\begin{definition}[Definition:Preimage/Mapping/Element]
Let $S$ and $T$ be sets
Let $f: S \to T$ be a mapping.
Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:
:$f^{-1} = \set {\tuple {t, s}: \map f s = t}$
Every $s \in S$ such that $\map f s = t$ is called '''a preimage''' of $t$.
'''The preimage''' of an element $t \in T$ is defined as:
:$\map {f^{-1} } t := \set {s \in S: \map f s = t}$
This can also be expressed as:
:$\map {f^{-1} } t := \set {s \in \Img {f^{-1} }: \tuple {t, s} \in f^{-1} }$
That is, the '''preimage of $t$ under $f$''' is the image of $t$ under $f^{-1}$.
\end{definition}