Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 4 / 2C / iii / Solution

This is not an equivalence relation. For $x=1/2$, $x\overline{x}=1/4\notin \mathbb{Z}$, so reflexivity fails. It is symmetric, because $y\overline{x}$ is the conjugate of $x\overline{y}$, and an integer is real. It also fails transitivity: take $x=z=1/2$ and $y=2$. Then $x\overline{y}=y\overline{z}=1$ are integers but $x\overline{z}=1/4$ is not. A single failure would suffice, but these examples identify exactly which properties break.