Definition:One-to-Many Relation

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

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

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.