Let $A$ be any set . Let $\wp(A)$ be the power set of $A$. Then which of the following are true
1) $\wp(A) = \emptyset$ for some $A$
2) $\wp(A) $ is a finite set for some $A$
3) $\wp(A)$ is a countable set for some $A$.
4) $\wp(A)$ is uncountable for some set $A$
for 1) take $A= \emptyset$, then $\wp(A) = \emptyset$.
for 2) take $A$ is a finite set, then $\wp(A)$ is a finite set.
for 3) If $A$ is finite then $\wp(A)$ is a finite set, if $A$ is infinite , then $\wp(A)$ is uncountable set. I think this is false.
for 4) Take $A = \mathbb N$, then $\wp(A)$ is uncountable.
This is an exam problem . I would be thankful if someone check my problem, if feel any error then correct.
For (1), $\wp(\emptyset)$ is not equal to $\emptyset$, because $\wp(\emptyset)=\{\emptyset\}$. The set $\{\emptyset\}$ is not an empty set, because the set $\{\emptyset\}$ contains one element, and that element is $\emptyset$.
For (2), you are correct, but you need to prove that is the case.
For (3) again, you have the right idea, but you need to prove it. To prove it, I suggest you prove that if $A$ is infinite, it contains a countable subset, and the power set of $A$ must contain the power set of the countable subset.
For (4), you are correct, and if you have proven in lectures that $\wp(a)$ is uncountable, this is enough. Otherwise, you need to prove it as well.