Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
Venn diagrams can be a bit awkward when dealing with power sets. You could look at a couple of small examples, but in this case your best approach may be simply to try to prove it, and either succeed or see where you run into difficulties; the latter often gives a clue towards finding a counterexample.
So try to show first that $\wp(A\cap B)\subseteq\wp(A)\cap\wp(B)$. Suppose that $X\in\wp(A\cap B)$; then $X\subseteq A\cap B$. Therefore ...
If you can complete that argument successfully, you can try showing the opposite inclusion.