Definition:Left-Total Relation

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 $$T$$.

It is also clear that 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.