Give a counterexample to show that the following construction fails to prove $\text{Theorem 1.49,}$ the closure of the class of regular languages under the star operation$.$ Let $N_{1} = (Q_1, Σ, δ_1, q_1, F_1)$ recognize $A_{1}.$ Construct $N = (Q_1, Σ, δ, q_1, F)$ as
follows$.$ $N$ is supposed to recognize $A_{1}^{*}.$
- The states of N are the states of N1.
- The start state of N is the same as the start state of N1.
- $F = \{q_{1}\}\cup F_{1}.$ The accept states $F$ are the old accept states plus its start state.
- Define $δ$ so that for any $q ∈ Q_{1}$ and any $a ∈ Σ{ε},$
In other words, you must present a finite automaton, $N1,$ for which the constructed automaton $N$ does not recognize the star of $N_{1}^{’s}$ language.
