Definition:Many-to-One Relation
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Definition
A relation $\mathcal R \subseteq S \times T$ is many-to-one if:
- $\mathcal R \subseteq S \times T: \forall x \in \operatorname{Dom} \left({\mathcal R}\right): \left({x, y_1}\right) \in \mathcal R \land \left({x, y_2}\right) \in \mathcal R \implies y_1 = y_2$
That is, every element of the domain of $\mathcal R$ relates to no more than one element of its codomain.
If in addition, every element of the domain relates to one element in the codomain, the many-to-one relation is known as a mapping (or function).
Such a relation is also referred to as:
- a functional relation;
- a right-definite relation;
- a right-unique relation;
- a partial mapping.
Sources
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$: Definition
