Inverse the mapping from a subset $B$ of $X$ to its characteristic function $\chi_B: X \to \{0,1\}$

Question

I am trying to find that the the power set P(X) of a set X is isomorphic to the set of maps from X to {0,1} That map should be given by the mapping of a subset $B$ of $X$ to its characteristic function $\chi_B: X \to \{0,1\}$

How would the inverse of this mapping be?

Answer 1

The inverse is the function that maps each function $f:X\to\{0,1\}$ to $f^{-1}(1)$.