$(71)$ Recall that a regular grammar is a CFG $G=(V,,R,S)$ in which all rules are of the form

$($I$)BcA,($II$)BA,($III$)Bc,$ and $($IV$)B,$ where $,$BinV and cin$.$ What

are the languages of regular grammars having

$($a$)$ rules of types $($I$),($II$),$ and $($III$),$ but none of type $($IV$)?$

$($b$)$ rules of types $($I$),($II$),$ and $($IV$),$ but none of type $($III$)?$

$($c$)$ rules of types $($I$),($III$),$ and $($IV$),$ but none of type $($II$)?$

$($d$)$ rules of types $($II$),($III$),$ and $($IV$),$ but none of type $($I$)?$

$($e$)$ rules of types $($I$)$ and $($II$),$ but none of type $($III$)$ or $($IV$)?$

$($f$)$ rules of types $($I$)$ and $($III$),$ but none of type $($II$)$ or $($IV$)?$

$($g$)$ rules of types $($I$)$ and $($IV$),$ but none of type $($II$)$ or $($III$)?$

Determine the answer for each, with a justification that you are correct.

$(72)$ Another type of rule, type $($I$\text{'}),$ is those of the form $BAc.$

$($a$)$ My friend Reggie got types $($II$),($III$),$ and $($IV$)$ correct, but used type $($I$\text{'})$ instead of

type $($I$).$ Reggie insists that it is ok$,$ because he still gets exactly the regular languages.

Is he correct? Explain carefully.

$($b$)$ My friend Fred uses rules of types $($I$),($II$),($III$),$ and $($IV$)$ as well as rules of type $($I$\text{'}).$

Fred insists that it is ok$,$ because he still gets exactly the regular languages. Is he

correct? Explain carefully.