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\begin{document}

\begin{center}
{\Large{\color{red}EXAM I}}
\end{center}

{\bf \it Name: }

{\bf Instructions:} Show your work 
and explain every step. You may not be given credit at all for incomplete solutions. Answers with no explanations will 
not be accepted. Do not write the numbers in the decimal form (keep them as fractions). Do not write on the back of the 
page.

\begin{enumerate}

\item (6 points) Find the negation, contrapositive, and the converse of the following statement (make it clear 
which one is which). 

\begin{center}
If $x$ is a prime number and $x > 2$, then $x$ is odd.
\end{center}

\vspace{0.5 in}

\item (6 points) Express 
the statement as a mathematical formula and find its negation.

\begin{center}
There exist a real number $y$ such that, for every real number $x$, $2 x^2 + 1 > x^2 y$.
\end{center}

\vspace{0.5 in}

\item (10 points) Let $B=\{5,4\}$, $C=[1,4]$, $D=(2,6]$, $E=\{4,a\}$. Find the following

\begin{enumerate}

\item $\mathcal{P}(B)$.

\item $B \times E$.

\item $C \cup D$.

\item $D - C$.

\item $D \cap 2\Z$.

\end{enumerate}

\item (6 points) Prove or disprove

\[p \longrightarrow (q \vee r) \equiv (p \longrightarrow q) \vee (p \longrightarrow r).\]


\vspace{1 in}

\item (4 points) What is the negation of $p \longleftrightarrow q$?

\vspace{.5 in}

\item (20 points) Decide whether the following are true or false

\begin{enumerate}

\item Let $A$ and $B$ be sets. Then, $|A \cup B| = |A| + |B|$.

\item Let $A$ and $B$ be sets. Then, $|A - B| = |A| - |A \cap B|$.

\item $\overline{\overline{q} \longrightarrow \overline{p}} \equiv p \wedge \overline{q}$.

\item $\{3\} \in \{1,2,3\}$.

\item $\{3\} \subseteq \{1,2,\{3\}\}$.

\item $\phi \in \{2,3,4,\{5\}\}$.

\item  $\phi \subseteq \{2,3,4,\{5\}\}$.

\item The sets $\{2,3\}$, $\{4,5\}$, and $\{3,7\}$ are pairwise disjoint.

\item The sets $\{a,b,c\}$, $\{d,e\}$, and $\{f,g\}$ form a partition of $\{a,b,c,d,e,f,g\}$.

\item It's possible to have $\{4,5,6\}$ and $\{5,7,8\}$ as equivalence classes for an equivalence relation on 
$\{4,5,6,7,8,9,10,11\}$.

\item It's possible to have an equivalence relation that is both symmetric and antisymmetric.

\item It's possible to have an equivalence relation that is both equivalence relation and a partial order.

\item If 5 and 7 are elements of set $A$, $\mathcal{R}$ is an equivalnce relation on $A$, and the equivalence 
class of 5 is $\{5,7,8\}$, then the equivalence class of 7 is $\{5,7,8\}$.

\item If 5, 6, and 7 are elements of set $A$ and $\mathcal{R}$ is an equivalnce relation on $A$, then it's possible to have the 
equivalnce class of 5 to be $\{6,7\}$.

\item Let $A=\Z^+$, $\mathcal{R}=\{(x,y) \in A \times A ~|\text{     } x | y)\}$. Then 3 and 5 are incomparable.

\item If $a$ and $b$ are nonzero real numbers and $a|b$ and $b|a$, then $|b| = |a|$.

\item If $A$ and $B$ are sets, then $A \oplus B = (A \cup B) - (A \cap B)$.

\item $p \longrightarrow q \equiv \overline{q} \longrightarrow \overline{p}$. 

\item If $A$, $B$, and $C$, are any sets, then $\overline{A \cup (B \cap C)} = \overline{(A \cup B)} \cup (\overline{A \cup C)}$.

\end{enumerate}

\item (25 points) Prove or disprove the following:

\begin{enumerate}
\item The set $S=\left\{\frac{1}{3n} ~|~ n \in \Z^+ \right\}$ has a smallest element.

\vspace{2 in}

\item If $a$ and $b$ are irrational, then $a+b$ is irrational.

\vspace{.5 in}

\item If $a$ and $c$ are in $\R$ with $a < c$, then $\exists b \in \R$ such that $a < b < c$.

\vspace{1 in}

\item If $n^2$ is odd, then $n$ is odd.

\vspace{1 in}

\item If $a$ and $b$ are both irrational, then $a^b$ is irrational.

\end{enumerate}

\newpage

\item (10 points) Prove the following by mathematical induction:

\[1 + 3 + \cdots + (2n-1) = n^2,~ \forall n \geq 1.\]

\newpage

\item (12 points)

\begin{enumerate}

\item Let $A=\Z$, $\mathcal{R}=\{(x,y) \in A \times A ~|\text{     }|2x - 3y| < 5)\}$. 
Prove by a counterexample that $\mathcal{R}$ is not reflexive.

\vspace{.5 in}

\item Let $A=\Z$, $\mathcal{R}=\{(x,y) \in A \times A ~|\text{     }|2x - y| < 5)\}$. 
Prove by a counterexample that $\mathcal{R}$ is not transitive.

\vspace{.5 in}


\item Let $A=\Z$, $\mathcal{R}=\{(x,y) \in A \times A ~|\text{     }x - y > 5)\}$. Prove 
that $\mathcal{R}$ is antisymmetric.

\vspace{1 in}

\item Let $\mathcal{R}=\{(x,y) \in \Z \times \Z ~|\text{     } x-y = 2k, ~k \in \Z\}$. Find $[7]$.

\vspace{1 in}

\item Let $A=\Z-\{0\}$, $\mathcal{R}=\{(x,y) \in A \times A ~|\text{     } x | y)\}$. 
Prove by a counterexample that $\mathcal{R}$ is not symmetric.

\vspace{.4 in}

\item Let $A=\Z-\{0\}$, $\mathcal{R}=\{(x,y) \in A \times A ~|\text{     } xy > 0)\}$. 
Prove that $\mathcal{R}$ is transitive.

\end{enumerate}

\end{enumerate}

\end{document}
