1. List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4, 5} to B = {3,4,5,6}, where (a,b) R if and only if a) a > b b) b = (a 2)2 + 2 c) the standard English name (e.g. O's name is zero, l's name is one, etc.) of a and b have the same length in terms of the number of letters 2. Consider the relation on A = {1,2,3,4} defined by R= = {(1,1), (1, 2), (1,3), (2, 1), (2, 4), (3, 1), (3, 3), (4, 2), (4,4)}. Answer each of the following questions and provide clear justification of your answer. a) Is R reflexive? b) Is R symmetric? c) Is R antisymmetric? d) Is R transitive? 3. Let R be a relation on Z * 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 = {9192 - 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 v1 +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 {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