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

Mathematics Asked by TheCreator on January 7, 2021

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

2 Answers

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

Correct answer by Emmanuel C. on January 7, 2021

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

Answered by Lee Mosher on January 7, 2021

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