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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 Exam 1}
\\
{\bf Friday, February 20}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} examination contains 4 problems.  Counting
the front cover and blank pages, the exam consists of 7 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      &  30  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  3      &  39  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
  4      &   1  &       \rule[-.3cm]{0cm}{0.9cm} \\ \hline
Total  & 100\mbox{\  \ } & \rule[-.3cm]{0cm}{0.9cm} \\ \hline
\end{tabular}
\end{center}

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

\begin{problems}

\item Do each of the following (6 points each).  
\begin{subproblem}
\item Define \emph{prime number}.
\vspace*{1.5in}

\item State the \emph{well-ordering property}.
\vspace*{1.5in}

\item State the contrapositive of the following assertion: 
\emph{If $x_1 \neq x_2$, then $f(x_1)\neq f(x_2)$.} 
\vspace*{1.5in}

\item Without using words of negation, state the opposite (i.e. negation) 
of the following assertion: \emph{For all $x\in A$ there exists $b\in B$
such that $b>x$.}
\vspace*{1.5in}

\item Define \emph{increasing function}.
\end{subproblem}
\newpage

\item Four of the following seven assertions are false.  
Identify three of them and give counterexamples on this page and (if
necessary) the next.  If you give counterexamples to more than
three statements, I will simply grade the first three.  
Note that you do not have to justify your counterexamples---only present 
them.  
(10 points each)
\begin{subproblem}
\item Let $f:A\to B$ and $g:B\to C$ be functions.  If $f$ is injective and 
      $g$ is injective then $g\circ g$ is injective.
\item Let $f:A\to B$ and $g:B\to C$ be functions.  If $g\circ f$ is injective 
      then $g$ is injective.
\item The function $f:\N\times\N\to\N$ given by $f(a,b) = a+b$ is
      surjective.
\item If $A, B, C$ are sets then $(A\cup B)\cap (A\cup C) = A\cup (B\cap C)$
\item Let $S = \{(a,b)\in\N\times \N: 2a+b\leq 5\}$.  Then 
      $S\subset [2]\times[2]$.
\item Let $A$ and $B$ be sets and $f:A\to B$, $g:B\to A$ be injective
      functions.  Then the cardinality of $A$ is the same as that of $B$.
\item Let $A$ and $B$ be sets.  Suppose that $A$ is a subset of $B$,
      but $B$ is not a subset of $A$.  Then the cardinality of $A$
      is not the same as the cardinality of $B$.

\end{subproblem}
\newpage

\item On the remaining pages of this exam, do three of the following four 
problems. If you turn in solutions to all four, I will simply grade the 
first three.  (13 points each)
\begin{subproblem}

\item Let $n\in\N$ be a natural number whose base 3 decimal expansion 
is $122_{(3)}$.  Give the base 7 decimal expansion of $n$.  

\item Prove that there are infinitely many prime numbers.

\item Let $A,B,C$ be sets.  Prove that $A - B \subset A - (B\cap C)$ 

\item Prove by induction that $\sum_{i=1}^n 2n-1 = n^2$. 
\end{subproblem}

\vfill
\item How many points should Professor Diller give you for problem 4? 
(1 point total)
\end{problems}

\end{document}

