Let G $=($Q$,,$ $,$ q$0,\{$qf$\})$ be an $-$NFA such that there are no transitions into q$0$ and no transitions out

of qf $.$ Formally represent the language accepted by each of the following modifications of G in terms of

L$($G$).$

a$.$ The $-$NFA constructed from G by adding and $-$transition from qf to q$0.$

b$.$ The $-$NFA constructed from G by adding and $-$transition from q$0$ to every state reachable from

q$0($along a path, whose labels may include symbols of $$ as well as $).$

c$.$ The $-$NFA constructed from G by adding and $-$transition to qf from every state that can reach

qf along some path.

d$.$ The automaton constructed from G by doing both $($b$)$ and $($c$).$