# Definition:Domain (Set Theory)

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*This page is about the concept of domain in set theory. For other uses, see Definition:Domain.*

## Definition

### Relation

Let $\mathcal R \subseteq S \times T$ be a relation.

The **domain** of $\mathcal R$ is defined as:

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

and can be denoted $\Dom {\mathcal R}$.

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

### 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 the set $S$ 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 $\operatorname{Dom} \left({\circ}\right)$.