Definition:One-to-Many Relation

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


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

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

Such a relation is also referred to as:
 * an injective relation;
 * a left-unique relation.

Also see

 * Many-to-One Relation
 * Injection