From Cantor we know that R = 2^Z. That is, R is equal to the number of subsets of Z. Is it also true that R is equal to the number of infinite subsets of Z?
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Yes, because there are only countably many finite subsets of Z. 


The set of subsets of $\mathbb{Z}$ can be seen as the set of all functions $f:\mathbb{Z}\to\{0,1\}$. And this last set can be seen as the set of all sequences of 0 and 1. With this last point of view, it is easy to see that there exists a bijection between this set and, for instance, the interval $[0,1]$. 

