Definition:One-to-Many Relation

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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


Also see