Definition:Inverse Image Mapping/Relation/Definition 2

Definition
Let $S$ and $T$ be sets.

Let $\powerset S$ and $\powerset T$ be their power sets.

Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

The inverse image mapping of $\mathcal R$ is the direct image mapping of the inverse $\mathcal R^{-1}$ of $\mathcal R$:
 * $\mathcal R^\gets = \paren {\mathcal R^{-1} }^\to: \powerset T \to \powerset S$

That is:
 * $\forall Y \in \powerset T: \map {\mathcal R^\gets} Y = \set {s \in S: \exists t \in Y: \tuple {t, s} \in \mathcal R^{-1} }$

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Relation