October 11th, 2011, 03:27 AM  #1 
Member Joined: Sep 2011 Posts: 97 Thanks: 0  find binary relation
let Z be integer Find a binary relation on Z251 that is reflexive, symmetric, antisymmetric and transitive. what is the meaning of Z251? 
October 11th, 2011, 05:20 AM  #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: find binary relation
Integers mod 251, I imagine.

October 11th, 2011, 07:38 AM  #3 
Member Joined: Sep 2011 Posts: 97 Thanks: 0  Re: find binary relation
then isn't the binary relation would be {0,0}? since that's the only one that's reflexive

October 11th, 2011, 08:28 AM  #4 
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: find binary relation
{0, 0} isn't a relation at all: relations are sets of ordered pairs, and 0 isn't an ordered pair. If you mean {(0, 0)} then that's not reflexive over Z_251 since, for example, 1 is not related to 1. 
October 11th, 2011, 07:48 PM  #5 
Member Joined: Sep 2011 Posts: 97 Thanks: 0  Re: find binary relation
so it would be soemthing like A = A X A?

October 11th, 2011, 09:18 PM  #6 
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: find binary relation
That's not antisymmetric. I suggest starting with the empty set and adding ordered pairs only as required by the axioms. Start with reflexivity. 
October 12th, 2011, 03:10 AM  #7 
Member Joined: Sep 2011 Posts: 97 Thanks: 0  Re: find binary relation
ok, i get the idea, but i don't know how to find a relation that is equivalence, and anti symmetry at the same time

October 12th, 2011, 04:46 AM  #8 
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: find binary relation
I gave my suggestion on how to do it. Don't forget to check that, after meeting some requirements, you check earlier properties to make sure they still apply.

October 12th, 2011, 05:38 AM  #9 
Member Joined: Sep 2011 Posts: 97 Thanks: 0  Re: find binary relation
ok, so is it something like: ?a, b, c ? {Z = 0 mod 251 a = b = c ^ (aRb > bRc) ^ (aRb ^ bRc > aRc)} i basically just merge those together, i don't knwo how to write it in binary relation 
October 12th, 2011, 05:56 AM  #10 
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: find binary relation
I don't know what that statement means. I'm just suggesting starting with the empty relation and adding in all pairs required by reflexivity (how many pairs? what is the relation now?), then adding in those required by symmetry, those by antisymmetry, then those required by transitivity. 1. { } 2. ? 3. ? 4. ? 5. ? Once you do this, repeat the earlier steps until the relation stays the same through a whole cycle. (If you don't do that you might find that making the relation transitive has caused it to lose symmetry or something like that.) 

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