For real numbers $x,y$, define $x\,R\,y$ iff $x-y+\sqrt{2}$ is irrational.
Then the relation $R$ is:
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For reflexivity: $xRx$ means $x-x+\sqrt{2}=\sqrt{2}$ (irrational) ⇒ true.
For symmetry: $xRy⇒x-y+\sqrt{2}$ irrational, but $y-x+\sqrt{2}=-(x-y)+\sqrt{2}$ may be rational. Not always true ⇒ not symmetric.
For transitivity: fails similarly.