1

How can you prove that "if $f$ and $g$ are surjective functions, $g\circ f$ will be surjective"?

Thanks for any help.

Martin Sleziak
  • 53,687
user2979879
  • 55
  • 3
    By applying the definition. There is not really any trick here. – Tobias Kildetoft Jan 14 '14 at 15:27
  • See http://math.stackexchange.com/questions/75246/surjectivity-of-composition-of-surjective-functions and http://math.stackexchange.com/questions/22572/injective-and-surjective-functions Maybe you can find some more among related links on the right. – Martin Sleziak Jan 14 '14 at 16:34

1 Answers1

2

Let $f:A\longrightarrow B$ and $g:B\longrightarrow C$. Take $c\in C$. Since $g$ is surjective, there exists $b\in B$ such that $g(b)=c$. Since $f$ is surjective, there exists $a\in A$ such that $f(a)=b$. Putting together, you have $(g\circ f)(a)=c$. Since $c$ was arbitrary, you have the surjectivity.

Danae Kissinger
  • 2,143