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

 * Definition:Preimage of Mapping


 * Definition:Domain of Relation
 * Definition:Codomain of Relation
 * Definition:Range


 * Definition:Image of Relation