Is it true that for any sets A and B,

False.

Let $X \in P(A) \cup P(B)$

$\Rightarrow X \in P(A)$ or $X \in P(B)$

$\Rightarrow X \subset A$ or $X \subset B$

$\Rightarrow X \subset(A \cup B)$

$\Rightarrow X \in P(A \cap B)$

$\therefore P(A) \cup P(B) \subset P(A \cup B)$   ...(1)

Again, let $X \in P(A \cup B)$

But $X \notin P(A)$ or $x \notin P(B)$

[For example let $A=\{2,5\}$ and $B=\{1,3,4\}$ and take $X=\{1,2,3,4\}$ ]

So, $X \notin P(A) \cup P(B)$

Thus, $P(A \cup B)$ is not necessarily a subset of $P(A) \cup P(B)$.