Bijection from $\mathbb {Z}^3$ to $\mathbb {Z}$

Question

I am not a mathematician. Let $\mathbb {Z}$ be a positive integer set. I need to know whether there exist a bijection from $\mathbb {Z}^3$ to $\mathbb {Z}$, what might be a possible mapping?

I know that bijection exists from $\mathbb {R}^3$ to $\mathbb {R}$.

Answer 1

You have an injective map $Z\rightarrow N$ defined by $f(n)=2^n, n>0$ and $f(n)=3^{-n}, n\leq 0$, this induces an injective map $g:Z^3\rightarrow N$ defined by $g(a,b,c)=2^{f(a)}3^{f(b)}5^{f(c)}$. The image of $g$ is in bijection with $N$, now take a bijection between $N$ and $Z$ and compose with $g$.