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\begin{document}
\headers{Math303}{Quiz 6}{Mon. Oct. 5, 2026}

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\noindent
\textbf{Directions:} Solve the following problems.  Give supporting work/justification where appropriate.

\begin{enumerate}

\item\ppoints{3}{1} Translate the following sentences to symbolic logic as directly and simply as possible.  Is the statement true or false?  Write the entire word.
\begin{enumerate}
	\item The product of two real numbers is zero if and only if one of the factors is zero.
	\vfill 
	
	\item Every integer divided by a non-zero integer is an integer.
	\vfill
		
	\item There is no maximum integer.
	\vfill
\end{enumerate}

\item\ppoints{3}{1} Translate the following statements in symbolic logic to English sentences as simply as possible.  Is the statement true or false?  Write the entire word.
\begin{enumerate}
	\item $\forall x,y,z\in \RR, (x < y) \wedge (y < z) \implies x < z$
	\vfill
	
	\item $\forall x,y\in \ZZ, (\exists k\in \ZZ, x = 2k \vee y = 2k) \implies (\exists \ell\in \ZZ, x+y = 2\ell)$
	\vfill

	\item $\forall A\subseteq\QQ, (\exists s,t\in\QQ, \forall a\in A, s\le a \le t) \implies (\exists n\in\NN, |A|\le n)$  
	\vfill
\end{enumerate}

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\item\points{2} Give an English sentence which asserts the negation of the sentence $P$ below as naturally as possible.  P: Either there are finitely many primes or every infinite set of positive integers contains at least one prime.
\vspace{6cm}

\item\points{2} Prove that the sum of two even integers is even. 
\vfill

\end{enumerate}
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