Definition:Preimage of Subset under Mapping/Definition 2

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

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:


 * $f^{-1} = \set {\tuple {t, s}: \map f s = t}$

Let $Y \subseteq T$.

The preimage of $Y$ under $f$ can be seen to be an element of the codomain of the inverse image mapping $f^\gets: \powerset T \to \powerset S$ of $f$:


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

Thus:
 * $\forall Y \subseteq T: f^{-1} \sqbrk Y = \map {f^\gets} Y$

If no element of $Y$ has a preimage, then $f^{-1} \sqbrk Y = \O$.