Characteristic Function of Preimage

Theorem

Let $X$ and $Y$ be sets.

Let $f : X \to Y$ be a function.

Let $A \subseteq Y$.

Then:

$\chi_{f^{-1} \sqbrk A} = \chi_A \circ f$

where $\chi_{f^{-1} \sqbrk A}$ denotes the characteristic function of $f^{-1} \sqbrk A$.

Proof

We show that if $x \in f^{-1} \sqbrk A$, then:

$\map {\paren {\chi_A \circ f} } x = 1$

and if $x \in X \setminus f^{-1} \sqbrk A$, then:

$\map {\paren {\chi_A \circ f} } x = 0$

Let $x \in f^{-1} \sqbrk A$.

Then $\map f x \in A$, and so:

$\map {\paren {\chi_A \circ f} } x = \map {\chi_A} {\map f x} = 1$

Now let $x \in X \setminus f^{-1} \sqbrk A$.

Then $\map f x \in Y \setminus A$, and so:

$\map {\paren {\chi_A \circ f} } x = \map {\chi_A} {\map f x} = 0$

$\blacksquare$