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, for example, 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.

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