MHB Proof of Sets: Proving (i) and (ii)

[Mathick]

If $X$ is a set, then the power set $P(X)$ of a set is the set of all subsets of $X$.

I need to decide whether the following statements are true or false and prove it:
(i) If $Z = X \cup Y$ , then $P(Z) = P(X) \cup P(Y)$.

(ii) If $Z = X \cap Y$ , then $P(Z) = P(X) \cap P(Y)$.

By examples I think that (ii) is true but (i) is false. However, I have no idea how to prove it.

Note: Example which I used was: $X=\left\{a,b\right\}$ and $Y=\left\{b,c\right\}$.

(i) $Z = X \cup Y =\left\{a,b,c\right\}$ , so $P(Z) = \left\{\left\{\right\},\left\{a\right\},\left\{b\right\},\left\{c\right\},\left\{a,b\right\},\left\{a,c\right\},\left\{b,c\right\},\left\{a,b,c\right\}\right\}$ .

However, $P(X) \cup P(Y) = \left\{\left\{\right\},\left\{a\right\},\left\{b\right\},\left\{c\right\},\left\{a,b\right\},\left\{b,c\right\}\right\}$.

That's why I disagree with a statement (i).

(ii) $Z = X \cap Y =\left\{b\right\}$ , so $P(Z) = \left\{\left\{\right\},\left\{b\right\}\right\}$ . However, $P(X) \cap P(Y) = \left\{\left\{\right\},\left\{b\right\}\right\}$ as well.

That's why I think a statement (ii) is true.

I would be grateful for help!

 

[Evgeny.Makarov]

That's why I disagree with a statement (i).

You are absolutely correct.

If $Z=X\cap Y$, then $A\subseteq Z\iff A\subseteq X\land A\subseteq Y$ for any $A$, which means that $A\in P(Z)\iff A\in P(X)\land A\in P(Y)\iff A\in P(X)\cap P(Y)$.