Definition:Preimage/Mapping/Mapping

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

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:


 * $f^{-1} = \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

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 $f$.

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

 * Domain
 * Codomain
 * Range


 * Image


 * Preimage of a Relation