My Math Forum Equivalence Relation question

 Number Theory Number Theory Math Forum

 October 30th, 2011, 07:05 PM #1 Member   Joined: Nov 2010 Posts: 78 Thanks: 0 Equivalence Relation question Hey all, need some help with the following question... Let ? be the relation on Z given by m ? n if and only if 4 divides m - n + 2. Prove or disprove the following: i) ? is reflexive So here we need to obviously show m ? m iff 4 divides m - m + 2. This would mean 4 divides 2, which is not true. So it is not reflexive, correct? ii) ? is symmetric This would mean m ? n implies n ? m....not sure where to go from here though.. if 4 divides m - n + 2 then 4 divides n - m + 2? iii) ? is transitive I would think here we have to show that if m ? n and n ? p then m ? p.... iv) Is ? an equivalence relation? It isn't, right? Since it does not satisfy all the above requirements..
October 30th, 2011, 07:39 PM   #2
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: Equivalence Relation question

Quote:
 Originally Posted by jstarks4444 Hey all, need some help with the following question... Let ? be the relation on Z given by m ? n if and only if 4 divides m - n + 2. Prove or disprove the following: i) ? is reflexive So here we need to obviously show m ? m iff 4 divides m - m + 2. This would mean 4 divides 2, which is not true. So it is not reflexive, correct?
That means that m is not equivalent to m for any m, so it is not reflexive unless the set is empty. Z is nonempty, so you're right, the relation is not reflexive.

Quote:
 Originally Posted by jstarks4444 ii) ? is symmetric This would mean m ? n implies n ? m....not sure where to go from here though.. if 4 divides m - n + 2 then 4 divides n - m + 2?
I prefer to write this m - n + 2 = 0 mod 4 and n - m + 2 = 0 mod 4. If you like, you can test each of the four possible values for m and see if the two statements have the same truth value in each case.

Quote:
 Originally Posted by jstarks4444 iv) Is ? an equivalence relation? It isn't, right? Since it does not satisfy all the above requirements..
Correct.

 Tags equivalence, question, relation

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post tom33 Algebra 3 January 17th, 2014 05:30 PM Taladhis Abstract Algebra 2 February 11th, 2013 09:20 AM page929 Abstract Algebra 1 October 11th, 2010 01:33 PM Dontlookback Abstract Algebra 1 April 20th, 2010 12:52 PM tinynerdi Abstract Algebra 1 January 11th, 2010 10:24 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top