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3. Let A be a set and R be a binary relation on A. For a positive integer n, we define an
R—cycle of length n to be a sequence a1, a2, . . . , an of n distinct elements of A such that for
all 'i = 1,2,...,n— 1 we have (a§,ai+1)e R and (amal) E R. (a) (5 points) Draw a diagram of a binary relation R on the set A = {a, b, c, d, e} which has
an R—cycle of length 4. (b) (20 points) Prove by mathematical induction on n that for all natural numbers 'n. 2 2, if
R is a relation on a set A and ($1,122,. . . , a,” is an R—cycle then R is not a partial order.
Please state and label the predicate PC”), the base case(s), and the statement of the
inductive step.

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