injective

Injective (one-to-one) function

A function $f:A\to B$ is called injective if $\forall x_1,x_2\in A:f(x_1)=f(x_2)⟹x_1=x_2$
Or conversely: $\forall x_1,x_2\in A:x_1\ne x_2⟹f(x_1)\ne f(x_2)$

(an injective but not surjective function)

EXAMPLE

$f_1:\mathbb{R}\to {\mathbb{R}}_0^{+}:x\mapsto x^2$ is not injective, because $f_1(-1)=f_1(1)$ and it is not surjective, because $f_1(x)\ge 0\forall x\in \mathbb{R}$ (negative real numbers are not in the codomain).
$f_2:{\mathbb{R}}_0^{+}\to \mathbb{R},x\mapsto x^2$ is injective, but not surjective because we canot reach negative numbers.
$f_3:\mathbb{R}\to {\mathbb{R}}_0^{+},x\mapsto x^2$ is not injective but surjective ($x^2=z$ has a solution for all $z\in {\mathbb{R}}_0^{+}$).
$f_4:{\mathbb{R}}_0^{+}\to {\mathbb{R}}_0^{+},x\mapsto x^2$ is bijective.

Link to original