Definition:Many-to-One Relation

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


 * $\forall x \in \operatorname{Dom} \left({\mathcal R}\right): \left({x, y_1}\right) \in \mathcal R \land \left({x, y_2}\right) \in \mathcal R \implies y_1 = y_2$

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

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).

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 is called a mapping.

Some approaches, for example, 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 and limited to this mention.