Power Set Explanation The following is from the Wikipedia page on the Power set:

By identifying a function in $2^S$ with the corresponding preimage of $1$, we see that there is a bijection between $2^S$ and $\mathcal P(S)$, where each function is the characteristic function of the subset in $\mathcal P(S)$ with which it is identified.

I don't understand the start of the sentence,"By identifying a function in $2^S$ with the corresponding preimage of $1$".  I can make out some sense out of the rest.  Please help.

It means that given a function $f\colon S\to\{0,1\}$, we can define $A_f=\{s\in S\mid f(s)=1\}=f^{-1}(1)$, and that the function $f\mapsto A_f$ is a bijection.
