Consider the state diagram of the following DFA.

$1.$ Give the formal definition of the DFA.

$2.$

$($a$)$ List $2$ strings the DFA accepts

$($b$)$ List $2$ strings the DFA rejects

$($c$)$ What is the language of this DFA $($the set of accepted strings$)?$

$3.$Design a DFA for each of the following languages. The alphabet is $\{$x$,$y$\}$

$$ L$1=\{$w $|$ w has substring xyyx$\}$

$$ L$2=\{$w $|$ w doest not have substring xyyx$\}$

$($Hint: Do not start this DFA from scratch, how can you modify the previous DFA$).$

$$ L$3=\{$w $|$ w has exactly $2$ yrs$\}$

$$ L$4=\{$w $|$ w has at least $2$ yrs$\}$