If $f,g: \mathbb{N}\rightarrow\mathbb{N}$ and $f\circ g=id_\mathbb{N}$, $f$ is invertible?

Question

If $f,g: \mathbb{N}\rightarrow\mathbb{N}$ and $f\circ g=\operatorname{id}_\mathbb{N}$

Is $f$ necessarily invertible?

I can't prove it formally, but also can't come up with two functions that contradict it.

$f(x) = 1$ if $x$ is odd, and $f(x) = x/2$ if $x$ is even. This is surjective, but not injective. — Prahlad Vaidyanathan, Aug 06 '21 at 10:25

score 5 · Answer 1 · answered Aug 06 '21 at 10:25

5

Take the function $$f:\Bbb{N}\to \Bbb{N} $$ by $f(1)=1, f(n)=n-1 \forall n >1$. Take $g(n)=n+1$

answered Aug 06 '21 at 10:25

epsilon_delta

1 Answers1