Definition:One-to-One Relation

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 range, and every element of the image is related to by exactly one element of its domain.

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