# Countable or uncountable question

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