[Math] Prove Injectivity and Surjectivity of functions like $g \circ f$

discrete mathematicsfunctions

I am trying to prove the following by the given:

  1. $g \circ f$ Surjective
  2. $f:A\rightarrow B$
  3. $g:B\rightarrow C$

1) Assuming that $g$ is Injective I want to prove the $f$ is Surjective.
2) there is option to say that $f$ is surjective without the assumption that $g$ is Injective?

Any suggestions? Thanks!

Best Answer

You could prove that $g$ is surjective by the fact that $g\circ f$ is surjective. Then, if $g$ is assumed to be injective, it is bijective and thus has an inverse $g^{-1}:C\to B$. Now $f$ can be written as $g^{-1}\circ(g\circ f)$. Can you deduce that $f$ is surjective?

2) Try to find $f$ and $g$ such that $f$ and $g\circ f$ are surjective, but $g$ is not injective. Hint: If $C=\{c\}$, then $g\circ f$ is always surjective, but $g$ is only injective if $B$ has only one element.

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