Problem $1$: $(25$ points$)$ Consider the following regular languages and answer the following questions,

$A=0^{**}(0+1{)}^{**},B=(0+10**{)}^{**}.$

Is AsubeB $?$ Why?

Is BsubeA $?$ Why?

$01101$

Compute the minimal finite automaton describing $B.$

Problem $2$: $(25$ points$)$ Let L$.$ be the formal language of all strings that start and end with the same symibol.

Construct a DFA describing $L.$

Can you construct an NFA describing $L$ having strictly a smaller number of states than your DFA for the first part?

Provide the minimal DEA that describes $L^R.$

Problem $3$: $(25$ points$)$ Which one of the following formal languages are regular?

$\{0^n{1}^{n+1}{2}^{n+2}|ninN\}sube\{0,1,2{\}}^{**}.$

The set of all strings in $\{0,1{\}}^{**}$ not having any substring of the form $11.$

Problem $4$: $(30$ points$)$ Consider the following $-$NFA describing a language $L_{?}$ and construct a minimal DFA describing $L_{?}^R($explicitly describe your method and write down the details of your computations$).$

Figure $1$: The $-$NFA of Problem $4.$