Use the construction in the proof of Theorem to give the state diagrams of NFAs recognizing the star of the languages $\{$w$|$w contains at least two $0$s and at most one $1\}.$

Here is the theorem :

THEOREM $1\mathrm{.}49$

The class of regular languages is closed under the star operation.

PROOF IDEA We have a regular language $A_1$ and want to prove that $A_1^{**}$ also is regular. We take an NFA $N_1$ for $A_1$ and modify it to recognize $A_1^{**},$ as shown in the following figure. The resulting NFA $N$ will accept its input whenever it can be broken into several pieces and $N_1$ accepts each piece.

We can construct $N$ like $N_1$ with additional $$ arrows returning to the start state from the accept states. This way, when processing gets to the end of a piece that $N_1$ accepts, the machine $N$ has the option of jumping back to the start state to try to read another piece that $N_1$ accepts. In addition, we must modify $N$ so that it accepts $,$ which always is a member of $A_1^{**}.$ One $($slightly bad$)$ idea is simply to add the start state to the set of accept states. This approach certainly adds $$ to the recognized language, but it may also add other, undesired strings. Exercise $1\mathrm{.}15$ asks for an example of the failure of this idea. The way to fix it is to add a new start state, which also is an accept state, and which has an $$ arrow to the old start state. This solution has the desired effect of adding $$ to the language without adding anything else.