\begin{definition}[Definition:Preimage/Relation/Subset]
Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:
:$\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
Let $Y \subseteq T$.
The '''preimage of $Y$ under $\RR$''' is defined as:
:$\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR}$
That is, the '''preimage of $Y$ under $\RR$''' is the image of $Y$ under $\RR^{-1}$:
:$\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$
If no element of $Y$ has a '''preimage''', then $\RR^{-1} \sqbrk Y = \O$.
\end{definition}