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\headers{Math375}{Homework 7}{Due: Oct. 1, 2015}

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\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  \textbf{Show your work}; See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item Let $\Sigma = \{0,1\}$, let $A=\bigcup_{k=0}^2 \Sigma^k$, and let $B=\bigcup_{k=0}^4 \Sigma^k$.
\begin{enumerate}
	\item List the strings in $A$.  What is $|A|$?
	\item What is $|B|$?
	\item Recall that $AB = \{xy~|~\mbox{$x\in A$ and $y\in B$}\}$.  Describe the members of $AB$.  What is $|AB|$?
\end{enumerate}

\item Let $\Sigma=\{0,1\}$ and let $A$ be the language defined recursively as follows: (1) $\lambda \in A$ and $1\in A$,  (2) if $x\in A$ then $x0\in A$,  (3) if $x\in A$, then $x01\in A$.
\begin{enumerate}
	\item True or False:  $0\in A$.  Justify your answer.
	\item True or False:  $011\in A$.  Justify your answer.
	\item Give a simple, non-recursive description of $A$.
\end{enumerate}


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