# Definition:Field of Relation

This page is about the field of a relation in relation theory. For other uses, see Definition:Field.

## Definition

Let $S$ and $T$ be sets.

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

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

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

That is, it is the union of the preimage of $\mathcal R$ with its image.