# Definition:Field of Relation

## Definition

Let $S$ and $T$ be sets.

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

The field of $\RR$ is defined as:

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

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

### 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 field of $\RR$ is defined as:

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

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