Problem 1 (2 points each)

Given X = {1,2,3,4,5,6,7,8,9}. Determine which of the following are partitions of X. If it is not a partition, why not? a. { {1, 3, 6}, {2, 8}, {5, 7, 9} } b. { {1, 5, 7}, {2, 4, 8, 9}, {3, 5, 6} } c. { {2, 4, 5, 8}, {1, 9}, {3, 6, 7} }

d. { {1, 3}, {5, 7}, {9}, {2, 4, 6, 8} }

e. { {6, 5, 4, 3}, {1, 7}, {8, 9} }

Problem 2 (3 points each)

Given A = {a, b, c, d}, list the tuple(s) that are missing from the following relations to fulfill the property listed.

a. R1 = {(a, a), (a, b), (c, b), (d, b), (b, b), (d, c), (d, d)} - reflexive

b. R2 = {(a, a), (b, a), (c, a), (d, d), (d, c), (a, b)} - symmetric

c. R3 = {(a, b), (a, d), (b, c), (d, b)} - transitive

d. R4 = {(a, a), (b, b), (c, d), (d, d)} - antisymmetric

e. R5 = {(a, a), (c, c), (d, d), (a, b)} - equivalence