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Suppose $f : X → Y$ and $g : Y → Z$ are functions. If $g ◦ f$ is bijective and $f$ is surjective. Then what would $g$ be? Would it be bijective or invective? I know that $g ◦ f$ is injective then $f$ is injective and that if $g ◦ f$ is surjective then $g$ is surjective. And I know how to prove these but how would I show if $g ◦ f$ is bijective and if $f$ is surjective what $g$ would be?

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Hint:

  • If $g$ isn't surjective, can $g\circ f$ be surjective?
  • If $g$ isn't injective, can $g\circ f$ be injective? (Here you need to use that $f$ it's surjective.)
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  • $\begingroup$ No g must be surjective for the composition to be surjective. I know this proof! And f is injective for the composition to be injective. So would that mean that for this case g is both surjective and injective hence bijective? $\endgroup$
    – user535425
    Commented Mar 20, 2019 at 17:42

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