Construct DFAs that recognize the following languages over the alphabet $\{$a$,$ b$\}$:

$($a$)\{$w $|$; w contains exactly two b$$s$,$ or an even number of a$$s$\}$

$($b$)\{$w $|$ w contains as substring ababb and$/$or bbb $\}.$

$($c$)\{$w $|$ w is any string except the two strings abba and aba$\}$

Draw the state diagram for all; add a formal description of the DFA for $($b$),$ and add a proof that

your DFA recognizes exactly the specified language for $($c$)).$ To do this, you will need to prove that

$(1)$ your DFA accepts all strings in the language and $(2)$ any string accepted is in the language