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## #1 2006-04-21 11:32:40

Kazy
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### Countably Infinite

I need to prove that the following sets are countably infinite:

a) Q (intersect) [0,1]   -  Rationals intersected with [0,1]
b) Q+ U {e^x | x ∈ Z}
c) The rational points on the unit circle:
{(x,y) | x² + y² = 1, x ∈ Q, y ∈ Q}

I know c has to do with pythagorean triples, but other than that, i'm lost. Can anyone help?

## #2 2006-04-21 22:31:17

Ricky
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### Re: Countably Infinite

Are you allowed to use that the rationals are countably infinite?  Are you allowed to use that the union between two countably infinite sets is countably infinite?  Are you allowed to use that a subset of a countably infinite set is countably infinite (or countably finite)?

If so, then the proofs become very easy.  If not, they are fairly difficult.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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