It must be posted somewhere, but I can't find it. I've been working on it for a while too without getting anywhere. Does there exist a bijection between $\mathbb{R}\times\mathbb{R}$ and $\mathbb{R}$? Is it possible to give an explicit bijection?

NOTE: This question is not a duplicate of the link suggested. Please see comments for further detail.


marked as duplicate by Zev Chonoles, Lord_Farin, Pedro Tamaroff, Amzoti, Ma Ming Jun 21 '13 at 21:54

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • 2
    $\begingroup$ Yes, there is a isomorphism as vector spaces over $\mathbb{Q}$. $\endgroup$ – user40276 Jun 21 '13 at 21:24
  • $\begingroup$ @Paul S. The sentences before the question hint that you're looking for a bijection rather than just wanting to know if a bijection exists. What exactly do you want? And I suggest you clarify this fast, before it gets closed as a duplicate and it actually isn't (if that's the case). $\endgroup$ – Git Gud Jun 21 '13 at 21:25
  • $\begingroup$ @GitGud Ok I just read the link that Zev Chonoles posted. I see that they prove a bijection exist, but don't show a bijection. Is it possible to show a bijection? $\endgroup$ – Paul S. Jun 21 '13 at 21:31
  • 1
    $\begingroup$ @dfeuer Don't they show injections in each direction, and then invoke the Schroeder–Bernstein theorem? $\endgroup$ – Paul S. Jun 21 '13 at 21:36
  • 1
    $\begingroup$ I believe it should be possible to get more explicit than that, but it will require a lot of care about sign encoding. $\endgroup$ – dfeuer Jun 21 '13 at 22:11