Power set of any set. Question:

Let $A$ be any set. Let $\mathbb{P}(A)$ be the power set of $A$. Then which one is true?

*

*$\mathbb{P}(A)=\emptyset$ for some $A$.


*$\mathbb{P}(A)$ is finite for some $A$.


*$\mathbb{P}(A)$ is countable for some $A$.


*$\mathbb{P}(A)$ is uncountable for some $A$.

Now 1. is not true, since if $A=\emptyset$, $\mathbb{P}(A)=\{\emptyset\}\neq \emptyset$. 3. is not true since let $A=\mathbb{N}$, then $\mathbb{P}(A)=2^\mathbb{N}$ is uncountable. So answer will be 2 and 4.
But when I saw the answer book, it says answer is (2,4) or (2,3,4). That or is for answer is not decided yet.
My question is why is their doubt!!! The answer should be $(2,4)$.... this is obvious, isn't? I am very bad at set theoritic arguments. Can someone clarify it?

Assuming countable means "countably infinite" (so finite sets are not considered countable), 3 is false. If $A$ is finite, then $\mathbb{P}(A)$ is finite.
If $A$ is infinite it contains a countably infinite subset $B$ (mild form of choice used here). And then $2^{\mathbb{N}} = |\mathbb{P}(B)| \ge |\mathbb{P}(A)|$ so that $\mathbb{P}(A)$ is uncountable. So a power set cannot be countable.
