injective relation

An injective relation is a relation where every element of the codomain of definition is uniquely assigned, i.e. has at most one associated element in the domain of definition.

Formally,
a binary relation $R$ over sets $X$ and $Y$ is injective if for all $x_a,x_b\in X$ and all $y\in Y$, if $(x_a,y)\in R$ and $(x_b,y)\in R$, then $x_a=x_b$.

i.e., no two elements of the domain can have the same image.