\begin{definition}[Definition:Preimage/Mapping/Mapping]
Let $f: S \to T$ be a mapping.
The '''preimage of $f$''' is defined as:
:$\Preimg f := \set {s \in S: \exists t \in T: f \paren s = t}$
That is:
:$\Preimg f := f^{-1} \sqbrk T$
where $f^{-1} \sqbrk T$ is the image of $T$ under $f^{-1}$.
In this context, $f^{-1} \subseteq T \times S$ is the the inverse of $f$.
It is a relation but not necessarily itself a mapping.
\end{definition}