# 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?

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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 $$g$$.

JDZ
October 10, 2019 05:03 AM

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