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Let $f$ be a function from a set $A$ to a set $B$, $g$ a function from $B$ to $C$ , and $h$ a function from $A$ to $C$, such that $h(a) = g(f(a))$ for all $a ∈ A$. Which of the following statements is always true for all such functions $f$ and $g$ ?

  • g is onto => h is onto
  • h is onto => f is onto
  • h is onto => g is onto
  • h is onto => f and g are onto

I know how to solve,so I am not giving those details. I just want to confirm the wording of the question.

I am getting $C$ option true as question asks for always true.

Had it been the statement:

Which of the following statements can be true for all such functions $f$ and $g$

Well, in this case I think option $D$ is more preferrable.

Am I thinking right ?

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    Commenting on your question, not the question asked.I don't understand your rephrasing. How does "can be true for all" differ from "is always true"? Further note: it's possible that more than one of these statements is correct for the question as asked. It doesn't make sense to "prefer" one of the option. – 2017-01-14
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    @EthanBolker Because I am thinking that D). can even be true here but its just that phrasing follows C). option. – 2017-01-14
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    Ethan is right - the two statements are the same. Please also note that there exist a function f and a function g s.t. all four "options" are true (e.g. take f=g=identity function). – 2017-01-14

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