Definition:Domain (Relation Theory)/Relation

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

The domain (sometimes seen as domain of definition) of $\mathcal R$ is defined as:
 * $\operatorname{Dom} \left({\mathcal R}\right) := \left\{{s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R}\right\}$

and can be denoted $\operatorname{Dom} \left({\mathcal R}\right)$.

That is, it is the same as what is defined here as the preimage of $\mathcal R$.

Also defined as
Some sources define the domain of $\mathcal R$ as the whole of the set $S$.

Using this definition, $s \in \operatorname{Dom} \left({\mathcal R}\right)$ whether or not $\exists t \in T: \left({s, t}\right) \in \mathcal R$.

Most texts do not define the domain in the context of a relation, so this question does not often arise.

Even if it does, the domain is often such that either it coincides with $S$ or that it doesn't actually matter that much.

Also see

 * Definition:Codomain of Relation
 * Definition:Range


 * Definition:Image of Relation
 * Definition:Preimage of Relation