\begin{definition}[Definition:Preimage of Subset under Mapping/Definition 1]
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 $f$''' is defined as:
:$f^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \map f s = t}$
That is, the '''preimage of $Y$ under $f$''' is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.
If no element of $Y$ has a '''preimage''', then $f^{-1} \sqbrk Y = \O$.
\end{definition}