Can anyone please help me prove P(AnB) = P(A) n P(B)
for power sets?!
we can prove this by showing that $\displaystyle \mathcal{P}(A \cap B) \subseteq \mathcal{P}(A) \cap \mathcal{P}(B)$ and $\displaystyle \mathcal{P}(A) \cap \mathcal{P}(B) \subseteq \mathcal{P}(A \cap B)$
For the first, assume $\displaystyle X \in \mathcal{P}(A \cap B)$. then $\displaystyle X \subseteq (A \cap B)$, but that means $\displaystyle X \subseteq A$ and $\displaystyle X \subseteq B$, and thus we have $\displaystyle X \in \mathcal{P}(A)$ and $\displaystyle X \in \mathcal{P}(B)$ respectively. and so $\displaystyle X \in \mathcal{P}(A) \cap \mathcal{P}(B)$
i leave the other direction to you