Definition:Domain of Relation/Also defined as

Definition
It is usual, as here, to define the domain of $\RR \subseteq S \times T$ as the subset of $S$ that bears some element of $S$ to $T$.

However, it appears to make sense to define it to be the whole of the set $S$.

Using this definition, $s \in \Dom \RR$ whether or not $\exists t \in T: \tuple {s, t} \in \RR$.

It would then be possible to refer to $\set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$ as the preimage of $\RR$.

This appears never to be done in the literature that has so far been studied as source work for.

Most texts do not define the domain in the context of a relation in the first place, 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 appears to be of small importance.