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February 28th, 2012, 08:35 PM   #1
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on relationships within the power set of Z+

Now that I understand the definition of power set, I was thinking this: If you have a bijection between Z+ and the set of all finite subsets of Z+, then you also have a "natural" bijection between Z+ and the set of all infinite subsets of Z+ with a finite number of positive integers missing. But since the cardinality of the power set is greater than the cardinality of Z+, then we have that the cardinality of the set of all infinite subsets of Z+ with an infinite number of positive integers missing is also greater than the cardinality of Z+. Without proof, is my reasoning sound?
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February 29th, 2012, 10:23 AM   #2
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Re: on relationships within the power set of Z+

Quote:
Originally Posted by icemanfan
Without proof, is my reasoning sound?
No. Before you were assuming that all subsets of the natural numbers were finite; now you're assuming that they're all either finite or cofinite (missing only a finite number of elements). But these as only a tiny fraction of the total. What about the even (positive) numbers? They're neither finite nor cofinite, so you're missing them.
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February 29th, 2012, 11:23 AM   #3
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Re: on relationships within the power set of Z+

I think you misunderstood what I am saying. I am saying that the cardinality of sets which are neither finite nor cofinite (e.g. the numbers divisible by a given number, the set of all primes, etc.) is greater than the cardinality of the positive integers, because (as you say) the other sets which are either finite or cofinite are only a fraction of the total.
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March 1st, 2012, 04:26 PM   #4
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Re: on relationships within the power set of Z+

Oh, sorry. In that case you're right: almost all of the members of the power set of Z+ are neither finite nor cofinite, and such members have cardinality greater than that of Z+.
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