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August 25th, 2010, 02:15 AM   #1
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Equivalence Relations

OK, I'm asked if the following relation is an equivalence relation:

S=Z, and a~b <=> ab=0. (Z is the integers)

For it to be an equivalence relation it must exhibit,reflexivity: a~a.
If a=0 then aa=0 and hence a~a. But if a=x, x>0 then a~/~a (a does not relate to a). So by providing an example where a~a does this make the relation reflexive? Or is it not reflexive because I have an example where a does not relate to a?

The relation is symmetric but not transitive so is not an equivalence relation but my issue is with the reflexivity of the relation.
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August 25th, 2010, 03:13 AM   #2
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Re: Equivalence Relations

Just looking at this problem again,

In terms of transitivity an example where ab=0 and bc=0 and ac=0 exists where a=0 or c=0 (or both). But if a and c are both not 0 but b=0 then ab=0 and bc=0 but ac=/=0 meaning that the relation is not transitive. It is this counterexample of transitivity that tells me the relation is not transitive. So I suppose by following that same reasoning the relation is not reflexive either since I provided a counterexample there too.

Am I reasoning correctly here or am I on the wrong track?
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August 25th, 2010, 03:26 AM   #3
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Re: Equivalence Relations

I think I have it,

In terms of transitivity: b=0, a and c non-zero. ab=0, bc=0 both satisfy the relation and since ac=/=0 then it is not transitive.

In terms of reflexivity: a=0 then aa=0, satisfies the relation. a=/=0 then aa=/=0, does not satisfy the relation and hence is not to be considered. Therefore the relation is reflexive!

...right?
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August 25th, 2010, 05:03 AM   #4
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Re: Equivalence Relations

No.

If even one element of your set isn't related to itself, the relation isn't reflexive.

Your relation is symmetric (by the symmetry of multiplication), but neither reflexive nor transitive.
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August 25th, 2010, 05:15 AM   #5
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Re: Equivalence Relations

Thankyou
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