\begin{definition}[Definition:Inverse of Mapping]
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping
The '''inverse''' of $f$ is its inverse relation, defined as:
:$f^{-1} := \set {\tuple {t, s}: \map f s = t}$
That is:
:$f^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in f}$
That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies:
:$\forall s \in S: \forall t \in T: \tuple {t, s} \in f^{-1} \iff \tuple {s, t} \in f$
\end{definition}