0
$\begingroup$

I have a bijection problem that I cannot get my head around. It goes like this:

let f: A -> B and g: B -> C be functions such that g o f is a bijection. Prove that f must be one-to-one and that g must be onto. And also an example showing that it is possible for neither f nor g to be a bijection.

This is the problem. I do not even know where to begin. Any help is appreciated.

$\endgroup$
  • 2
    $\begingroup$ As any math problem, start from definition. What is bijection? What is 1-1? What is onto? what is g o f? $\endgroup$ – scaaahu Oct 1 '13 at 4:33
2
$\begingroup$

Hints: First part: Suppose that $f$ is not one-to-one. That implies that $f(x) = f(y)$ for some $x \neq y$. Can $g \circ f$ be a bijection? Similar idea for the other half. Second part: We know that $f$ must be one-to-one and $g$ must be onto. So if $f$ and $g$ are not bijections, then $f$ must not be onto and $g$ must not be one-to-one. See how these two conditions mesh together.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.