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If you have an infinite set $X$, how do you prove that $|$ $X$ $\times$ $X$ $|$=$|$ $X$ $|$. I believe AC is needed but I don't quite see how to write the proof. Thanks.

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While this is a great question that everyone should see at some point of their mathematical career, I somewhat doubt this is close enough to research level to warrant a post here. As a hint, note two sets have the same cardinality if you can find a bijection between them. –  mathic Dec 11 '10 at 10:09
Please read the FAQ. The question is not appropriate for MO. Just two remarks: 1) For infinite cardinals $\kappa$ there is a bijection $\kappa \to \kappa \times \kappa$ which is given by the type of a well-ordering of the product. This works without AC. 2) The statement you cited is equivalent to AC in ZF. –  Martin Brandenburg Dec 11 '10 at 10:11
Yes, fair enough. I am a maths research student (though not in set theory) and I just wanted some clarification on this point made in a paper I'm reading. Since it's not anywhere near the area I'm working on I didn't realise it was not a high enough level to post here. Guess I'll go somewhere else! Sorry! –  G-Unit Dec 11 '10 at 10:21
No problem. Check out math.stackexchange.com. –  Martin Brandenburg Dec 11 '10 at 11:16
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closed as too localized by Martin Brandenburg, Chandan Singh Dalawat, Robin Chapman, Simon Thomas, Ryan Budney Dec 11 '10 at 14:33

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