(c) Consider the equivalence relation R on the set X = {1,2,3,4,5,6,7} given by R = {(1,1),(1,6), (2, 2), (2,5), (2,7), (3, 3), (3,4), (4,3), (4,4), (5,2), (5,5), (5,7), (6, 1), (6,6), (7,2), (7,5), (7,7)} Write out the equivalence classes of R and find the partition of X induced by R (you are not required to check that R is an equivalence relation). (d) Prove the following statement. Let R be an equivalence relation on X and let x, y € X. If (x, y) = R, then the equivalence classes R(x) and R(y) are equal. In symbols, (x,y) ≤R ⇒ R(x) = R(y).

- then 10. ) (a) Find all partitions of the set {3,5). (b) For each partition P that you found in (a), find the corresponding equivalence relation ~p.

If R = {(1, 2), (1, 4), (2, 3). (3, 1), (4, 2)}, what is the symmetric closure of R? O {(1, 2), (2, 1), (1, 4), (4, 1), (2, 3), (3, 2), (3, 1), (1, 3), (4, 2), (2, 4)} O {(1, 1), (1, 2), (1, 4), (2, 2), (2, 3), (3, 1). (3, 3), (4, 2), (4, 4)} O {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)} O {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1). (3, 2). (3, 3), (3, 4), (4. 1), (4, 2), (4, 3), (4, 4)}

Which of the following relations is the a partial ordering on the set A = {3,7,9}? {(3,3),(7,7), (3,9),(9,7), (9,9)} O {(3,3),(7,7),(7,9),(9,9), (9,7)} O 3,3,7,7,9,9,3,7,7,9,3,9 O None of the others O {(3,3),(3,7),(7,9),(7,7),(9,9)}