Definition:Preimage/Relation/Relation

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

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse relation to $\mathcal R$, defined as:


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

The preimage of $\mathcal R \subseteq S \times T$ is:


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

Also known as
Some sources, for example, call this the domain of $\mathcal R$.

However, this term is discouraged, as it is also seen used to mean the entire set $S$, including elements of that set which have no images.

Also see

 * Preimage of a Mapping


 * Domain
 * Codomain
 * Range


 * Image