Definition:Injection/Definition 4

Definition
Let $f$ be a mapping.

$f$ is an injection :
 * $\forall y \in \operatorname{Im} \left({f}\right): \left\vert{f^{-1} \left({y}\right)}\right\vert = \left\vert{f^{-1} \left[{\left\{{y}\right\}}\right]}\right\vert = 1$

where:
 * $\operatorname{Im} \left({f}\right)$ is the image set of $f$
 * $\left\vert{\cdot}\right\vert$ denotes the cardinality of a set
 * $f^{-1} \left({y}\right)$ is the preimage of $y$
 * $f^{-1} \left[{\left\{{y}\right\}}\right]$ is the preimage of the subset $\left\{{y}\right\} \subseteq \operatorname{Im} \left({f}\right)$.

That is, the preimage of $y$ is a singleton for all $y$ in the image set of $f$.

Also see

 * Equivalence of Definitions of Injection