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%\pagestyle{fancyplain} \chead{Math 3345, Autumn 2021, Kutler}


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{\large{\textsc{Homework 6}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
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Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item \textbf{{[Falkner Section 2 Exercise 17]}} Use the method of conditional proof to explain why the sentence
\[ (P \Rightarrow Q) \, \Rightarrow \, \big\{ [P \Rightarrow (Q \Rightarrow R)] \, \Rightarrow \, (P \Rightarrow R) \big\} \]
is a tautology. \textbf{You do NOT need to use the book's method of ``discharging assumptions.'' However, please be clear about which assumptions you are making, and why you may make these assumptions.}

\item Let $A(x,y)$ be the conditional sentence
\[ \text{If $x = 3$ and $y = 5$, then $xy = 15$.} \]
\begin{enumerate}
\item Use one of the generalized De Morgan's laws to write the negation of the sentence $(\forall x \in \RR)(\forall y \in \RR) \, A(x,y)$.
\item Is $(\forall x \in \RR)(\forall y \in \RR) \, A(x,y)$ true or false? Explain your answer.
\item Let $B(x,y)$ be the converse of $A(x,y)$. Is $(\forall x \in \RR)(\forall y \in \RR) \, B(x,y)$ true or false? Explain your answer.
\end{enumerate}

\item \textbf{{[Falkner Section 3 Exercise 10]}} For each of the following sentences, write out what it means in words, state whether it is true or false, and prove your statement.
\begin{enumerate}
\item $(\exists y \in \RR) (\forall x \in \RR) (x + y = x)$.
\item $(\forall x \in \RR) (\exists y \in \RR) (x + y = x)$.
\item $(\exists y \in \RR) (\forall x \in \RR) (x + y = 0)$.
\item $(\forall x \in \RR) (\exists y \in \RR) (x + y = 0)$.
\item $(\exists y \in \RR) (\forall x \in \RR) (xy = 1)$.
\item $(\forall x \in \RR) (\exists y \in \RR) (xy = 1)$.
\end{enumerate}

\end{enumerate}

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\textbf{Practice Problems}
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It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item \textbf{{[Falkner Section 2 Exercise 22]}} Let $A$ be the sentence \newline $(P \Rightarrow Q) \, \Rightarrow \, \big\{ [P \Rightarrow (Q \Rightarrow R)] \, \Rightarrow \, (P \Rightarrow R) \big\}$. We saw in Exercise 17 that $A$ is a tautology. Let $B$ be the converse of $A$. Write out $B$ in terms of $P$, $Q$, and $R$. Then show that $B$ is not a tautology, by finding a combination of truth values for $P$, $Q$, and $R$ that makes $B$ false. You should be able to do this without writing out a truth table.

\item \textbf{{[Falkner Section 3 Exercise 9]}} See book for problem statement.
\end{enumerate}




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