be an element of P(A intersection B). By definition it means that X is a subset of (A intersection B). Hence, X is a subset of A and X is a subset of B. Hence, X is an element of P(A) and X is an element of P(B). Hence, X belongs to (P(A) intersection P(B)).

Suppose that X belongs to (P(A) intersection P(B)). Then X belongs to P(A) and X belongs to P(B) or in other words X is a subset of A and X is a subset of B and hence, X is a subset of (A intersection B) and hence, X belongs to P(A intersection B).

The proof is complete.