The question is to show that $ \mathcal{P}(A)$~$ \mathcal{P}(B) $ and $\mathcal{F}(A,C)$~$\mathcal{F}(B,D)$ where A,B,C, and D are sets, $A$~$B$, $C$~$D$, and $\mathcal{F}(A,C)$ and $\mathcal{F}(B,D)$ are the sets of all functions for their respective sets.
I don't exactly know how to prove the two power sets are equal in cardinality when the sets themselves are equal in cardinality. It seems intuitive, but I don't know how to really explain it.
The set of all functions question, I don't even know really how to begin
Equal cardinality means there is a bijection. Take bijections between A and B, and between C and D, and use those to construct bijections of the power sets and the function sets.