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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.

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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 $x\in[n,n+1)$.

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  • $\begingroup$ 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

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