Definition:Left-Total Relation

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

Then $\RR$ is left-total if and only if:

$\forall s \in S: \exists t \in T: \tuple {s, t} \in \RR$

That is, if and only if every element of $S$ relates to some element of $T$.

Also known as

A left-total relation $\RR \subseteq S \times T$ is also sometimes referred to as:

A total relation, but $\mathsf{Pr} \infty \mathsf{fWiki}$ already has a definition for such a concept
A relation on $S$, but this can be confused with an endorelation
A multiple-valued function or multifunction, but the latter is usually reserved for complex functions

The term left-total relation is usually preferred.


In the field of complex analysis, a left-total relation is usually referred to as a multifunction.

Also see

  • Results about left-total relations can be found here.