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

That is, iff every element of $$S$$ relates to some element of $$T$$.

From Inverse of Right-Total is Left-Total, if $$\mathcal R$$ is left-total, then its inverse $$\mathcal R^{-1}$$ is right-total.

It is also sometimes referred to as a total relation, but this can be confused with a connected relation, so it is probably best not to use that term.

Multifunction
A left-total relation is also known as a multifunction, or multi-valued function.

It is usually used in this context in the field of complex analysis.

Strictly speaking, it is not a function (that is, a mapping) at all, as (by implication) there exist elements in the domain which are mapped to more than one element in the codomain.

Also see

 * Right-Total Relation