elementary set theory - Help understanding Cantor's Theorem

I am having trouble understanding the proof of Cantor's Theorem:

https://proofwiki.org/wiki/Cantor%27s_Theorem

http://www.whitman.edu/mathematics/higher_math_online/section04.10.html

The part that confuses me is this set

$$A = \{ x \in S \mid x \not \in f(x) \}$$

Can someone explain what this is saying in plain English. I am reading it as the set of elements $x$ that are in $S$ but not in $P(S)$, or that are in $S$ but cannot be mapped to $P(S)$.

My thinking is that every element in $S$ is a subset of $P(S)$, by the definition of power set. Every element in $S$ CAN be mapped to $P(S)$, so that set $A$ must be the empty set? Why not?

Can someone help clarify this?

Answer

The point is that $f$ takes an element of $S$ and returns a subset of $S$. And now we define $A$ to be the set of those which are not elements of the sets they are mapped to.

What is $A$ exactly? Well that depends greatly on the set $S$ and the function $f$. Let's consider two examples.

The first, take $f(x)=\{x\}$, then for every $x\in S$ it is true that $x\in f(x)$. Therefore $A=\varnothing$. But it is true that there is no $x$ such that $f(x)=\varnothing$!

In the second example, take $f(x)=S\setminus\{x\}$, then for every $x\in S$ it is true that $x\notin f(x)$. Therefore $A=S$. But in this case it is true that there is no $x$ such that $f(x)=S$!

In both examples we see that $A$, which depends very much on the function $f$, is not in its range. And that's the point of the theorem. If $f\colon S\to\mathcal P(S)$, then $f$ is not surjective.