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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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closed as too localized by Todd Trimble, Peter Shor, J.C. Ottem, Joel David Hamkins, coudy Mar 5 '11 at 20:22

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This question should be closed; it is more appropriate for math stack exchange. – Todd Trimble Mar 5 '11 at 20:04
up vote 1 down vote accepted

Yes, because there are only countably many finite subsets of Z.

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That was actually going to be my next question. Thanks :D – wjomlex Mar 5 '11 at 20:02
A slightly more difficult problem is to construct continuum many (infinite) subsets of N such that any two have a finite intersection. I leave it to you as an exercise! Hint: change N to Q. – GH from MO Mar 5 '11 at 20:24

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 $[0,1]$.

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