There are two functions' spaces with the same cardinality, but I was asked to find a bijection between them. So, the first space is $\mathbb{R}^\mathbb{R}$ and the second is $(\mathbb{R}^{[0,1)})^{\mathbb{Z}}$. (When $\mathbb{R}$ stands for the reals and $\mathbb{Z}$ for the integers).

I tried separating each functions from $\mathbb{R}^\mathbb{R}$ space to $\mathbb{R}^{[z,z+1)}$ and work from their, but I couldn't find a way to define one value for each function.

I would appreciate your help with this problem.


Hint. Given $f:\mathbb{R}\to\mathbb{R}$, then for any $n\in\mathbb{Z}$, define $f_n:[0,1)\to \mathbb{R}$ as $f_n(x)=f(x+n)$.

On the other hand, given for $n\in\mathbb{Z}$, $f_n:[0,1)\to \mathbb{R}$, then define $f:\mathbb{R}\to\mathbb{R}$ as $f(x)=f_n(x-n)$ for $x\in[n,n+1)$.

  • $\begingroup$ thanks!!, I havn't fully got it yet, but I hope ill understand it soon! $\endgroup$ – dan Sep 25 '16 at 15:33
  • $\begingroup$ @dan Well, let me know if you need further explanations. $\endgroup$ – Robert Z Sep 25 '16 at 21:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.