Definition:Inverse Image Mapping/Relation/Definition 1

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 mapping $\mathcal R^\gets: \powerset T \to \powerset S$ that sends a subset $X \subseteq T$ to its preimage $\mathcal R^{-1} \paren X$ under $\mathcal R$.

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Relation