Definition:Many-to-One Relation

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


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

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

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.

None of these names is as intuitively obvious as many-to-one relation, so the latter is the preferred term on.

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.

Also see

 * Definition:One-to-Many Relation

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).