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\begin{document} 
\paper{qu4}{}{Quiz, Week~4}

\begin{problem}
Notes 2 contains a proof by induction that:
%
\[
1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}.
\]
Give a proof of this theorem using the well-ordering principle. 
\end{problem}

\newpage

\begin{problem}
Consider two integers $a$ and $b$. We define the set $L$ recursively,
as follows: 
\begin{itemize}
\item 
\textbf{Base case:} $0\in L$.
\item 
\textbf{Constructor cases:} if $x\in L$, then
\begin{itemize}
\item[$\bullet$] $x+a\in L$,
\item[$\bullet$] $x-a\in L$,
\item[$\bullet$] $x+b\in L$,
\item[$\bullet$] $x-b\in L$.
\end{itemize}
\end{itemize}
Use structural induction to prove that all members of $L$ are linear
combinations of $a$ and $b$. In other words, prove that 
\[
\text{for every $x\in L$: }\quad 
x = sa + tb
\;\text{ for some integer coefficients $s$ and $t$.}
\]
\end{problem}

\end{document} 
