\begin{definition}[Definition:Inverse Image Mapping/Relation/Definition 2]
Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $\RR \subseteq S \times T$ be a relation on $S \times T$.
The '''inverse image mapping''' of $\RR$ is the direct image mapping of the inverse $\RR^{-1}$ of $\RR$:
:$\RR^\gets = \paren {\RR^{-1} }^\to: \powerset T \to \powerset S$
That is:
:$\forall Y \in \powerset T: \map {\RR^\gets} Y = \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$
\end{definition}