Definition:Domain (Set Theory)/Relation

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

The domain 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 $\mathcal R$.

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:

$\displaystyle \operatorname{Dom} \left({\mathcal R}\right) := \left\{{\left({s_1, s_2, \ldots, s_{n-1}}\right) \in \prod_{i \mathop = 1}^{n-1} S_i: \exists s_n \in S_n: \left({s_1, s_2, \ldots, s_n}\right) \in \mathcal R}\right\}$

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

$\displaystyle \operatorname{Dom} \left({\mathcal R}\right) := \left\{{\left({s_1, s_2, \ldots, s_{n-1}}\right) \in S^{n-1}: \exists s_n \in S_n: \left({s_1, s_2, \ldots, s_n}\right) \in \mathcal R}\right\}$

Also defined as

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 is of small importance.

Also known as

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

Also see