Proof of invertibility of injective functions

• Dec 30th 2011, 08:18 PM
itpro
Proof of invertibility of injective functions
Would someone please help me with construction of a proof that every invertible function is injection.

Thank you.
• Dec 30th 2011, 11:45 PM
FernandoRevilla
Re: Proof of invertibility of injective functions
Quote:

Originally Posted by itpro
Would someone please help me with construction of a proof that every invertible function is injection.

If $f:A\to B$ is invertible, there exists $g:B\to A$ such that $g\circ f =I_A$ . Now,

$f(x_1)=f(x_2)\Rightarrow g[f(x_1)]=g[f(x_2)]\Rightarrow$

$(g\circ f)(x_1)=(g\circ f)(x_2)\Rightarrow I_A(x_1)=I_A(x_2)\Rightarrow x_1=x_2$