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

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

That is, iff:
 * $$\operatorname{Im} \left({\mathcal R}\right) = T$$

where $$\operatorname{Im} \left({\mathcal R}\right)$$ denotes the image of $$\mathcal R$$.

Such a relation can also be called surjective.

It is clear from this definition that a right-total mapping is a surjection, which explains the alternative use of language.

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

Also see

 * Left-Total Relation