Understanding this theorem and how to go about proving it surjectivity?

Question

I don't follow understand what this theorem is saying: "Every function $f: A \to{\cal P}(A)$ is not surjective."

Is this saying so for example let A = {1,2,3} then the power set of A is {},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3} Is this saying that there is no surjective function that takes every element in each of its power sets to atleast 1 element in A?

Is my understanding correct?

Answer 1

No, that's not what it's saying.

It's saying that there is no surjective function taking elements of $A$ to subsets of $A$. That is, if $f$ is any function that takes elements of $A$ and returns subsets of $A$, it can't return all subsets of $A$.

Answer 2

Considering a map $f : A \to P(A)$ from a set to its power set. What do you think about the subsets of all elements which are not in their image $\{x \in A | x \notin f(x)\}$ ? For example, give it a name and suppose that some element $y\in A$ is sent to it, then see what happens.