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\fancyhead[L,LO]{
\textit{\small Math 387 Group problems, Monday Week 1}
}
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\begin{document}
\thispagestyle{firststyle}

Except for Exercise~\ref{ex:dfa}, you are asked to create DFAs (deterministic
finite automata) for the listed languages on the 
alphabet $\Sigma = \{0,1\}$.
\begin{enumerate}[label=\arabic*.\ , ref=\arabic*]
  \item $\Sigma^{\ast}$, the collection of all words, including the empty
    word~$\varepsilon$.
  \item The empty language, $\emptyset$.
  \item $\{\varepsilon\}$.
  \item $\{\varepsilon, 110\}$.
  \item Words of length~$4$.
  \item Words with length a multiple of $4$.
  \item Nonempty words whose beginning and ending characters differ.
  \item\label{ex:dfa} Consider the following DFA.
    \begin{center}
      \begin{tikzpicture}[fa,baseline={([yshift=-1em] current bounding
	box.north)},scale=0.8]
	\node[state, initial] (x) {$x$};
	\node[state, accepting, right of=x] (y) {$y$};
	\node[state, below right of=x] (z) {$z$};
	\draw (x) edge[loop above] node{$b$} (x);
	\draw (x) edge[above, bend left] node{$a$} (y);
	\draw (y) edge[right, bend left] node{$a$,$b$} (z);
	\draw (z) edge[left, bend left] node{$b$} (x);
	\draw (z) edge[left, bend left] node{$a$} (y);
      \end{tikzpicture}
    \end{center}
    \begin{enumerate}[label=(\roman*)]
      \item Give a formal description $(Q,\Sigma,\delta,q_{0},F)$ for this
	language, including a table defining $\delta$.
      \item What language is recognized by this DFA?
    \end{enumerate}
  \item Construct DFAs $A$ and $B$ such that $L(A)$ is the language of all words with length
    divisible by~$2$ and $L(B)$ is the language of all words with length divisible by~$3$.
    Use the Cartesian product construction from the text to create a DFA $C$
    such that $L(C)=L(A)\cup L(B)$, labeling the states accordingly.
  \item Words with length divisible by either $2$ or $3$ but not both.
  \item Words with the same number of $0$s as $1$s.
\end{enumerate}
\end{document}
