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\centerline{\bf Honors Homework 3} 

\centerline{Morally Due Mon April 1 at 10:00 AM. }

\vspace{1cm}
Prove using \textbf{Structural Induction}:
\begin{enumerate}
    \item (30 points) 
Define a set of strings $S$ by

\begin{itemize}
    \item $\epsilon \in S$
    \item If $\sigma \in S$, then
    \begin{itemize}
        \item $b\sigma \in S$
        \item $\sigma b \in S$
        \item $\sigma aa \in S$
        \item $aa \sigma \in S$
    \end{itemize}
\end{itemize}
Prove that every string in $S$ contains an even number of $a$'s.
    

\item 
(35 points) 
Let $S$ be a set of ordered pairs of integers defined recursively by

\begin{itemize}
    \item $(0,0) \in S$

    \item If $(a,b) \in S$, then $(a+2,b+3) \in S$ and $(a+3,b+2) \in S$.
\end{itemize}

Prove $5|a+b$ when $(a,b) \in S$

\item 
(35 points) 
Prove by induction that a full binary tree has an odd number of nodes. 

\end{enumerate}
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