# Definition:Domain (Set Theory)/Relation

## Definition

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

The domain of $\mathcal R$ is defined as:

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

and can be denoted $\Dom {\mathcal R}$.

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

### General Definition

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

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

The domain of $\mathcal R$ is the set defined as:

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

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

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

## Also defined as

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

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

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 is of small importance.

## Also known as

Some sources refer to this as the domain of definition of $\mathcal R$.