Proof of invertibility of injective functions

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December 30th 2011, 07:18 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.
December 30th 2011, 10:45 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,

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

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

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