Definition:Domain (Relation Theory)/Relation/Class Theory

Definition

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation in $V$.

The domain of $\RR$ is defined and denoted as:

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

That is, it is the class of all $x$ such that $\tuple {x, y} \in \RR$ for at least one $y$.

Also known as

Some sources refer to the domain of $\RR$ as the domain of definition of $\RR$.

Some sources use a distinctive typeface, for example, $\map {\mathsf {Dom} } \RR$.

Also see

• Definition:Image of Relation (Class Theory)

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 8$ Relations