Prove that this is incorrect
$g \circ f$ is surjective, but $g$ is not surjective.
$f$ and $g$ are not injective, but $g \circ f$ is injective.
I know these both don't exist but, I don't really know how to formally prove them. Do I use proof by contradiction?
- We know $g \circ f$ is surjective and we want to show that $g$ is surjective. Let $y \in\mathbb R$. since $g \circ f$ is surjective, there exists an $b \in \mathbb R$ such that $g(f(b)) = y$. We set $c = f(b) \in\mathbb R,$ then $g(f(b))=g(c)=y$. So $g$ is surjective.
Is this how you're suppose to prove the question? I feel like I just proved that $g \circ f$ is surjective then $g$ is surjective but not the question above.
- I don't really know where to start with this one. Am I suppose to use proof by contradiction here?