bijective

Bijective function

A function $f:A\to B$ is called bijective if it is both injective and surjective.
A bijection basically means that there is a one-to-one correspondence between all elements of the domain and the codomain: For every unique element in $A$, there is a unique element in $B$.

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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.

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set equality can be defined via bijectivity

$|A|=|B|⟺\exists f:A\to B,f\text{ bijective}$

A bijection can also be composed of injections and surjections, e.g.: