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September 21st, 2010, 07:07 PM   #1
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countable, uncountable

so we are told that if E1, E2, ... are finite sets, and
E:=E1 x E2 x ... :={(x1, x2, ...) : xj in Ej for all j in N}
then E is countable.
We are to prove this or show a counterexample to disprove it.


So in the back of the book it tells us that this is false.

I am totally confused about this problem, and how to go about it.
Any help would be appreciated.
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September 21st, 2010, 07:38 PM   #2
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Re: countable, uncountable

Can you prove that the real numbers between 0 and 1 are uncountable? Can you prove that the real numbers between 0 and 1, expressed in binary, are uncountable? Can you take it from there?
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September 22nd, 2010, 05:38 AM   #3
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Re: countable, uncountable

Yes, I can prove that numbers from 0 to 1 are uncountable. However, I do not see how this can help me.
Isnt the set [0,1] infinite?
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September 22nd, 2010, 05:46 AM   #4
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Re: countable, uncountable

Quote:
Originally Posted by wannabe1
Isnt the set [0,1] infinite?
Yes, but more relevant is the fact that the set {0, 1} is finite.
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September 22nd, 2010, 02:41 PM   #5
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Re: countable, uncountable

hmm, so {0,1} is finite, so since it is an infinite series of finite sets times each other, does that mean like cantor's proof to prove (0,1) is infinite, we are able to prove it the same way saying suppose each element y1=x11x12x13........, y2=..........,... and so on
then show that there exists an element which is not in there so therefore we have a cotnradiction and our set is uncountable?
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September 22nd, 2010, 03:18 PM   #6
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Re: countable, uncountable

Right. (Or just show that there's a bijection between one such sequence and the real numbers between 0 and 1 and call that a proof, since you know the reals are uncountable.)
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