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\begin{document}

\begin{center}
{\large{\textsc{Homework 7}}} \\
\textsc{Math 3345 -- Spring 2024 -- Kutler}
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\begin{center}
\textbf{Exercises}
\end{center}

Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item Let $A(x,y)$ be the conditional sentence
\[ \text{If $x = 3$ and $y = 5$, then $xy = 15$.} \]
Although most people probably read this sentence as a straightforwardly true statement, it is technically a function of the variables $x$ and $y$ (it's just that ``$x = 3$ and $y = 5$'' is false for almost all values of $x$ and $y$). To produce an unambiguously true or false sentence, we should bound these variables with quantifiers.
\begin{enumerate}
\item Is $(\forall x \in \RR)(\forall y \in \RR) \, A(x,y)$ true or false? Explain your answer.
\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 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 14]}} For each of the following sentences, write out what it means in words, state whether it is true or false, and prove your answer.
\begin{enumerate}
\item $(\exists ! \, x \in \RR) (2x + 7 = 3)$.
\item $(\exists ! \, x \in \RR) (x^2 - 4x + 3 < 0)$.
\item $(\exists ! \, x \in \ZZ) (x^2 - 4x + 3 < 0)$.
\item $(\exists ! \, x \in \RR) (x^2 - 4x + 4 = 0)$.
\item $(\exists ! \, x \in \RR) (x^2 - 4x + 5 = 0)$.
\item $(\forall x \in \RR) (\exists ! \, y \in \RR) (x + y = 0)$.
\item $(\forall x \in \RR) (\exists ! \, y \in \RR) (xy = 1)$.
\item $(\forall x \in \RR) [\text{if $x \neq 0$, then $(\exists ! \, y \in \RR) (xy = 1)$}]$.
\item $(\forall x \in \RR) (\exists ! \, y \in \RR) (xy = 0)$.
\item $(\forall x \in \RR) [\text{if $x \neq 0$, then $(\exists ! \, y \in \RR) (xy = 0)$}]$.
\end{enumerate}

\end{enumerate}

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\begin{center}
\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 Read the (incorrect) proof given in \textbf{{[Falkner Section 5 Exercise 8]}}, which purports to prove by induction that all horses have the same color.
\begin{enumerate}
\item Modify this proof to obtain (incorrect) proofs of the following (false) statements:
\begin{enumerate}
\item All students at OSU have the same favorite food.
\item All songs are in the same key.
\item All rivers flow in the same direction.
\item All mountains have the same height.
\end{enumerate}
\item Each of these proofs makes the same error. What is it?
\end{enumerate}
\end{enumerate}




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