Talk:Paradoxical set

Latest comment: 12 years ago by Max Longint in topic Natural numbers example

Natural numbers example

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For the example given for the natural numbers, the given function, as of may 26, 2009, (n maps to n/2 or (n-1)/2 depending on parity) is not injective, so it is not clear how it generates a group. —Preceding unsigned comment added by 204.60.65.162 (talk) 03:31, 27 May 2009 (UTC)Reply

Therefore I removed the following section from the article here, sice it is a chat, rather than encyclopedic text. And unreferenced, too. Max Longint (talk) 00:53, 17 December 2011 (UTC)Reply

Natural numbers

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An example of a paradoxical set is the natural numbers. They are paradoxical with respect to the group of functions   generated by the natural function  :

 

Split the natural numbers into the odds and the evens. The function   maps both sets onto the whole of  . Since only finitely many functions were needed, the naturals are  -paradoxical. This example does not work: the natural function is not injective, and as such has no well defined inverse. Therefore it does not generate a group of functions.


So, what's the deal here? Max Longint (talk) 00:53, 17 December 2011 (UTC)Reply

Assessment comment

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The comment(s) below were originally left at Talk:Paradoxical set/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The example with the   is not good.

The function   specified is not one-to-one and hence not invertible, and so it is not clear what 'the group of functions   generated by the natural function  ' means. (Also, a little quibble, the specified function doesn't map the odd numbers onto   in the more frequent definition of   that contains zero.)

Perhaps one should say that the 'universe set' is the real numbers  , and the group   is the group of bijections of   generated by the two invertible functions

 

and   .

Last edited at 00:51, 7 April 2009 (UTC). Substituted at 02:26, 5 May 2016 (UTC)