Definition:One-to-One Relation

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A relation $\mathcal R \subseteq S \times T$ is one-to-one if it is both many-to-one and one-to-many.

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

Also see

Compare this with a one-to-one mapping, in which every element of the domain is mapped to an element of the codomain.