Definition:Preimage

Let $$\mathcal{R} \subseteq S \times T$$ be a relation.

$$\mathcal{R}$$ can also be (and usually is in this context) a mapping or function.

Preimage of an Element
Every $$s \in \operatorname{Dom} \left({\mathcal{R}}\right)$$ such that $$t \in \mathcal{R} \left ({s}\right)$$ is called a preimage of $$t$$.

In some contexts, it is not individual elements that are important, but all elements of $$\operatorname{Dom} \left({\mathcal{R}}\right)$$ which are of interest.

Thus the preimage of an element $$t \in \operatorname{Rng} \left({\mathcal{R}}\right)$$ is defined as:


 * $$\mathcal{R}^{-1} \left ({t}\right) = \left\{{s \in \mathrm {Dom} \left({\mathcal{R}}\right): s \mathcal{R} t}\right\}$$

The preimage is therefore also known as the inverse image.

The preimage of $$t \in \operatorname{Rng} \left({\mathcal{R}}\right)$$ is also known as the fiber of $$t$$.

Note that:
 * $$t \in \operatorname{Im} \left ({\mathcal{R}}\right)$$ may have more than one preimage.
 * It is possible for $$t \in \operatorname{Rng} \left({\mathcal{R}}\right)$$ to have no preimages at all, in which case $$\mathcal{R}^{-1} \left ({t}\right) = \varnothing$$.

Preimage of a Subset
The preimage of $$Y \subset \operatorname{Rng} \left({\mathcal{R}}\right)$$ is:

$$\mathcal{R}^{-1} \left ({Y}\right) = \left\{{s \in \operatorname{Dom} \left({\mathcal{R}}\right): \exists y \in Y: s \mathcal{R} y}\right\}$$

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

Preimage of a Relation
The preimage of a relation $$\mathcal{R}$$ is:

$$\operatorname{Im}^{-1} \left ({\mathcal{R}}\right) = \mathcal{R}^{-1} \left ({\operatorname{Rng} \left({\mathcal{R}}\right)}\right) = \left\{{s \in \operatorname{Dom} \left({\mathcal{R}}\right): \exists t \in \operatorname{Rng} \left({\mathcal{R}}\right): \left({s, t}\right) \in \mathcal{R}}\right\}$$