# Definition:Field of Relation

This page is about Field of Relation. For other uses, see Field.

## Definition

Let $S$ and $T$ be sets.

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

The field of $\RR$ is defined as:

$\map {\operatorname {Field} } \RR := \set {x \in S: \exists t \in T: \tuple {x, t} \in \RR} \cup \set {x \in T: \exists s \in S: \tuple {s, x} \in \RR}$

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