Definition:One-to-Many Relation

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.

Also known as

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

an injective relation

a left-unique relation

a single-rooted relation

a one-many correspondence

Also see

• Definition:Many-to-One Relation
• Definition:Injection

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

Sources

• 1939: E.G. Phillips: A Course of Analysis (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Number: $1.2$ Fundamental notions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): one-many correspondence
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): one-to-many