What is an equivalence relation? For each of the following pairs $(X, \sim)$, determine whether or not $\sim$ is an equivalence relation on $X$ :

(i) $X=\mathbb{R}, x \sim y$ iff $x-y$ is an even integer;

(ii) $X=\mathbb{C} \backslash\{0\}, x \sim y$ iff $x \bar{y} \in \mathbb{R}$;

(iii) $X=\mathbb{C} \backslash\{0\}, x \sim y$ iff $x \bar{y} \in \mathbb{Z}$;

(iv) $X=\mathbb{Z} \backslash\{0\}, x \sim y$ iff $x^{2}-y^{2}$ is $\pm 1$ times a perfect square.