TITLE: Let $A, B$ be sets. Show that $\mathcal P(A ∩ B) = \mathcal P(A) ∩ \mathcal P(B)$.
QUESTION [5 upvotes]: Let $A, B$ be sets. Show that $\mathcal P(A \cap B) = \mathcal P(A) \cap \mathcal P(B)$.

I understand what this question is asking. The power set of an intersection equals the intersection of two power sets. I just have no idea how to prove it.

REPLY [3 votes]: $\mathcal{P}(A\cap B)$ is the set of all and only the sets which are at the same time subsets of $A$ and $B$. But by definition these are exactly the elements of $\mathcal{P}(A)\cap\mathcal{P}(B)$.