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Rohit Kakkar July 23rd, 2014 07:29 AM

there is a relation defined by xRy iff x-y+(2)^(1/2) is an irrational number the R is:
1) an equivalence relation
2) symmetric
3) transitive
4) none of these

ZardoZ July 23rd, 2014 08:43 AM

xRx this means that $x-x+\sqrt{2}=\sqrt{2}$ is irrational, so the relation is reflexive.

xRy this means that $x-y+\sqrt{2}$ is irrational but $y-x+\sqrt{2}$ is irrational too so the relation is symmetric.

To prove that this relation is not transitive use $x=1$, $y=-\sqrt{2}$ and $z=\sqrt{2}$.

$1\mathcal{R}\left(-\sqrt{2}\right)$, because $1-(-\sqrt{2})+\sqrt{2}=1+2\sqrt{2}$ is irrational.

$\left(-\sqrt{2}\right)\mathcal{R}\left(\sqrt{2}\right)$ because $-\sqrt{2}-\sqrt{2}+\sqrt{2} =-\sqrt{2}$ which is irrational

but $1\cancel{\mathcal{R}}\sqrt{2}$, $1-\sqrt{2}+\sqrt{2}=1$ is not irrational.

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