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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
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\def\HW{5}
\def\DUE{10/2/2018}
\def\NAME{YOUR NAME HERE}
\title[Homework \HW]{Homework \HW \\
Math 308\\
\small Due: \DUE}
\author{}
%\date{}                                           % Activate to display a given date or no date

\begin{document}



Add on to homework 5:

\begin{enumerate}[1.]
\setcounter{enumi}{7}
\item Rewrite the following using $\forall$ and $\exists$.
\begin{enumerate}[(a)]
\item For all integers $x$, we have $x$ is odd or even. (Your answer should include a definition of even/odd using $\exists$.)

%\begin{ans}
%\end{ans}


\item There exist two positive numbers such that their sum is negative.

%\begin{ans}
%\end{ans}

\end{enumerate}





\item Consider the statement ``If $a$ and $b$ are real numbers with $a \neq 0$, then $ax+b=0$ has a solution.'' 
\begin{enumerate}[(a)]
\item Rewrite this statement  using symbolic notation $\forall$ and $\exists$.

%\begin{ans}
%\end{ans}

\item Negate this statement, giving your answer both in symbolic notation, and in words. 

%\begin{ans}
%\begin{enumerate}[\qquad\qquad\qquad]
%\item[\bf In symbols:]
%\item[\bf In words:]
%\end{enumerate}
%\end{ans}

\end{enumerate}

\item Negate the following.
\begin{enumerate}[(a)]
\item There exists a grey cat.

%\begin{ans}
%\end{ans}

\item Every cat has an owner. 

%\begin{ans}
%\end{ans}

\item Some of the students in the class are not here today.

%\begin{ans}
%\end{ans}

\item For all $x, y \in \ZZ_{>0}$ there exists $z \in \ZZ_{>0}$ such that $x=y+z$.

%\begin{ans}
%\end{ans}

\item The number $\sqrt{x}$ is rational if $x$ is an integer.

%\begin{ans}
%\end{ans}

\end{enumerate}

\item For each of the following,
\begin{enumerate}[\qquad\qquad(i)]
\item restate in words;
\item decide whether it's true or false; and 
\item prove or disprove accordingly.
\end{enumerate}

\begin{enumerate}[(a)]
\item $\forall x \in \ZZ, \exists y \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\item $\forall y \in \ZZ, \exists x \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\item $\exists x \in \ZZ, \forall y \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\item $\exists y \in \ZZ, \forall x \in \ZZ (x^2 = y)$

%\begin{ans}
%\begin{enumerate}[(i)]
%\item \textbf{In words:} 
%\item This statement is \fbox{true/false}. 
%\item \begin{proof}
%
%
%\end{proof}
%\end{enumerate}
%\end{ans}

\end{enumerate}


\end{enumerate}



\end{document}