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\begin{document}
\headers{Math303}{Quiz 5}{Mon. Sep. 28, 2026}

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

\begin{enumerate}

\item\ppoints{6}{1} We define the following statements and open sentences.
\begin{align*}
	P &\st\mbox{$0=1$}   & Q(x) &\st\mbox{$x>0$} \\
	R(a,b,c) &\st\mbox{$a^2+b^2=c^2$}  &  S(A) &\st\mbox{$A$ is a finite set.} 
\end{align*}
Decide whether the following are true or false; indicate your answer by writing the entire word ``true'' or the entire word ``false''.  Give brief justifications for partial credit.
\begin{enumerate}
	\item $\sim P$
	\vfill
	
	\item $P \implies Q(1)$
	\vfill
	
	\item $(P \vee R(3,4,5)) \implies Q(0)$
	\vfill
	
	\item $Q(0) \iff P$
	\vfill 
	
	\item $Q((-1)^2) \implies S(\PS(\{1,2,3,4,5\}))$
	\vfill
	
	\item $\sim S(\{x\in\RR\st x^2 -6x + 2 = 0\}) \iff (R(3,2,1) \implies Q(2))$
	\vfill
\end{enumerate}

\newpage

\item\ppoints{2}{1} Truth tables and logical equivalence.
\begin{enumerate}
	\item Write a truth table for $P\implies (Q\implies R)$
	\vfill
	
	\item Give a simple statement using only logical operands in $\{\sim,\vee,\wedge\}$ which is logically equivalent to $P\implies (Q\implies R)$.
	\vspace{2cm}
\end{enumerate}

\item\ppoints{2}{1} Let $P$, $Q$, and $R$ be statements.  Use the standard logical operands $\sim, \lor, \land, \implies, \iff$ to express the following statements.
\begin{enumerate}
	\item If any one of the statements in $\{P,Q,R\}$ is true, then they all are true.
	\vfill
	
	\item Whenever two of the statements in $\{P,Q,R\}$ are true, the third is also true.
	\vfill
\end{enumerate}

\end{enumerate}
\end{document}