Definition:Field of Relation
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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:
- $\Field \RR := \set {x \in S: \exists t \in T: \tuple {x, t} \in \RR} \cup \set {x \in T: \exists s \in S: \tuple {s, x} \in \RR}$
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.
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations