Define an equivalence relation. Which of the following is an equivalence relation on the set of non-zero complex numbers? Justify your answers. (i) $x \sim y$ if $|x-y|^{2}<|x|^{2}+|y|^{2}$. (ii) $x \sim y$ if $|x+y|=|x|+|y|$. (iii) $x \sim y$ if $\left|\frac{x}{y^{n}}\right|$ is rational for some integer $n \geqslant 1$. (iv) $x \sim y$ if $\left|x^{3}-x\right|=\left|y^{3}-y\right|$.