\section{Constructing minimal state DFA}

\begin{tabular}{|*{7}{c|}}
                               \cline{1-1}
  $\underline{A}$                     \\ \cline{1-2}
  $\underline{B}$ & $x_1$                 \\ \cline{1-3}
  C & $x_1$ & $x_1$           \\ \cline{1-4}
  D & $x_1$ & $x_1$ & $x_2$         \\ \cline{1-5}
  $\underline{E}$ & $x_1$ & $x_1$ & $x_1$ & $x_1$    \\ \cline{1-6}
  $\emptyset$ & $x_1$ & $x_1$ & $x_2$ & $x_2$ & $x_1$ \\ \hline
    & $\underline{A}$ & $\underline{B}$ & C & D & $\underline{E}$ & $\emptyset$ \\ \hline
\end{tabular}

\begin{enumerate}
  \item First I marked (with $x_1$) all the pairs in which at least one of them were final state:
  \[ ([\underline{A}, \emptyset], [\underline{A}, \underline{E}], [\underline{A}, D], [\underline{A}, C], [\underline{A}, \underline{B}]) \]
    \[ ([\underline{B}, \emptyset], ([\underline{B}, \underline{E}], ([\underline{B}, D], ([\underline{B}, C]) \]
    \[ ([\underline{E}, \emptyset], [\underline{E}, C], [\underline{E}, D])\]

  \item We are left with the pairs:
  \[ ([\emptyset , C], [\emptyset , D], [D , C] ) \]
  For pair: $ [\emptyset , C] $ C goes to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$
  For pair: $ [\emptyset , D] $ D goes to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$
  For pair: $ [D , C] $ both C and D go to final state $\underline{E}$ on transition 'b' therefore we mark it with $x_2$
\end{enumerate}

No states could be minimized! Therefore our final minimal state DFA looks like this:

\begin{figure}[!htb]
\centering
\begin{tikzpicture} [node distance = 2cm, on grid, auto]

\node (q0) [state, accepting, initial, initial text = {}] {A};
\node (q1) [state, accepting, above right = of q0] {B};
\node (q2) [state, below right = of q0] {C};
\node (q3) [state, accepting, above right = of q2] {E};
\node (q4) [state, below right = of q2] {D};
\node (q5) [state, right = of q1] {$\emptyset$};

\path [-stealth, thick]
(q0) edge node {$a$}   (q1);

\path [-stealth, thick]
(q0) edge node {$b$}   (q2);

\path [-stealth, thick]
(q1) edge [loop] node {$a$}   (q1);

\path [-stealth, thick]
(q2) edge node {$a$}   (q4);

\path [-stealth, thick]
(q2) edge node {$b$}   (q3);

\path [-stealth, thick]
(q4) edge node {$b$}   (q3);

\path [-stealth, thick]
(q1) edge [bend left] node {$a, b$}   (q5);

\path [-stealth, thick]
(q3) edge [below loop] node {$a$}   (q3);

\path [-stealth, thick]
(q3) edge node {$a$}   (q5);

\path [-stealth, thick]
(q4) edge [bend right] node {$a$}   (q5);

\path [-stealth, thick]
(q5) edge [loop] node {$a, b$}   (q5);





\end{tikzpicture}
\caption{DFA graph after minimalization} \label{dfaMinimalization}
\end{figure}