May 5th, 2012, 07:36 PM  #1 
Joined: May 2012 Posts: 3 Thanks: 0  Maximal element
Stuck on this problem , how do I prove Every finite set of real numbers has a maximal element by induction 
May 6th, 2012, 02:15 AM  #2 
Joined: Apr 2010 Posts: 215 Thanks: 0  Re: Maximal element
Start with a set of 1: , obviously is the maximal element. If we have a set of n: with a maximal element for some i, then obviously the set has a maximal element of . By induction every set of finite n has a maxmial element. 
May 6th, 2012, 08:24 AM  #3 
Joined: May 2012 Posts: 3 Thanks: 0  Re: Maximal element
thank sooooo much! 
May 6th, 2012, 11:21 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 14,533 Thanks: 564 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic  Re: Maximal element
Moved to Set Theory. Quote:
 
May 7th, 2012, 04:59 AM  #5 
Math Team Joined: Apr 2012 Posts: 1,547 Thanks: 14  Re: Maximal element
This is what I find so strange about some proofs. To me, the fact that a set of two integers, ie the the set of the maximal element from the inductively antecedent set and whatever element is being added to the next set, has to have a maximal element is indeed obvious, because it is immediately obvious that ANY finite set has a maximal element. The case of a two member set strikes me as just as special case of the more general proposition ostensibly being proven. FWIW! Probably not much!

May 7th, 2012, 05:02 AM  #6  
Math Team Joined: Apr 2012 Posts: 1,547 Thanks: 14  Re: Maximal element Quote:
 
May 7th, 2012, 05:37 AM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 14,533 Thanks: 564 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic  Re: Maximal element Quote:
Quote:
 
May 7th, 2012, 10:47 AM  #8 
Math Team Joined: Apr 2012 Posts: 1,547 Thanks: 14  Re: Maximal element
Thanks for the replies, CRG. On further refelctions, noting that >/= are binary relations, I realized that it's obvious that finite set of integers (which I incorrectly thought the question concerned) simply because it's easy to do the induction in one's head, almost automatically. So indeed, writing out the proof is a matter of making explicit and conscious the sort of automatic reasoning we do in simple cases. That's cool. As for the reals and their mysteries, I avoid them as much as I can, but you always eventually butt heads with them. I just know I have to tread extra, extra carefully when I find myself in the realm of the reals! 
May 7th, 2012, 11:04 AM  #9 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 14,533 Thanks: 564 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic  Re: Maximal element
I don't avoid the reals but I think it's wise to tread carefully with them. Most people don't, and you can eventually get into trouble that way.


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