# 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 if and only if:

$\forall t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R$

That is, if and only if every element of $T$ is related to by some element of $S$.

That is, if and only if:

$\Img {\mathcal R} = T$

where $\Img {\mathcal R}$ denotes the image of $\mathcal R$.

## Also known as

A right-total relation can also be referred to as surjective or onto.