# Definition:One-to-Many Relation

## Definition

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

- $\mathcal R \subseteq S \times T: \forall y \in \operatorname{Im} \left({\mathcal R}\right): \left({x_1, y}\right) \in \mathcal R \land \left({x_2, y}\right) \in \mathcal R \implies x_1 = x_2$

That is, every element of the image of $\mathcal R$ 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**