Definition:Left-Total Relation

Definition

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 $T$.

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.

A multifunction is a left-total relation $\RR$ which is specifically not many-to-one or one-to-one.

That is, for each element $s$ of the domain of $\RR$, there exists more than one $t$ in the codomain of $\RR$ such that $\tuple {s, t} \in \RR$.