Problem 3. (20 points) Let R be an equivalence relation over set A. The set of all elements that are related to element a E A is called the equivalence class of a with respect to R, and the equivalence class is denoted [al In other words, [a]R -{b: (a,b) E R). A) (5 points) Let R [(a, b): a, b E N, a mod 7b mod 7) (the remainders of both numbers when divided by 7 are equa i) Describe the sets [O and[1 i) What is the equivalence class of 14? iii) What is the equivalence class of 75? iv) How many distinct equivalence classes of the natural numbers are created by R? v) Are the equivalence classes pairwise disjoint? B) (15 points) Let R be an equivalence relation over set A. Let a and b be any two elements in A i) Prove that (a, b) E R[a [b]. (Hint: show that [a] S b and [b] S [al.) ii) Prove that [a]-[b-[a] n [b] # . iii) Prove that [a] n [b] # -, (a, b) E R. This establishes that every equivalence relation partitions the set it is defined over into a set of pairwise disjoint equivalence classes. Every element of the set belongs to exactly one equivalence class