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.

Some sources, for example, call the domain the departure set.

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

Also see

 * Codomain
 * Range


 * Image
 * Preimage