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\markright{MATH 243, section 2, Midterm 1}

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\textbf{College of the Holy Cross, Fall Semester, 2017} \\
\textbf{MATH 243, Section 2 -- Midterm 1} \\
\textbf{Friday, September 29}
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\vfil

\noindent\textbf{Your Name}: \Blank
\vskip 0pt plus .3fil

\paragraph{Instructions} Please write your answers in the spaces
provided, and show work on the test itself. Use the back of the
preceding page if you need more space for scratch work.  There are
100 possible points distributed as below.

\vfil

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

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\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1  & \hfil/\  20\hfil \Strut \\
\hline
2  & \hfil/\  20\hfil \Strut \\
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3  & \hfil/\  20\hfil \Strut \\
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4  & \hfil/\  20\hfil \Strut \\
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5  & \hfil/\  20\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\

\hline
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\begin{enumerate}

\item{} 
\begin{enumerate}
\item[(a)] (15) Construct the truth table for the statement 
$$(({\rm not}\ Q)\ {\rm and}\ (P\ {\rm implies}\ Q))\ {\rm implies}\ ({\rm not}\ P)$$

\vskip 4.0truein

\item[(b)] (5)  Suppose you know that $x \in {\bf R}$ implies that 
$x^2 \ge 0$ and someone one hands you a mathematical object $x$ for 
which you can define $x^2$, and for which $x^2$ is a real number
with $x^2 < 0$.  What can you conclude?
\end{enumerate}
\vfil
\eject 
\item{} Let 
\begin{align*}
A &= \{x \in \mathbb{Z} : x = 3k + 1, {\rm some}\ k \in \mathbb{Z}\}\\
B &= \{x \in \mathbb{Z} : x = 4k, {\rm some}\ k \in \mathbb{Z}\}\\
C &= \{-9,-8,-7,-6,-5,-4,-3,-2,-1\}.
\end{align*}
\begin{enumerate}
\item[(a)] (7)  What is the set $(A \cup B)\cap C$?
\vskip 2.0truein
\item[(b)] (7)  What is the set $B^c \cap C$?  
\vskip 2.0truein
\item[(c)] (6)   Prove or disprove:  $4A \subseteq A \cap B$, where 
$4A = \{4x : x \in A\}$. 
\end{enumerate}
\vfil
\eject

\item{}  Consider the statement about $x \in \mathbb{Z}$:  If $x^2 \notin 1 + 4\mathbb{Z}$, 
then $x$ is even.   
\begin{enumerate}
\item[(a)] (10)  What is the contrapositive form of this implication? 
\vskip 3.0truein
\item[(b)] (10)  Prove the contrapositive form.  What does this imply about
the original statement?   
\end{enumerate}
\vfil\eject

\item{} (20) Let 
\begin{align*}
A &= \{x \in \mathbb{Z} : x = -2 + 5k\ {\rm for\ some}\ k\in \mathbb{Z}\}\\
B &= \{x \in \mathbb{Z} : x = 3 + 5k\ {\rm for\ some}\ k \in \mathbb{Z}\}.
\end{align*}
Is $A = B$?  Prove your assertion.   
\vfil\eject

\item{} (20) Prove by mathematical induction:  If $a,d$ are any constants and
$n \ge 0$ is a natural number, then
$$a + (a + d) + (a + 2d) + \cdots + (a + nd) = \frac{n+1}{2}\cdot\left(2a + nd\right).$$ 
\vfil\eject
\end{enumerate}


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