# Check if ρ is an equivalence relation

Check if $$xρy \iff (x^2-y^2)(x^2y^2 - 1) = 1$$ is an equivalence relation.

I know that for it to be an equivalence relation, a relation must have these properties: reflexivity, symmetry and transitivity.

For reflexivity, I tried proving that the left side equals 1, but I failed.

Can someone help me? I really have no idea how these are done.

• $x\rho x$ does not hold. – Dietrich Burde Nov 14 '15 at 12:30

It means that $x \rho x$. That translates to $$\left(x^2 - x^2\right)\left(x^2x^2 - 1\right) = 0 \neq 1$$
So $\rho$ is not an equivalence relation. We don't need to check the other properties.