$14\mathrm{.}10$ Eliminating a State

Let D be a DFA with alphabet $\{$a$,$ b$\},$ state set $\{$p$,$ q$,$ r$\},$ start state p$,$ and final state set $\{$q$,$ r$\}.$ The transition function has $\backslash$delta $($p$,$ a$)=$ q$,\backslash$delta $($p$,$ b$)=$ r$,\backslash$delta $($q$,$ a$)=$ r$,\backslash$delta $($q$,$ b$)=$ p$,\backslash$delta $($r$,$ a$)=$ p$,$ and $\backslash$delta $($r$,$ b$)=$ q$.$ Then, after making D into an r$.$e$.-$NFA according to our rules, if we first remove state p$,$ we will create four new transitions. $($If we change an existing transition by adding a new term to its regular expression, this counts as a new transition.$)$