# Definition:Many-to-One Relation/Defined

## Definition

Let $f \subseteq S \times T$ be a many-to-one relation.

### Defined at Element

Let $s \in S$.

Then $f$ is defined at $s$ if and only if $s \in \Dom f$, the domain of $f$.

### Defined on Set

Let $R \subseteq S$.

Then $f$ is defined on $R$ if and only if it is defined at all $r \in R$.

Equivalently, if and only if $R \subseteq \Dom f$, the domain of $f$.