Prove union/intersect of two powersets is/is not equal to powerset of their union? I am supposed to show whether or not the union/intersection of two powersets is equal to the powerset of the union/intersection, respectively. 
I found these two links that have both the proofs, and say that the union is not always equal, but the intersection is always equal. The proofs seem very similar to me. Can someone help me understand why one shows they are always equal and the other is only sometimes?
https://proofwiki.org/wiki/Union_of_Power_Sets
https://proofwiki.org/wiki/Intersection_of_Power_Sets

The powerset of intersection is equal to the intersection of powersets, i.e. $P(A\cap B)=P(A)\cap P(B)$, can be proven as follow:
1) $P(A \cap B) \subseteq P(A) \cap P(B)$


*

*$\forall E \in P(A \cap B)$

*$[\forall x \in E(x \in A \cap B \equiv x \in A \land x \in B)]$

*$(E \subseteq A) \land (E \subseteq B)$

*$(E \in P(A)) \land (E \in P(B))$

*$E \in P(A) \cap P(B)$

*$\therefore P(A\cap B)\subseteq P(A) \cap P(B)$
2) $P(A) \cap P(B) \subseteq P(A \cap B)$


*

*$\forall E \in P(A) \cap P(B)$

*$[\forall x \in E (x \in A \land x \in B \implies x \in (A \cap B))]$

*$E \in P(A \cap B)$

*$\therefore P(A) \cap P(B) \subseteq P(A \cap B)$
Together, these give $P(A) \cap P(B) = P(A \cap B)$

To see that $P(A\cup B)\neq P(A)\cup P(B)$, simply consider the cardinality of each sides - $2^{|A|+|B|} = |P(A)\cup P(B)|\leq 2^{|A|}+2^{|B|}$ doesn't necessarily hold. 
One example would be when $A=\{1\},B=\{2\}$
$P(A\cup B)=\{\{\},\{1\},\{2\},\{1,2\}\}$
$P(A)\cup P(B)=\{\{\},\{1\},\{2\}\}$
