Homework: Exercise 10 : a b d c Exercise 11 : a) R2 contains the pair (a,b) if R contains the pair (b,b) b) R3 contains the pair (a,b) if R2 contains the pair (b,b) R2 contains the pair (b,b) if R contains the pair (b,a) So, R3 contains the pair (a,b) if R contains the pair (b,a) Exercise 12 : a) Reflexive closure of R is {(0, 1); (1, 1); (1, 2); (2, 0); (2, 2); (3, 0); (0, 0); (3, 3)} b) Symmetric closure of R is {(0, 1); (1, 1); (1, 2); (2, 0); (2, 2); (3, 0); (1, 0); (2, 1); (0, 2); (0, 3)} c) Transitive closure of R is {(0, 1); (1, 1); (1, 2); (2, 0); (2, 2); (3, 0); (0, 2); (1, 0); (2, 1); (3, 1); (3, 2); (0, 0)} Exercise 13 : The symmetric closure of R is {(a, a) | a ∈ Z} Exercise 14 : a) {(1, 2); (1, 4); (3, 3); (4, 1); (4, 2); (4, 4); (1, 1); (2, 2)} b) {(1, 2); (1, 4); (3, 3); (4, 1); (4, 2); (4, 4); (2, 1); (2, 4)} c) {(1, 2); (1, 4); (3, 3); (4, 1); (4, 2); (4, 4); (2, 1); (2, 4); (1, 1); (2, 2)} Exercise 15 : a) This relation is equivalence b) This relation isn’t equivalence because it isn’t transitive and reflexive c) This relation is equivalence d) This relation isn’t equivalence because it isn’t transitive Exercise 16 : a)These collections of subsets aren’t partitions of {1,2,3,4,5,6} b)These collections of subsets are partitions of {1,2,3,4,5,6} c)These collections of subsets are partitions of {1,2,3,4,5,6} d)These collections of subsets aren’t partitions of {1,2,3,4,5,6} Exercise 17 : a) {(0, 0); (1, 1); (2, 2); (1, 2); (2, 1); (3, 3); (4, 4); (5, 5); (3, 4); (4, 3); (3, 5); (5, 3); (4, 5); (5, 4)} b) {(0, 0); (1, 1); (0, 1); (1, 0); (2, 2); (3, 3); (2, 3); (3, 2); (4, 4); (5, 5); (4, 5); (5, 4)} Exercise 18 : a) Yes R is a equivalence relation b) [1]R is set of integers c) [ 12 ]R = {a + 12 | a ∈ Z} Exercise 19 : a) Yes R is an equivalence relation b) [(1, 3)] = {(1, 3); (3, 1); (2, 2)} [(2, 4)] = {(1, 5); (5, 1); (2, 4); (4, 2); (3, 3)} [(1, 1)] = {(1, 1)} c) ?? Exercise 20 : This relation is a partial ordering This relation isn’t a partial ordering because it isn’t antisymmetric and transitive This relation is a partial ordering This relation is a partial ordering This relation isn’t a partial ordering because it isn’t antisymmetric and transitive Exercise 21 : a) (Z, =) is a poset b) (Z, 6=) isn’t a poset c) (Z, >) is a poset d) (Z, -) is ??