Answer to Question #295346 in Discrete Mathematics for Zakwan

Question

2.

Find the transitive closures of these relations on {1, 2, 3, 4}.

a) {(1, 2), (2,1), (2,3), (3,4), (4,1)}

b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)}

c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)}

d) {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)}

3.

Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is

a) reflexive and transitive.

b) symmetric and transitive.

c) reflexive, symmetric, and transitive.

4.

Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack.

a) {(0, 0), (1, 1), (2, 2), (3, 3)}

b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}

c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}

d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2),(3, 3)}

e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0),(2, 2), (3, 3)}