
Z > ZxZ Function
Hello,
I would like to define a function from Z > ZxZ that is both onto and one to one. I think it is going to have to be a piecewise function, but I am totally lost on how to get started. What I do know is that the function can be represented graphically by a spiral where the inputs and outputs look something like this:
f(0) = (0,0)
f(1) = (1,0)
f(2) = (1,1)
f(3) = (0,1)
f(4) = (1,1)
f(5) = (1,0)
f(6) = (1,1)
f(7) = (0,1)... so on
I know this is only for positive Z, and this is part of the reason I am stuck. I am also not sure how to represent this info as a function. Any help would be appreciated. This is bugging the crap out of me. Thank you.

Re: Z > ZxZ Function
It may be easier to write a bijection $\displaystyle f:\mathbb{Z}\to \mathbb{N}$, so that $\displaystyle g=(f,f):\mathbb{Z}\times \mathbb{Z} \to \mathbb{N}\times \mathbb{N}$ is a bijection. For example, take
$\displaystyle f(x)=\begin{cases} 2x, & \text{if }x\geq 0 \\ 2x+1, & \text{if }x<0\end{cases}$
It remains to find a bijection $\displaystyle \varphi: \mathbb{N}\times \mathbb{N} \to \mathbb{N}$. The standard one is the Cantor pairing function
$\displaystyle \varphi(x,y)= \frac{(x+y+1)(x+y)}{2}+x$
This last function makes precise the usual snakelike enumeration diagram for $\displaystyle \mathbb{N}\times \mathbb{N}$.
The function you want is $\displaystyle g^{1} \circ \varphi^{1} \circ f$. Since each function in the composition is a bijection, the result is also a bijection.