\begin{definition}[Definition:Inverse Mapping/Definition 1]
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let $f^{-1} \subseteq T \times S$ be the inverse of $f$:
:$f^{-1} := \set {\tuple {t, s}: \map f s = t}$
Let $f^{-1}$ itself be a mapping:
:$\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$
and
:$\forall y \in T: \exists x \in S: \tuple {y, x} \in f$
Then $f^{-1}$ is called the '''inverse mapping of $f$'''.
\end{definition}