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\begin{document}

\begin{center}
{\large{\textsc{Homework 7}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
\end{center}

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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 3 Exercise 7]}} Let $P$ be the sentence
\[ (\exists x \in \RR) \big( x \geq 0 \ \text{and} \ \sqrt{x+2} < \sqrt{x} + \sqrt{2} \big). \]
\begin{enumerate}
\item Use one of the generalized De Morgan's laws and one of the ordinary De Morgan's laws to show that $\neg P$ is logically equivalent to
\[ (\forall x \in \RR) \big( x < 0 \ \text{or} \ \sqrt{x+2} \geq \sqrt{x} + \sqrt{2} \big). \]
\item Is $P$ true or false? Provide a proof for 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}

\item \textbf{{[Falkner Section 5 Exercise 2]}} Prove by induction that for each $n \in \NN$,
\[ 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}. \]

\end{enumerate}

\vspace{.2 in}
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\begin{center}
\textbf{Practice Problems}
\end{center}

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 3 Exercise 8]}} Which of the variables $x$ and $y$ is free in the sentence $P(x,y)$. Answer the same question about each of the four sentences $(\exists y) P(x,y)$, $(\forall x) (\exists y) P(x,y)$, $(\forall x) P(x,y)$, and $(\exists y) (\forall x) P(x,y)$.

\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}




\end{document}
