Inverse Function

Definition 1. Given any injection $f$, its inverse relation $f^{-1}$ is also a function, called its inverse function.

Proposition 1. For any injection $f$ from a set $X$ onto a set $Y$:

1. \[f \circ f^{-1} = \id_X.\]
2. \[f^{-1} \circ f = \id_Y.\]

Proposition 2. Let $A$ and $B$ be finite sets. For any surjection $f : A \to B$,

\[\lvert A \rvert = \sum_{b \in B} \big\lvert f^{-1}[\{ b \}] \big\rvert.\]

Proposition 3. Let $A$, $B$ be finite sets and $f : A \to B$ be a surjection. If for all $b \in B$,

\[\big\lvert f^{-1}[\{ b \}] \big\rvert = k,\]

then

\[\lvert A \rvert = k \lvert B \rvert.\]