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\begin{document}




{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Jeffrey Diller

\begin{center}
{\bf Math 223: Introduction to Mathematical Reasoning}
\\
{\bf Spring Semester 2004}
\\
{\bf Final Exam}
\\
{\bf Tuesday, May 4}
\end{center}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} examination contains 6 problems.  Counting
the front cover and blank pages, the exam consists of 9 sheets
of paper.  No calculators allowed.
\\[1.5ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.3cm]{0cm}{0.9cm} \\ \hline\hline
  1      & 30   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  2      & 40   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      & 10   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      & 10   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  5      & 30   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  6      & 30   &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
Total  & 150\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
\newpage

\begin{problems}

\item State the following (6 points each).  
\begin{subproblem}
\item Definition of \emph{convergent sequence}.
\vspace*{1in}
\item Definition of \emph{cauchy sequence}.
\vspace*{1in}
\item Definition of \emph{prime number}.
\vspace*{1in}
\item \emph{Bolzano-Weierstrass Theorem}.
\vspace*{1in}
\item Definition of \emph{countable set}.
\end{subproblem}

\newpage
\item 
Five of the following eight assertions are false.  In the space
at the bottom of this page, identify four of them and give counterexamples.
Note that you do not have to justify your counterexamples---only present 
them. (10 points each)
\begin{subproblem}
\item Given $a,b,c\in\N$ such that $a|bc$ it follows that $a|b$ or $a|c$. 
\item A sequence $\langle x\rangle$ cannot have more than one limit.
\item If $f:A\to B$, $g:B\to C$ are functions such that $g\circ f$ is
injective, then $g$ is injective.
\item A bounded sequence is convergent.
\item If $S$ is a set of rational numbers that is bounded above, then
the least upper bound of $S$ is a rational number. 
\item If $A\subset B$ and $A\subset C$, then $A\subset B\cap C$.
\item If $\langle y\rangle$ is a subsequence of 
$\langle x\rangle$ and $\lim_{n\to\infty} x_n= L$,
then $\lim_{n\to\infty} y_n = L$.
\item If $2x \equiv 2\mod 4$, then $x\equiv 1\mod 4$.
\end{subproblem}
\newpage

\item 
Compute the $4$--ary (base 4) expansion for $1/5$. (10 points)
\vspace*{3in}
\item Prove that $\R$ is uncountable (10 points).
\newpage


\item On the next two pages, do two of the following three problems
(15 points each + 5 points extra credit for correct solutions to (b) and (c)).
\begin{subproblem}
\item 
Prove for every $n\in\N$ that 
$\sum_{j=1}^n j = \frac{n(n+1)}{2}$.
\item 
Let $A,B,C$ be sets. Prove that $A\cap (B\cup C) \subset (A\cap B) \cup C$.  
Venn diagrams do not constitute a proof!
\item 
Let $p$ be a prime number and $n\in\Z$ satisfy $n^2 \equiv 1 \mod p$.
Show that $n \equiv \pm 1 \mod p$.
\end{subproblem}
\newpage

\item On the next two pages, do two of the following three problems
(15 points each + 5 points extra credit if you do (b) and (c) and get them
both right).
\begin{subproblem}
\item 
Show using the definition of limit that 
$\lim_{n\to\infty} \frac{n-1}{n+1} = 1$.
\item
Suppose that $\langle x\rangle$ is a sequence that converges to a limit $L$.  
Show
using the definition of limit that $\lim_{n\to\infty} x_n^2 = L^2$.
\item
Let $L\in\R$ be a given number and $x_1\in\R$ be some number larger
than $L$.  Then for every $n\in\N$, let $x_{n+1} = \frac{x_n+L}{2}$.
Show that $\lim_{n\to\infty} x_n = L$.  
\end{subproblem}
\newpage

\ignore{
\item Extra Credit: I won't even look at this unless you've turned in
reasonable solutions to the other problems on this exam (10 points).  
Let $p,q\in\N$
be natural numbers such that $1\leq p<q<10$.  What is the maximum number
of digits that one must find in the base 10 expansion of $p/q$ before 
the digits begin to repeat?  Justify your answer, \emph{without actually
computing any decimal expansions}.
Let $k\in\N$ be larger than $1$, and $n\in\N$ be smaller than $k-1$.
Determine the $k$-ary expansion of $\frac{n}{k-1}$ and prove that your
answer is correct. }
\end{problems}

\end{document}
\end
