Power Set Proofs

**rookie23** · February 16th 2009, 03:25 PM

Can anyone please help me prove P(AnB) = P(A) n P(B)

**mathaddict** · February 16th 2009, 07:08 PM

Originally Posted by rookie23

Can anyone please help me prove P(AnB) = P(A) n P(B)

Maybe you can use the distributive laws .

**Jhevon** · February 16th 2009, 09:08 PM

Originally Posted by mathaddict

Maybe you can use the distributive laws .

for power sets?!

Originally Posted by rookie23

Can anyone please help me prove P(AnB) = P(A) n P(B)

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 $X \in \mathcal{P}(A) \cap \mathcal{P}(B)$

i leave the other direction to you

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