Definition:Injection/Definition 4

Definition
Let $f$ be a mapping.

$f$ is an injection :
 * $\forall y \in \Img f: \card {\map {f^{-1} } y} = \card {\set {f^{-1} \sqbrk {\set y} } } = 1$

where:
 * $\Img f$ denotes the image set of $f$
 * $\card {\, \cdot \,}$ denotes the cardinality of a set
 * $\map {f^{-1} } y$ is the preimage of $y$
 * $f^{-1} \sqbrk {\set y}$ is the preimage of the subset $\set y \subseteq \Img f$.

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