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


\begin{document}

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

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 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 $n \in \NN$,
\[ 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}. \]

\item In class, we saw a \textbf{correct} proof of the following statement: For each $n \in \NN$,
\[ 1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \]

Consider the \textbf{incorrect} proof of the same statement given on the next page.

\noindent\fbox{
\parbox{5.5in}{
\begin{proof}
\textbf{Base Case:} Since $1 = \frac{1 \cdot 2}{2}$, $P(1)$ is true. \\

\textbf{Inductive Step:} Let $n \in \NN$. Then
\begin{align*}
1 + 2 + \cdots + n + (n+1) &= \frac{(n+1)(n+2)}{2} \\
\frac{n(n+1)}{2} + (n+1) &= \frac{(n+1)(n+2)}{2} \\
n+1 &= \frac{(n+1)(n+2)}{2} - \frac{n(n+1)}{2} \\
n+1 &= \bigg(\frac{n+1}{2}\bigg) \big( (n+2) - n \big) \\
n+1 &= n+1.
\end{align*}
Thus, $P(n+1)$ is true.
\end{proof}
}
}\\

Explain, in complete sentences, what is wrong with this proof. How would you fix it?





\end{enumerate}

\vspace{.2 in}

\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 What is the error in all of these proofs?
\end{enumerate}
\end{enumerate}




\end{document}
