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Determine whether the following set is countable or uncountable. If it is countably infinite, exhibit a one-to-one correspondence between the set of positive integers and the set.

The set of irrational numbers between $\sqrt{2}$ and $\pi\over 2$.

I think this set is countable, but not sure what is the right way to solve it and how to exhibit the one to one correspondence. Can anyone help me?

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The set is uncountable –  Yan Yau Oct 4 '13 at 3:24
    
Can you explain why? –  Layla Oct 4 '13 at 3:29
    
i forgot the proof, so i will let someone else explain it :-/ –  Yan Yau Oct 4 '13 at 3:35
    
Is it like if the set of irrational numbers between √2 and π/2 were countable, then the union of these two countable sets would be countable. However, that would mean the set of all real numbers between √2 and π/2 is countable. It's a fact that the set of all real numbers between any two numbers is an uncountable set, so we have a contradiction. Therefore, the set of irrational numbers between √2 and π/2 is uncountable. –  Layla Oct 4 '13 at 3:39

1 Answer 1

up vote 5 down vote accepted

Since the set of all rational numbers is countable, the set of all rational numbers between those two is countable. If the set of all irrational numbers between those two were countable, then the set of all numbers between those two would be the union of two countable sets. It's not hard to prove that the union of two countable sets is countable.

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how to exhibit the one to one correspondence then? –  Layla Oct 4 '13 at 3:33
2  
@Layla: Michael is trying to convey to you that the set in question is not countable. –  Clive Newstead Oct 4 '13 at 3:36
    
@Layla : My point is that there is no such correspondence, because if there were, then a certain set would be countable that we already know is not countable. –  Michael Hardy Oct 4 '13 at 3:41
    
@Clive Newstead : Did I prove it right in the upper comment? –  Layla Oct 4 '13 at 3:43
    
@Michael Hardy: Did I prove it right in the upper comment? –  Layla Oct 4 '13 at 3:49

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