TITLE: How to show equinumerosity of  the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic?
QUESTION [9 upvotes]: How to show equinumerosity of  the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic?
Not homework, practice exercise.

REPLY [3 votes]: Arturo Magidin has given an answer in symbols. In words it might be something like:
For any subset $S$ of $A$ (i.e. element of the powerset) there is a unique function which sends each element of $S$ to $1$ and everything else in $A$ to $0$; conversely, for any function $f$ from $A$ to $\{ 0,1 \}$ there is a unique subset of $A$ which contains all the elements of $A$ sent to $1$ and none of those sent to $0$.