I'm asked to prove that the relation $R$ on $\mathbb{C},$ $xRy \iff x^3-yx^2 = y^3-xy^2$ is an equivalence relation. It's easily shown it's reflexive and symmetric, but I'm having problems with its transitivity. Any tips?
Is $x^3-yx^2 = y^3-xy^2$ transitive?
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2What is the domain of $R$? – 2011-10-30
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1@Chris Sorry. It's the set of complex numbers. – 2011-10-30
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0With a bit of rewriting you can show that $xRy$ if and only if $(x-y)^2 (x+y) = 0$. This should help. – 2011-10-30
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1@Michael Lugo: It rewrites to $(x^2+y^2)(x-y)=0$. I made the same mistake. – 2011-10-30
1 Answers
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The claim is false. We have $xRy \iff x^3-yx^2 = y^3-xy^2 \iff x^2(x-y)=y^2(y-x)$ $\iff (x^2+y^2)(x-y)=0 \iff x^2+y^2=0\ \mathrm{or}\ x=y$. So for example $1Ri$ and $iR{-1}$ but $1 \not{R} {-1}$.