# Definition:One-to-Many Relation

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## Contents

## Definition

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

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

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** - a
**single-rooted relation** - a
**one-many correspondence**

## Also see

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**one-many correspondence** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**one-to-many**