Definition:Inverse Image Mapping/Mapping/Definition 1

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

Also see

 * Equivalence of Definitions of Inverse Image Mapping of Mapping