# Math Help - Power Set Proofs

1. ## Power Set Proofs

2. Originally Posted by rookie23

Maybe you can use the distributive laws .

we can prove this by showing that $\mathcal{P}(A \cap B) \subseteq \mathcal{P}(A) \cap \mathcal{P}(B)$ and $\mathcal{P}(A) \cap \mathcal{P}(B) \subseteq \mathcal{P}(A \cap B)$
For the first, assume $X \in \mathcal{P}(A \cap B)$. then $X \subseteq (A \cap B)$, but that means $X \subseteq A$ and $X \subseteq B$, and thus we have $X \in \mathcal{P}(A)$ and $X \in \mathcal{P}(B)$ respectively. and so $X \in \mathcal{P}(A) \cap \mathcal{P}(B)$