Definition:Domain (Relation Theory)

This page is about Domain in the context of Set Theory. For other uses, see Domain.

Definition

Let $\RR \subseteq S \times T$ be a relation.

The domain of $\RR$ is defined and denoted as:

$\Dom \RR := \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$

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

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 $S$, and can be denoted $\Dom f$.

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

The domain of $\circ$ is the set $S$ and can be denoted $\Dom \circ$.