# Definition:Many-to-One Relation

## Definition

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

- $\forall x \in \Dom \RR: \forall y_1, y_2 \in \Cdm \RR: \tuple {x, y_1} \in \RR \land \tuple {x, y_2} \in \RR \implies y_1 = y_2$

That is, every element of the domain of $\RR$ relates to no more than one element of its codomain.

### Defined

Let $f \subseteq S \times T$ be a many-to-one relation.

#### Defined at Element

Let $s \in S$.

Then $f$ is **defined at $s$** if and only if $s \in \Dom f$, the domain of $f$.

#### Defined on Subclass

Let $R \subseteq S$.

Then $f$ is **defined on $R$** if and only if it is defined at all $r \in R$.

Equivalently, if and only if $R \subseteq \Dom f$, the domain of $f$.

## Also known as

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

- a
**rule of assignment** - a
**functional relation** - a
**right-definite relation** - a
**right-unique relation** - a
**partial mapping**.

Some sources break with mathematical convention and call this a **(partial) function**.

These sources subsequently define a **total function** to be what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called a mapping.

None of these names is as intuitively obvious as **many-to-one relation**, so the latter is the preferred term on $\mathsf{Pr} \infty \mathsf{fWiki}$.

However, it must be noted that a **one-to-one relation** is an example of a **many-to-one relation**, which may confuse.

The important part is the **to-one** part of the definition, which is as opposed to the **to-many** characteristic of a one-to-many relation and a many-to-many relation.

Some approaches, for example 1999: András Hajnal and Peter Hamburger: *Set Theory*, use this as the definition for a mapping from $S$ to $T$, and then separately specify the requisite left-total nature of the conventional definition by restricting $S$ to the domain.

However, this approach is sufficiently different from the mainstream approach that it will not be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ and limited to this mention.

## Also see

If in addition, every element of the domain relates to **exactly** one element in the codomain, the **many-to-one relation** is known as a mapping (or function).

- Results about
**many-to-one 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 - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 4$. Relations; functional relations; mappings - 1989: George S. Boolos and Richard C. Jeffrey:
*Computability and Logic*(3rd ed.) ... (previous) ... (next): $1$ Enumerability - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $10$: Definition $1.3$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.4$: Definition $\text{A}.23$