Definition:One-to-Many Relation

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


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

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

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

Compare the definition for injection.