Definition:One-to-Many Relation

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Definition

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

$\RR \subseteq S \times T: \forall y \in \Img \RR: \tuple {x_1, y} \in \RR \land \tuple {x_2, y} \in \RR \implies x_1 = x_2$


That is, every element of the image of $\RR$ 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.


Examples

Inverse Sine

Let $\RR \subseteq \R \times \R$ be the relation on $\R$ defined as:

$\forall \tuple {x, y} \in \R \times \R: \tuple {x, y} \in \RR \iff x = \sin y$

Then $\RR$ is a one-to-many relation.


Also known as

A one-to-many relation is also known as:

a one-many relation
an injective relation
a left-unique relation
a single-rooted relation
a one-many correspondence


Also see

  • Results about one-to-many relations can be found here.


Sources