Explanation Given: set of n -equivalence classes of states of M. Proof: We know that the equivalence classes of R k are a refinement of the equivalence classes of R k − 1 for each positive integer k. The equivalence classes are finite sets, and finite sets cannot be refined indefinitely (the most refined they can be is for each equivalence class to contain just one state). Therefore this sequence of refinements must stabilize and remain unchanged from some point onward. It remains to show that as soon as we have R n = R n + 1 , then R n = R m for all m > n, from which it follows that R n = R * , and so the equivalence classes for these two relations will be the same. By induction, it suffices to show that if R n = R n + 1 , then R n + l = R n + 2 ...

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Chapter 13.3, Problem 59E

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To show:

There is a nonnegative integer n such that the set of n -equivalence classes of states of M is the same as the set of ( n + 1)-equivalence classes of states of M.

Then show for this integer n, the set of n -equivalence classes of states of M equals the set of *-equivalence classes of states of M.

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Chapter 13.3, Problem 58E

Chapter 13.3, Problem 60E

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DISCRETE MATHEMATICS LOOSELEAF

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There is a nonnegative integer n such that the set of n -equivalence classes of states of M is the same as the set of ( n + 1)-equivalence classes of states of M. Then show for this integer n, the set of n -equivalence classes of states of M equals the set of *-equivalence classes of states of M.

Solution Summary: The author explains that the equivalence classes of R k are a refinement of the

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