q7 and q8

d) irreflexive? e) reflexive and symmetric? f) neither reflexive nor irreflexive? 5. Let R be a relation that is reflexive and transitive. Prove that Rn R for all positive integers n. 6. Let R1 and R2 be the "divides" and "is a multiple of" relations on the set of all positive integers, respectively. That is, R1=(a, b) | a divides b)and R2 ={(a, b) I a is a multiple of b). Find a) R1 U R2 b) R1nR2. c) R1-R2. d) R2-R1. e) RR. Let p(n) denote the number of different equivalence relations on a set with n elements (in other words the number of partitions of a set with n elements). Show that p(n) satisfies the recurrence relation p(n)= C(n-1, j)p(n-j-1) and the initial condition p(0) = 1. 7. 8. Find (S100 k! ) mod 12