One-to-one functions of 2 variables Are there any one-to-one functions of 2 variables? For each of the following prove or disprove whether there is a one-to-one function $f$ of 2 variables:


*

*$f$ is from $\Bbb{N}^2$ to $\Bbb{N}$

*$f$ is form $\Bbb Z^2$ to $\Bbb Z$

*$f$ is from $\Bbb R^2$ to $\Bbb R$

*$f$ is from $\Bbb Q^2$ to $\Bbb Q$

*Any other maps that may be more interesting like $\Bbb R^2$ to $\Bbb N$ or something.


I don't feel like thinking much right now because I'm quite tired, so I apologize if the 
question is obvious.
Edit: I must have been out of my mind earlier. By function I meant to say polynomial.

Yes there are, since , e.g., $\mathbb R, \mathbb R^N; N>1$ have the same cardinality. This is equivalent to saying that there is not just an injection, but a bijection between them.
