# Definition:Many-to-One Relation

## Definition

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

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

That is, every element of the domain of $\mathcal R$ 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 Set

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

Such a 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.