Give a counterexample to show that the following construction fails to prove Theorem 1.49, the closure of the class of regular languages under the star operation.7 Let N1 = (Q1,Σ, δ, q1, F1) recognize A1. Construct N = (Q1, Σ, δ1, q1, F1) as follows. N is supposed to recognize A*1 .
a. The states of N are the states of N1.
b. The start state of N is the same as the start state of N1.
c. F = {q1} [ F1.
The accept states F are the old accept states plus its start state.
d. Define δ so that for any q ∈ Q1 and any α ∈ Σε,
Show this construction graphically, as in Figure 1.50.
Theorem 1.49
The class of regular languages is closed under the star operation.
Figure 1.50
