May 5th, 2010, 04:41 AM  #1 
Newbie Joined: Oct 2009 Posts: 14 Thanks: 0  Equivalence Classes
Hi, I have a task here that ask me to find the equilvalence classes. The task is written as following: Let X = {2, ..... , 12} ? N and define a relation R on X with nRm <=> m/n ? N Now let S be the least (or smallest) equivalence relation that expand R so that we have a relation 2S3. Find S by describing all the equivalence classes to S. The problem for me here is that I can't see how we have a relation 2S3 at all. And I can't describe the equivalence classes without knowing what the relation S is here. I do know that S is an equivalence relation since it is said in the task's text, which also mean that S is a relation that is reflexive, symmetric and transitive. But still, I just can't see what the S relation is, especially when we have a relation 2S3 somehow. So I would appreciate any kind of help or suggestions here. 
May 5th, 2010, 08:57 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Equivalence Classes
What does "expand R" mean? I suspect that if I understood that I could answer the question.

May 5th, 2010, 09:14 AM  #3 
Newbie Joined: Oct 2009 Posts: 14 Thanks: 0  Re: Equivalence Classes
Thanks for reply, If I understand the text correctly, "expand R" might mean that S is an equvalence relation on X so that nRm <=> nSm, or something like that. There are no clear definitions of the word "expand" in the text so that's what I guess is the meaning of "expand". Maybe you have a different opinion on what S expanding R might be? 
May 5th, 2010, 09:36 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Equivalence Classes Quote:
Perhaps it means that S is a superset of R, viewed as a collection of ordered pairs (m, n)?  
May 5th, 2010, 09:44 AM  #5 
Newbie Joined: Oct 2009 Posts: 14 Thanks: 0  Re: Equivalence Classes
Can you give me a concrete example on that please, i don't quiet get it yet. Because you might be right, and if so, that can be what I have missed to solve this task.

May 5th, 2010, 10:08 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Equivalence Classes Quote:
If that was the case, then since 1Rn for n = 1,2,...,12 then 1Sn for those n, and the same if we change variables: 1Rm and 1Sm. Then since S is an equivalence relation, nS1 and mS1. By transitivity, mS1 and 1Sn give mSn. So if that interpretation is right, then S is the trivial relation where everything is related to everything else.  
May 5th, 2010, 10:58 AM  #7 
Newbie Joined: Oct 2009 Posts: 14 Thanks: 0  Re: Equivalence Classes
I see... But the relation is based on X, which was X = {2, ..... , 12} ? N You explained it with 1, and 1 is not in X, so would that give another meaning if we only consider the numbers in X? And if what you are saying is correct, wouldn't it be existing a lot of equivalence classes which is what we are looking for to begin with? 
May 5th, 2010, 11:26 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Equivalence Classes Quote:
2Rn for n even, 2Rm for m even, thus 2Sn, 2Sm, nS2, and mS2. By transitivity on mS2 and 2Sn, all even numbers are related to each other by S. We can then glob all the evens together; I'll call them [2]. Similarly, all multiples of three are Srelated: mSn for m,n divisible by 3 (call such elements [3]). Since 6 is divisible by both 2 and 3, we have [2] = [3]. So right now we have [2], 5, 7, and 11. I'll let you finish this off. So far we know that 5S5, 7S7, 11S11, and mSn for any m,n in X other than 5, 7, and 11. But do any other pairs hold?  
May 5th, 2010, 01:21 PM  #9 
Newbie Joined: Oct 2009 Posts: 14 Thanks: 0  Re: Equivalence Classes
Ok, think things are starting to get more clear for me, I think. So if I am gonna write the equivalence classes, we would have a class for 2, which you called [2]? If we take that as example, an equivalence class where we pair numbers with 2 would be something like this? E(2) = {2, 4, 6, 8 ,10, 12} ? Since the pairings with 2 are the even numbers in X? 
May 5th, 2010, 02:18 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Equivalence Classes Quote:
Edit: Of course you don't need to use this notation, but I imagine you'll cover it soon and it's good to recognize it.  

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