If we have a relation $\sim$ on $\mathbb{Z}/6\mathbb{Z}\times (\mathbb{Z}/6\mathbb{Z}\setminus\{0\})$ so that $(w,x)\sim(y,z)$ if $wz=xy$, how is $\sim$ not an equivalence relation?
How is this not an equivalence relation?
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elementary-number-theory
modular-arithmetic
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1Obviously $(w,x)\sim(w,x)$ because $wx=wx$. Next, if $(w,x)\sim(y,z)$, we see that $wx=yz$ and so $zy=xw$ and we see that $(y,z)\sim(w,x)$. Finally, if $(w,x)\sim (y,z)$ and $(y,z)\sim(a,b)$ then (after some work) $wzb=xza$ or $wb=xa$ which means that $(w,x)\sim(a,b)$. But I am told this is not an equivalence relation. Any help? – 2011-11-17
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0What do you mean by $wz=xy$? – 2011-11-17
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0Think about involving divisors of zero – 2011-11-17
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0To give a counterexample proving that the relation is not transitive – 2011-11-17