For any sets $A$ and $B$,show that $P(A \cap B) = P(A) \cap P(B).$

(N/A) Let $X \in P(A \cap B).$ Then $X \subset A \cap B.$
So,$X \subset A$ and $X \subset B.$
Therefore,$X \in P(A)$ and $X \in P(B),$ which implies $X \in P(A) \cap P(B).$
This gives $P(A \cap B) \subset P(A) \cap P(B).$
Let $Y \in P(A) \cap P(B).$ Then $Y \in P(A)$ and $Y \in P(B).$
So,$Y \subset A$ and $Y \subset B.$
Therefore,$Y \subset A \cap B,$ which implies $Y \in P(A \cap B).$
This gives $P(A) \cap P(B) \subset P(A \cap B).$
Hence,$P(A \cap B) = P(A) \cap P(B).$