Definition:Domain (Set Theory)

From ProofWiki
Jump to navigation Jump to search

This page is about Domain in the context of Set Theory. For other uses, see Domain.

Definition

Relation

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

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

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


That is, it is the same as what is defined here as the preimage of $\RR$.


Mapping

The term domain is usually seen when the relation in question is actually a mapping.


Let $f: S \to T$ be a mapping.

The domain of $f$ is $S$, and can be denoted $\Dom f$.

In the context of mappings, the domain and the preimage of a mapping are the same set.


Binary Operation

Let $\circ: S \times S \to T$ be a binary operation.

The domain of $\circ$ is the set $S$ and can be denoted $\Dom \circ$.


Also see