How do I prove this function is not surjective?

Question

\begin{align*} g: \mathbb{N} & \to \mathbb{Z} \\ g(n) &= \begin{cases} \frac{n+1}{2} & n \textrm{ is odd.}\\ -\frac{n}{2} & n \textrm{ is even.} \end{cases} \end{align*}

Since $0\notin \mathbb{N}$, I think this function is not surjective. How are you supposed to prove this? Pick a generic element $x$ of natural numbers and somehow show that it can never produce $g(n)=0$?

Thanks.

You have to show that $0 \in \mathbb{Z}$ has no preimage – tattwamasi amrutam Feb 08 '18 at 20:12 — tattwamasi amrutam, Feb 08 '18 at 20:12

score 1 · Accepted Answer · answered Feb 08 '18 at 20:13

1

I think that your idea is right, just observe that since

$\frac{n+1}{2}=0\iff n=-1\not \in\mathbb{N}$
$ -\frac{n}{2} \iff n=0\not \in\mathbb{N}$

$g(n)=0\in \mathbb{Z}$ is not reached.

answered Feb 08 '18 at 20:13

user

154,566

Hector Blandin · Answer 2 · 2018-02-08T20:28:28.047

1

It depends how you define $\mathbb{N}$. If $0\in\mathbb {N}$, then $g(0)=0$ since $0$ is even. Otherwise, you right. See here: http://mathworld.wolfram.com/NaturalNumber.html

edited Feb 08 '18 at 20:28

answered Feb 08 '18 at 20:19

Hector Blandin

2,471

How do I prove this function is not surjective?

2 Answers2