Definition:Preimage/Mapping/Subset

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

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


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

Let $Y \subseteq T$.

The preimage of $Y$ under $f$ is defined as:


 * $f^{-1} \left ({Y}\right) := \left\{{s \in S: \exists y \in Y: f \left({s}\right) = y}\right\}$

That is, the preimage of $Y$ under $f$ is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.

If no element of $Y$ has a preimage, then $f^{-1} \left ({Y}\right) = \varnothing$.

Also known as
As well as using the notation $\operatorname{Im}^{-1} \left ({f}\right)$ to denote the preimage of an entire mapping, the symbol $\operatorname{Im}^{-1}$ can also be used as follows:

For $Y \subseteq T$, we have:
 * $\operatorname{Im}^{-1}_f \left ({Y}\right) := f^{-1} \left ({Y}\right)$

but this notation is clumsy and generally not preferred.

Also see

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


 * Definition:Image of Subset under Mapping


 * Definition:Preimage of Relation

Clearly:
 * $\displaystyle f^{-1} \left ({Y}\right) = \bigcup_{y \mathop \in Y} f^{-1} \left({y}\right)$

... the union of the preimages of each of the elements of $Y$.