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\begin{document}
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\title{BMCC MAT100.722 Exam 1}
%\author{\textsc{D. Dakota Blair}}
\baselineskip=4ex
\maketitle
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\begin{abstract}

\end{abstract}
You will be given all class to complete this test. You are allowed to have one sheet of notes. Each problem is worth 10 points. Good luck!
\section{Vocabulary}
\begin{center}
\textbf{\large Match the concept in the first column with an example from the second.}
\begin{tabular}[width=6in]{llll}
(a) & $A \mbox{ is a proper subset of } B$ & \hspace{1in}(1) & $\symnot q \implies \symnot p$ \\
(b) & $\mbox{Contrapositive}$ & \hspace{1in}(2) & For all, none, for each\\
(c) & $\mbox{Universal Quantifier}$ & \hspace{1in}(3) & $A = B$\\
(d) & $\mbox{Contradiction}$ & \hspace{1in}(4) & $A = \{\}$\\
(e) & $A \mbox{ is a subset of }$ B & \hspace{1in}(5) & $p \symiff \symnot p$
\end{tabular}
\end{center}

\section{Sets}
\textbf{\large Replacing ??? with what symbol will make the following statement true?
\[
\emptyset \hspace{1pc}???\hspace{1pc} A
\]}
\begin{enumerate}
\item $=$
\item $\equiv$ 
\item $\subset$
\item $\in$ 
\end{enumerate}

\section{Sets}
Let the set of multiples of $4$ be named $F$.
Similarly let the set of multiples of $6$ be named $S$.
That is,
\begin{align}
F &= \{4, 8, 12, 16, \ldots\}\\
S &= \{6, 12, 18, 24, \ldots\}
\end{align}
\begin{enumerate}
\item \textbf{\large Are $F$ and $S$ equal, equivalent, both or neither?}
\begin{enumerate}
\item $F = S$.
\item $F \equiv S$.
\item $F \equiv S$ and $F = S$.
\item $F \neq S$ and $F \not\equiv S$.
\end{enumerate}
\item \textbf{\large What is the set $F \intersect S$?}
\begin{enumerate}
\item \{12, 24, \ldots\}\hspace{1pc}(The set of the multiples of 12) 
\item \{12\}.
\item \{2, 4, 6, 8, \ldots\}\hspace{1pc}(The set of even numbers).
\item \{1, 3, 5, 7, \ldots\}\hspace{1pc}(The set of odd numbers).
\end{enumerate}
\end{enumerate}

\section{Sets}
\textbf{\large Explain why ``the set of sets'' is not a well-defined set.}

\pagebreak
\section{Logic}
\begin{enumerate}
\item \textbf{\large If $\symnot(p \symand q)$ is false, what must be the truth
values of the component statements?}
\begin{enumerate}
\item T
\item F
\item $\symnot p \symor \symnot q$
\item $\symnot p \symand \symnot q$
\end{enumerate}
\item Identify a suitable converse of the statement ``No teachable kitten has green eyes.''
\begin{enumerate}
\item All green-eyed kittens are teachable.
\item If it is a teachable kitten, it has green eyes.
\item If it is a kitten with green eyes, it is teachable
\item If it is a kitten with green eyes, it is not teachable
\end{enumerate}
\end{enumerate}

\pagebreak
\section{Logic}
\begin{enumerate}
\item \textbf{\large For how many cases, i.e. rows in the truth table, is the following statement F?
\[
(p \implies q) \symand (\symnot q) \symand p
\]
}
\begin{enumerate}
\item 1 
\item 2 
\item 3 
\item 4
\end{enumerate}
\item \textbf{\large For how many cases is the following statment T?
\[
\symnot(\symnot p \symand \symnot q) \symand \symnot (p \symand q)
\]}
\begin{enumerate}
\item 0
\item 1 
\item 2 
\item 3 
\end{enumerate}
\end{enumerate}

\section{Logic}
\textbf{\large Explain why the statement ``A week has eight days if and only if October has forty days'' is true.}

\section{Number Systems}
List and explain some advantages of place value.

\pagebreak
\section{Number Systems}
\textbf{\large Identify the correct Egyptian multiplication work for the problem $33\cdot81$.}
\begin{enumerate}
\item $2673$
\item $1089$
\item \begin{align*}
\rightarrow 1 &\leftrightarrow 81\\
2 &\leftrightarrow 162\\
4 &\leftrightarrow 324\\
8 &\leftrightarrow 648\\
16 &\leftrightarrow 1296\\
\rightarrow 32 &\leftrightarrow 2592\\
33 = 32 + 1 &\leftrightarrow 33*81 = 81 + 2592 = 2673.
\end{align*}
\item \begin{align*}
\rightarrow 1 &\leftrightarrow 33\\
2 &\leftrightarrow 66\\
4 &\leftrightarrow 132\\
8 &\leftrightarrow 264\\
16 &\leftrightarrow 528\\
\rightarrow 32 &\leftrightarrow 1056\\
33 = 32 + 1 &\leftrightarrow 33*81 = 33 + 1056 = 1089.
\end{align*}
\end{enumerate}

\section{History}
Describe something you have learned about \'Evariste Galois and his contributions to mathematics.
\end{document}
