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

Many sources define the domain as:
 * $\operatorname{Dom} \left({\mathcal R}\right) = \left\{{s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R}\right\}$

that is, what is defined here as the preimage of $\mathcal R$.

This is the approach taken by:

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

Even if it does, the domain and preimage are often such that either they coincide or that it doesn't actually matter that much.

Also see

 * Codomain
 * Range


 * Image
 * Preimage