Show that function$ f : Z → Z × Z$ such that $f(n) = (−n, 2n − 1)$ is bijective (or not).

Question

I am having problems showing with this is bijective or not. i am mostly confused about how to deal with f(n) having two parts and how to show whether it is injective/surjective.

Answer 1

Let $(0,0)\in{\bf{Z}}\times{\bf{Z}}$ and assume that $f(n)=(0,0)$ for some $n\in{\bf{Z}}$, then $(-n,2n-1)=(0,0)$, so $-n=0$ and $2n-1=0$, then we have both $n=0$ and $n=1/2$, a contradiction, so $f$ is not onto, and hence not bijective.