I am not a mathematician, but I study it for fun. I know that Cantor showed that the infinite set of the real numbers is larger than the rationals since the real numbers are uncountable and the rationals are countable. The real numbers include rational and irrational numbers and the irrational numbers are uncountable. Is the set of irrational numbers smaller than the set of real numbers, the same size, or does this question not make sense? Thank you!
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2It depends what you mean by 'smaller'. The irrationals are a proper subset of the reals, so in this sense they are smaller. But there exists a bijection between the irrationals and the reals, so they have the same cardinality. In this sense they're the same size. – Jul 06 '20 at 00:30
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1See this question – lulu Jul 06 '20 at 00:31
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1(Note that while this question has been closed as a duplicate, that doesn't mean it's a bad question - it's a great question! It just means that it's already been asked/answered here before.) – Noah Schweber Jul 06 '20 at 00:36
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Thank you for your feedback! – Anne Jul 06 '20 at 00:37
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Take a look at Suber's "A Crash Course on the Mathematics of Infinite Sets" – vonbrand Jul 06 '20 at 02:40