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October 30th, 2011, 07:05 PM   #1
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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..
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October 30th, 2011, 07:39 PM   #2
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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.
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