Definition:Injection/Definition 2

Definition
An injection is a relation which is both one-to-one and left-total.

Thus, a relation $f$ is an injection :
 * $(1): \quad \forall x \in \operatorname{Dom} \left({f}\right): \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$
 * $(2): \quad \forall y \in \operatorname{Im} \left({f}\right): \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f \implies x_1 = x_2$
 * $(3): \quad \forall s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R$

Also see

 * Equivalence of Definitions of Injection