Can someone find a (two-sided) inverse function to the characteristic function χ? given...

Question

so i have given the folowing info:

In general, if $A$ is a set, for $S \in \mathcal P(A)$ define $\chi(S) : A \to \{0,1\}$ by $$\chi(S)(a) = \begin{cases} 1 & \textrm{if } a \in S, \\ 0 & \textrm{if } a \notin S. \end{cases}$$ Then $\chi : \mathcal P(A) \to 2^A$ is bijective.

But I would like to know the what the inverse is to $\chi$. I'm quite new with the definition "characteristic function", so I don't even now how to start. I normally calculate the inverse by replacing x with y and vice versa, but know I don't have a clue. Please help me.

Thank you in advance

Answer 1

If you have an $f\in 2^A$, then $f$ is a function from $A$ to $2=\{0,1\}$. So you need to associate $f$ with a subset $F\subseteq A$. As the comment above said, take the inverse image of 1 under $f$, that is, $F=\{a\in A : f(a)=1\}$. So, you have a function from $2^A$ to $\mathcal P (A)$.

Answer 2

Given $f \in 2^A$, define $\chi^{-1}(f) = \{a \in A \mid f(a)=1\} \in \mathcal P(A)$.