Definition:Domain (Relation Theory)

Relation
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 always 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.

Mapping
The term domain is usually seen when the relation in question is actually a mapping.

In the context of mappings, the domain and the preimage of a mapping are the same set.

This definition is the same as that for the domain of a function.

Morphism
If $f : X \to Y$ is a morphism, then the domain of $f$ is defined to be the object $X$, often written $X = \operatorname{dom}f$ or $X = D(f)$.

Alternative terms
The domain of (usually) a mapping is called by some sources, for example, the departure set.

Others refer to it on occasion as the source, but this is not recommended as there are other uses for that term.

Also see

 * Codomain
 * Range


 * Image
 * Preimage