1. (a) (i) In Z, let a ~ bif a = b mod 3. Show that is an equivalence relation on Z and determine the associated equivalence classes. (ii) Let ~ be an equivalence relation on a set S. Show that the equiva- lence classes form a partition of S. (iii) Let H be a subgroup of a finite group G. For r, YEG, let ~ be the relation defined by x~ y if x € Hy. Show that is an equivalence relation and determine the equivalence class [2] for~