Answer
A) M is a finite-state automaton, then the quotient automaton M recognizes the same language as M.
B)M is a finite-state automaton with the property that for every state s of M there is a string x ∈ I ∗ such that f (s0, x) = s, then the quotient automaton M has the minimum number of states of any finite-state automaton equivalent to M.
Work Step by Step
a) By the way the machine M was constructed, a string will drive M from the
start state to a final state if and only if that string drives M from
the start state to a final state.
b) For a proof of this theorem, see a source such as Introduction to Automata Theory, Languages, and Computation (2nd Edition) by John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman (Addison-Wesley, 2000).
