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\textbf{College of the Holy Cross, Fall Semester 2017} \\
\textbf{MATH 243 -- Mathematical Structures, section 2} \\
\textbf{Exam 3 -- December 7}

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\noindent\textbf{Your Name}: \underline{\phantom{Name Name Name Name Name}}

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\paragraph{Instructions} Please write your answers in the spaces
provided on the following pages, and show work on the test itself. 
{\it For possible partial credit, even if you cannot completely solve a problem, 
include definitions of terms involved, partial results you can do, etc.}  Use the 
back of the preceding page if you need more space for scratch work.



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\noindent\textbf{\noindent\textbf{Please do not write in the space below}}


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Problem & Points/Poss    \strut \\

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I  & \hfil/\  25\hfil \strut \\

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II  & \hfil/\  20\hfil \strut \\

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III  & \hfil/\  20\hfil \strut \\

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IV  & \hfil/\  15\hfil \strut \\
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V  & \hfil/\  20\hfil \strut \\

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Total   & \hfil/100\hfil \strut \\

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I. (25) Give the statement and proof of ``Fermat's Little Theorem.''
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II. (20)  An RSA public key encryption system has public key $m = 551, e = 11$.
``Crack the code'' by determining the private key information:  $p,q,d$.
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III.  Let $f : A \to B$ be a mapping.  
\begin{enumerate}
\item[(A)] (10) Show that if $U_1,U_2$ are subsets of $B$, then 
$f^{-1}(U_1 \cap U_2) = f^{-1}(U_1) \cap f^{-1}(U_2)$.  
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\item[(B)]  (10) If $f$ is injective, and $T_1,T_2$ are subsets of $A$, show
that $f(T_1) \cap f(T_2) \neq \emptyset$ implies $T_1 \cap T_2 \neq \emptyset$.
\end{enumerate}
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IV. (15) Let $R$ be the relation on $\mathbb{R} \backslash \{0\}$ defined by 
$a\ R\ b \Leftrightarrow \frac{a}{b} \in \mathbb{Q}$.  See below.\footnote{Here $\mathbb{Q} = \{m/n : m, n \in \mathbb{Z}, n \ne 0\}$ is the set of rational numbers.}  Is $R$ an equivalence relation?  Prove
your assertion.  
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V.  \begin{enumerate}
\item[(A)] (10) Show that $\mathbb{N}$ is not bounded above in the real numbers.
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\item[(B)] (10)  Use part (A) to show that for all real numbers $\varepsilon > 0$, there
exist $n \in \mathbb{N}$ such that $\left| 1 - (1 + \frac{(-1)^n}{\sqrt{n}})\right| < \varepsilon$
\end{enumerate}
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