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

Thank you.

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- Dec 30th 2011, 07:18 PMitproProof 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, 10:45 PMFernandoRevillaRe: Proof of invertibility of injective functions
If $\displaystyle f:A\to B$ is invertible, there exists $\displaystyle g:B\to A$ such that $\displaystyle g\circ f =I_A$ . Now,

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

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