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\begin{document} \pagestyle{empty} \begin{flushleft} 

\scs{Name:}

\vspace{.1in}

\scs{Math 301 \hfill Exam 1 (Chapters 1, 2, and 4) \hfill February 13, 2015}

\scs{Instructions:} 
Answer all 9 questions in the space provided.
Show any relevant work for problems whose solution is not a proof; solutions without justification may receive little or no credit.
Proofs should be written clearly using complete English sentences.
Additional scratch paper is available at the front of the room.
You may refer to the following definitions (as well as any others that you remember).


\dfn{
An integer $a$ is {\em even} if there is an integer $b$ such that $a=2b$.
Integer $a$ is {\em odd} if there is an integer $b$ such that $a = 2b+1$. 
}

\dfn{
Let $a, b \in \ZZ$.
We say that $a$ {\em divides} $b$ (written $a|b$) if there is an integer $c$ such that $b=ac$.
}

\dfn{
A natural number $p$ is {\em prime} if its only positive divisors are $1$ and $p$.
}

\dfn{
Let $a, b \in \ZZ$.
Then the {\em greatest common divisor of $a$ and $b$} (denoted $\gcd(a,b)$) is the largest integer that divides both $a$ and $b$.
The {\em least common multiple of $a$ and $b$} (denoted $\lcm(a,b)$) is the smallest positive integer multiple of both $a$ and $b$.
}

\dfn{
The {\em power set} of a set $A$ (denoted $\mathcal{P}(A)$) is the set of all subsets of $A$.
}

\prob{
Write out the following sets by listing out their elements between braces: 
\begin{itemize}

\item[a) ] $\displaystyle \{2x: x \in \ZZ \text{ and } \abs{3x} < 6\}$ 

\vfill

\item[b) ] $\displaystyle \{X \subseteq \RR: \abs{X} = 0\}$

\vfill

\item[b) ] $\displaystyle \mathcal{P}\left( \{ \emptyset, \RR \} \right)$

\vfill

\end{itemize}
}

\newpage

\prob{
Calculate the following cardinalities:
\begin{itemize}

\item[a) ] $\displaystyle \abs{\{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \} }$

\vfill

\item[b) ] $\displaystyle \abs{\mathcal{P}\left( \mathcal{P} (A) \right) \times A}$  where $A$ is a set with cardinality $n$.

\vfill

\end{itemize}
}

\prob{
Let $A = \{1, 2, 3, 5, 7\}$ and $B = \{2, 4, 8\}$ have universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
Write out the following sets by listing their elements between braces: 
\begin{itemize}

\item[a) ] $\displaystyle B \cap (B \cup A)$

\vfill

\item[b) ] $\displaystyle \overline{(\overline{A} \cup B)}$

\vfill

\end{itemize}
}

\newpage

\prob{
Suppose the statement $\neg\left[ P \implies (Q \lor R) \right]$ is true.
What are the truth values of $P$, $Q$, and $R$?

\vfill

}


\prob{
Determine if the statement $P \implies \neg Q$ is logically equivalent to the statement $\neg(P \land Q)$. 

\vfill

}

\newpage

\prob{
Determine if the following statements are True or False:
\begin{itemize}

\item[a) ] $\displaystyle \forall x \in \ZZ\ \exists y \in \ZZ,\ x+y = 0$.

\vfill

\item[b) ] $\displaystyle \exists x \in \ZZ\ \forall y \in \ZZ,\ x+y = 0$.

\vfill

\end{itemize}
}

\prob{
Let $n \in \ZZ$. 
Consider the statement ``If $3|n$ and $5|n$, then $15|n$.''
\begin{itemize}

\item[a) ] Write the converse of the statement.

\vfill

\item[b) ] Write the contrapositive of the statement.

\vfill

\end{itemize}
}

\newpage

\prob{
Let $a,b,c,d \in \ZZ$.
Prove that if $a | b$ and $c | d$, then $(ac)|(bd)$.
}

\newpage

\prob{
Let $n \in \NN$.
Prove that if $p$ is a prime number, then $\gcd(p,n) = 1$ or $\gcd(p,n) = p$. 
}

\end{flushleft}
\end{document}