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: 14,187 Thanks: 481 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: 14,187 Thanks: 481 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: 14,187 Thanks: 481 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.


Tags 
element, maximal 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Maximal Consistent  lain86  Applied Math  0  April 2nd, 2013 01:14 PM 
maximal ideal  cummings123  Abstract Algebra  1  February 27th, 2013 06:06 AM 
maximal N  mathLover  Algebra  0  April 17th, 2012 02:25 AM 
Maximal Ideal  julien  Abstract Algebra  1  November 19th, 2006 06:56 PM 
Maximal element  ag05  Number Theory  3  January 1st, 1970 12:00 AM 