July 9th, 2019, 01:33 PM  #1 
Newbie Joined: Jul 2019 From: Brazil Posts: 1 Thanks: 0  Definition and sets
The following sentence, a set is infinite if and only if the set have a bijection with itself, the negation of the affirmation is true? A set is finite if only if there doesn't exist any bijection with itself? The idea is what is a better definition of a bijection in this case. (Sorry, guys, I don't know English very well.) Last edited by skipjack; July 9th, 2019 at 03:29 PM. 
July 10th, 2019, 06:27 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,786 Thanks: 708 
Any finite set can have a bijection with itself. For example the elements form a finite list. Reverse the order to form another list. The two lists are in one to one correspondence.

July 10th, 2019, 06:37 PM  #3  
Senior Member Joined: Aug 2012 Posts: 2,342 Thanks: 731  Quote:
The definition is: A set is infinite if and only if it has a bijection with a proper subset of itself. So for example the set of positive integers is infinite because it can be bijected with the even positive integers via the map $f(n) = 2n$. Since this is a DEFINITION, it's automatically reversible. A set is finite if (and only if) there is NO bijection with a proper subset of itself. For example there is certainly no bijection between a set of five elements and any set of 1, 2, 3, or 4 elements. By generalizing that idea, we can prove that each of the usual counting numbers 0, 1, 2, 3, ... are finite. Quote:
For completeness I'll mention that this definition of infinity, that a set has a bijection to a proper subset of itself, is technically called Dedekindinfinite. There are other definitions of infinite sets that may or may not be equivalent to Dedekindinfinite sets, depending on which axioms of set theory you choose. Last edited by Maschke; July 10th, 2019 at 06:40 PM.  
July 10th, 2019, 06:41 PM  #4 
Senior Member Joined: Aug 2012 Posts: 2,342 Thanks: 731  Do infinite sets have bijections with themselves? If they did, would that make them finite?

July 11th, 2019, 01:46 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,786 Thanks: 708  
July 11th, 2019, 02:35 PM  #6  
Senior Member Joined: Aug 2012 Posts: 2,342 Thanks: 731  Quote:
 
July 12th, 2019, 12:18 PM  #7  
Senior Member Joined: Aug 2012 Posts: 2,342 Thanks: 731  Quote:
And what is the relevance of that to the OP's question?  

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