Chapter 13.3, Problem 58E

To determine

(a)

To show that for every nonnegative integer $k,R_k$ is an equivalence relation on $S$.

To determine

(b)

To show that $R_{+}$ is an equivalence relation on $S$.

To determine

(c)

To prove that $s\text{ and }t$ are also $k-1$ equivalent.

To determine

(d)

To show that equivalence classes of $R_k$ are the refinement of the equivalence classes of $R_{k-1}$.

To determine

(e)

To prove that if $s\text{ and }t$ are also $k$ equivalent, then they are also $*-$ equivalent.

To determine

(f)

To prove that all the states in the $R_{*}-$ equivalence classes are either final states or non-final states.

To determine

(g)

To show that $f(s,a)\text{ and }f(t,a)$ are also $R*-$ equivalent for all $a\in S$.