全単射

Proposition

全単射の必要十分条件

Statement

$S,T$:Set
$f:S \rightarrow T$
$f:\text{Bijection} \Leftrightarrow \exists f^{-1}$

Proof

$\Rightarrow$
$\forall t \in T, \exists s \in S, \text{ s.t. } f(s) = t \quad(\because f:\text{Surjective})$
$s$ is unique $\quad(\because f:\text{Injective})$
Define $g:T \rightarrow S$ as
$g:t \mapsto s \text{ s.t. }f(s) = t$
then
$f \circ g(t) = t$
$g \circ f(s) = s$
$\therefore f \circ g = g \circ f = id$
$\therefore g = f^{-1}$
$\Leftarrow$
$\forall t \in T, \exists s = f^{-1}(t)$
then $f(s) = t$
$\therefore f$:Surjective
let $f(s) = t, f(s') = t'$
then $s = f^{-1}(t), s' = f^{-1}(t')$
$\therefore t = t' \Rightarrow s = f^{-1}(t) = f^{-1}(t') = s'$
$\therefore f:$ Injective
QED.