Definition:Left-Total Relation

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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:

$\forall s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R$

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:

Therefore 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.


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

Also see