Definition:Inverse Image Mapping/Mapping/Definition 2

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

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

Let $f: S \to T$ be a mapping.

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

That is:
 * $\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Mapping