Definition:Mapping/Defined

Definition

A mapping $f \subseteq S \times T$ is defined at $x \in S$ if and only if:

$\exists y \in T: \tuple {x, y} \in f$

If for some $x \in S$, one has:

$\forall y \in T: \tuple {x, y} \notin f$

then $f$ is not defined or (undefined) at $x$, and indeed, $f$ is not technically a mapping at all.

Also see

• Definition:Well-Defined Mapping
• Definition:Partial Mapping

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Equivalence Relations: $\S 19$