3. Let R be a relation on Zx Z such that (a,b)R(c,d) if and only if a - c= 5(b- d). Prove that R is an equivalence relation. 4. Let Q = {q192 - 9n | n > 3 and qi {0,1,2,3} for i = 1, 2, ,n}, i.e. the set of quarternary strings of length three or more. Let R be a relation on Q such that vRw if and only if vi+v2 = wi+w2 and V3 = W3 (with the convention that X; represents the jth digit of the string x). Prove that R is an equivalence relation. - 5. Let Sn {dd2 ...dn | di E {0, 1, 2} for i 1, 2,..., n}, i.e., the set of ternary strings of length n. List (in full) the equivalence classes for each of the following equivalence relations on the given set. a) On S2, where a R b if and only if the last digit of a equals the last digit of b. b) On S3, where a R b if and only if the sum of the digits in a equals the sum of the digits in b. c) On S3, where a R b if and only if a is either b written in forwards order or b written in reverse order