Definition:Domain (Set Theory)

From ProofWiki

Jump to: navigation, search

This page is about the concept of domain in set theory. For other uses, see Definition:Domain.

Definition

Let $\mathcal R \subseteq S \times T$ be a relation.

The domain (sometimes seen as domain of definition) of $\mathcal R$ is defined as:

$\operatorname{Dom} \left({\mathcal R}\right) = \left\{{s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R}\right\}$

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

That is, it is the same as what is defined here as the preimage of $\mathcal R$.

Some sources define the domain of $\mathcal R$ as the whole of the set $S$.

Using this definition, $s \in \operatorname{Dom} \left({\mathcal R}\right)$ whether or not $\exists t \in T: \left({s, t}\right) \in \mathcal R$.

Most texts do not define the domain in the context of a relation, 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 doesn't actually matter that much.

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

Let $f: S \to T$ be a mapping.

The domain of $f$ is the set $S$ and can be denoted $\operatorname{Dom} \left({f}\right)$.

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

Let $\circ: S \times S \to T$ be a binary operation.

The domain of $\circ$ is the set $S$ and can be denoted $\operatorname{Dom} \left({\circ}\right)$.