(4) Given a set $S$, its power set is the set $\mathcal{P}(S) = \{U | U \subset S\}$. In other words, $\mathcal{P}(S)$ is the set of all subsets of $S$.
(a) Show that $S \subset T$ if and only if $\mathcal{P}(S) \subset \mathcal{P}(T)$.
(b) Prove or provide a counterexample: For any sets $A$, $B$, $\mathcal{P}(A \cap B) \subset \mathcal{P}(A) \cap \mathcal{P}(B)$.
(c) Prove or provide a counterexample: For any sets $A$, $B$, $\mathcal{P}(A) \cap \mathcal{P}(B) \subset \mathcal{P}(A \cap B)$.
(d) Prove or provide a counterexample: For any sets $A$, $B$, $\mathcal{P}(A \cup B) \subset \mathcal{P}(A) \cup \mathcal{P}(B)$.
(e) Prove or provide a counterexample: For any sets $A$, $B$, $\mathcal{P}(A) \cup \mathcal{P}(B) \subset \mathcal{P}(A \cup B)$.