# Definition:Left-Total Relation

## Definition

Let $S$ and $T$ be sets.

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

Then $\mathcal R$ is left-total if and only if:

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

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

## Also known as

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

A total relation, but this can be confused with a connected relation.
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 relation $\mathcal R: S \to S$ which is left-total is also referred to as a serial relation.

## Multifunction

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