# Definition:One-to-Many Relation

## Definition

A relation $\RR \subseteq S \times T$ is one-to-many if and only if:

$\RR \subseteq S \times T: \forall y \in \Img \RR: \tuple {x_1, y} \in \RR \land \tuple {x_2, y} \in \RR \implies x_1 = x_2$

That is, every element of the image of $\RR$ is related to by exactly one element of its domain.

Note that the condition on $t$ concerns the elements in the image, not the codomain.

Thus a one-to-many relation may leave some element(s) of the codomain unrelated.

## Also known as

A one-to-many relation is also referred to as:

an injective relation
a left-unique relation
a single-rooted relation
a one-many correspondence