Definition:Preimage/Mapping/Mapping

Definition
Let $f: S \to T$ be a mapping.

The preimage of $f$ is defined as:


 * $\operatorname{Im}^{-1} \left ({f}\right) := \left\{{s \in S: \exists t \in T: f \left({s}\right) = t}\right\}$

That is:
 * $\operatorname{Im}^{-1} \left ({f}\right) := f^{-1} \left [{T}\right]$

where $f^{-1} \left [{T}\right]$ 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.

Also known as
Consistently with the definition as the image of $T$ under $f$, $f^{-1} \left [{T}\right]$ can also be used instead of $\operatorname{Im}^{-1} \left ({f}\right)$.

Also see

 * Definition:Image of Mapping
 * Definition:Domain of Mapping
 * Definition:Codomain of Mapping
 * Definition:Range of Mapping

Generalizations

 * Definition:Preimage of Relation