Let f: S $\rightarrow$ T and g: T $\rightarrow$ S be two function. If gf is invertible, prove that f is one-to-one and g is onto.

by Simran Williams   Last Updated October 10, 2019 05:20 AM

I have tried doing this since the past 2 hours. How do I do this?

Answers 1

It's just a straightforward application of the definitions:

$\bullet \,$If $gf$ is injective, then $f$ is injective because $f(a)=f(b)\Rightarrow gf(a)=gf(b)\Rightarrow a=b$.

$\bullet \,$If $gf$ is surjective, then $g$ is surjective because $b\in S\Rightarrow \exists \,a\in S$ such that $gf(a)=b$, so $f(a)\in T$ gets sent to $b$ by $g$.

October 10, 2019 05:03 AM

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