Definition:Domain (Relation Theory)/Relation

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 $\RR$.

General Definition

Let $\ds \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$.

Let $\ds \RR \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\ds \prod_{i \mathop = 1}^n S_i$.

The domain of $\RR$ is the set defined as:

$\ds \Dom \RR := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \RR}$

The concept is usually encountered when $\RR$ is an endorelation on $S$:

$\ds \Dom \RR := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \RR}$

Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation in $V$.

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

$\Dom \RR := \set {x \in V: \exists y \in V: \tuple {x, y} \in \RR}$

That is, it is the class of all $x$ such that $\tuple {x, y} \in \RR$ for at least one $y$.

Also defined as

It is usual, as here, to define the domain of $\RR \subseteq S \times T$ as the subset of $S$ that bears some element of $S$ to $T$.

However, it appears to make sense to define it to be the whole of the set $S$.

Using this definition, $s \in \Dom \RR$ whether or not $\exists t \in T: \tuple {s, t} \in \RR$.

It would then be possible to refer to $\set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$ as the preimage of $\RR$.

This appears never to be done in the literature that has so far been studied as source work for $\mathsf{Pr} \infty \mathsf{fWiki}$.

Most texts do not define the domain in the context of a relation in the first place, 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 appears to be of small importance.

Also known as

Some sources refer to the domain of $\RR$ as the domain of definition of $\RR$.

Some sources use a distinctive typeface, for example, $\map {\mathsf {Dom} } \RR$.