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?
2 Answers
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Yes, because there are only countably many finite subsets of Z.
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$\begingroup$ That was actually going to be my next question. Thanks :D $\endgroup$– wjomlexMar 5, 2011 at 20:02
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$\begingroup$ 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. $\endgroup$ Mar 5, 2011 at 20:24
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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 $[0,1]$.