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Relationthere 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 |

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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