(a) Let R be the relation on A = {1, 2, 3, 4, 5, 6, 7}, where the directed graph associated with R consists of the two components, each a directed cycle, shown in Fig. 7.14. Find the smallest integer n > 1, such that Rn = R. What is the smallest value of n > 1 for which the graph of Rn contains some loops? Does it ever happen that the graph of Rn consists of only loops?

(b) Answer the same questions from part (a) for the relation R on A = {1, 2, 3, . . . , 9, 10}, if the directed graph associated with R is as shown in Fig. 7.15.

(c) Do the results in parts (a) and (b) indicate anything in general?

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4 Figure 7.14 6 10 9 Figure 715