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 or onto.

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
 * Surjection iff Image equals Codomain