\begin{definition}[Definition:Preimage/Relation/Element]
Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as:
:$\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
Every $s \in S$ such that $\tuple {s, t} \in \RR$ is called '''a preimage of $t$'''.
In some contexts, it is not individual elements that are important, but ''all'' elements of $S$ which are of interest.
Thus '''the preimage of $t \in T$''' is defined as:
:$\map {\RR^{-1} } t := \set {s \in S: \tuple {s, t} \in \RR}$
This can also be written:
:$\map {\RR^{-1} } t := \set {s \in \Img {\RR^{-1} }: \tuple {t, s} \in \RR^{-1} }$
That is, '''the preimage of $t$ under $\RR$''' is the image of $t$ under $\RR^{-1}$.
\end{definition}