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\begin{document}

\begin{center}
{\Large{\color{red}Assignment 1}}

{\it Due Wed, Feb 14, 06 at 11:00 AM in Class}

\end{center}


{\bf Remarks:} 
I may not grade all assignments, and may not grade all questions/parts on 
the assignments I choose to grade. 
You're welcome to ask me for help. 
Show your work and explain every step. If you don't provide enough 
explanation, you may get no credit or partial 
credit. You should do the assignment by yourself.


\begin{enumerate}

\item Express the following statements as formulas and find the negation 
of the formula:

\begin{enumerate}

\item There exists an integer $x$ such that for every integer $y$, $x + y 
= 7$.

\item For every integer $y$, there exists an integer $x$ such that $x + y 
= 7$.

\item $n$ is an odd integer if and only if $n^2$ is an odd integer.

\end{enumerate}

\item Decide if the following statements are true or false. Explain. 

\begin{enumerate}

\item $(x+1)^2 \geq x^2$, $\forall x \in \R$.

\item $\forall x, y \in \R$, $2 x^2 + 3 y^2 > 0$.

\item $\exists x, y \in \R$, $2 x^2 + 3 y^2 > 0$.

\item There exists an integer $x$ such that for every integer $y$, $x + y 
= 7$.

\item For every integer $y$, there exists an integer $x$ such that $x + y 
= 7$.

\end{enumerate}

\item Express the following as a formula and find its negation, its 
converse, and its contrapositive.

\begin{center}
If $x$ is rational and $y$ is irrational, then $x + y$ is irrational.
\end{center}

\item Determine if $(p \wedge \overline{q}) \vee (\overline{p} \wedge q)$ 
is logically equivalent to 
$(\overline{p} \vee \overline{q})$. Explain.


\item Prove or disprove the following (i.e. if the statement is true, 
prove it, and if the statement 
is false, write down a counterexample). 

\begin{enumerate}

\item The set $S = \{ \frac{1}{2n}~|~ n \in \N\}$ has no smallest element.

\item $2 x^2 - 8 x + 15 > 5$, $\forall x \in \R$.

\item If $n$ is a prime number, then $n^2 + n + 41$ is prime.

\item If $x$ is irrational, then $\frac{x}{2}$ is irrational.

\item For every integer $n$, $n^2 + n$ is even.

\item If $a$ and $d$ are real numbers such that $a < d$, then there exist 
real numbers 
$b$ and $c$, such that $a < b < c < d$.

\end{enumerate}

\end{enumerate}

\end{document}
