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Does a subset of $R$ contain equal number of rational and irrational numbers? How to prove?

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Your title doesn't match your body. Which question do you mean to ask? – Chris Eagle Jan 4 '13 at 13:50
I thought the probabilty must be 1/2. I am asking whether it is. – Swapnanil Saha Jan 4 '13 at 13:51
That's the title of your question, but not the body of your question. – Thomas Andrews Jan 4 '13 at 14:00
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You have asked two different questions.

A number chosen uniformly at random between 1 and 10 has probability zero of being rational.

An interval of reals contains a countable infinity of rationals, an uncountable (i.e., much larger) infinity of irrationals. The rationals have measure (i.e., length) zero; all the measure is in the irrationals.

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This is what I was asking. I have heard that there are more irrationals than rationals. But how do I show this? – Swapnanil Saha Jan 4 '13 at 13:53
That is yet a third different question, @SwapnanilSaha :) – Thomas Andrews Jan 4 '13 at 14:02
Swapnanil, I don't want to write out the whole story, for a couple of reasons. One, I'm sure it has been done on this site before --- see the "Related" questions on the right side of this page? There are a few that ask about the relations between rationals and irrationals. Two, I think the answer I have given has given you enough keyphrases to type into the internet to see what turns up. If you do that and all the sites are hard to understand, come back with a question about the place where you get stuck. – Gerry Myerson Jan 4 '13 at 14:13
Thank you Gerry. – Swapnanil Saha Jan 8 '13 at 18:30

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