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\lhead{MATH 311-02}
\chead{Problem Set \#4}
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\rfoot{\bf due March 4}
\begin{document}
\begin{enumerate}
\item \textbf{(12 points)} \emph{Below are three existence statements
which are either true or false. For each of them, either prove
them true (by either an example or a nonconstructive proof) or
prove them false (by a disproof of existence).}
\begin{enumerate}
\item \emph{There is a real number $x$ such that $x^6+x^4+1=2x^2$.}
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% This here is an example of an expected response, with key parts
% left out, of course. Similar responses will exist for the
% following two parts.
This statement is true (or false!). The proof (or disproof!) follows.
\begin{proof}
Suppose there is a bleepity bloppity bloop. We go on at some
length building our logical scaffolding. Maybe we have a direct
proof and we show what we want to show. Maybe we just assert the
existence of an example, and that's all there is to our
proof. Maybe we're disproving it by contradiction and we reach
an absurdity at the end. Whatever your proof method is, you
eventually finish it.
\end{proof}
\singlespacing
\item \emph{There is an integer $n$ such that $4\mid n^2+2$.}
\onehalfspacing
\singlespacing
\item \emph{There is an integer $n$ such that $n^3\equiv 6\pmod 7$.}
\onehalfspacing
\singlespacing
\end{enumerate}
\item \textbf{(6 points)} \emph{Prove that there is exactly one
solution to the equation $x=\cos x$.}
\onehalfspacing
\singlespacing
\item \textbf{(6 points)} \emph{Prove that
$1+\frac14+\frac19+\cdots+\frac1{n^2}\leq 2-\frac1n$ for every
positive integer $n$.}
\onehalfspacing
% This is of course a proof by induction. A basic template for
% inductive proof is below.
\begin{proof}
We shall prove the above statement by induction on $n$. First, we
demonstrate the base case: put your usually straightforward
investigation of some small $n$ here.
Now, we assume by the inductive hypothesis that this, that, and
t'other. Then we do a bunch of logic-type stuff until you come up
with a firmly demonstrated inductive step.
\end{proof}
\singlespacing
\item \textbf{(8 points)} \emph{Consider the sequence given by the
recurrence:}
% An align environment is sort of a hybrid of math mode and
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% equations, all lined neatly up at their equals signs. The only
% difference between "align" and "align*" is that the former has
% equations indivudually nubmered; feel free to take off the
% asterisks in the environemnt below as an experiment.
\begin{align*}
a_1&=1\\
a_2&=4\\
a_3&=9\\
a_n&=a_{n-1}-a_{n-2}+a_{n-3}+4n-6\text{ for }n\geq4
\end{align*}
\emph{Explore the next few values of the recurrence and conjecture a
formula to explain their values. Then, prove your conjecture.}
\onehalfspacing
\singlespacing
\item \textbf{(8 points)} \emph{The Fibonacci numbers are given by the
recurrence:
\begin{align*}
F_1&=1\\
F_2&=1\\
F_n&=F_{n-1}+F_{n-2}\text{ for }n\geq3
\end{align*}
Prove that $F_n$ is even if and only if $n$ is divisible by 3.}
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% You only need to do the following if you want to -- which is why
% it's commented out in the basic template, so that it doesn't show
% up unless you uncomment it and work on it.
% \item \textbf{(4 point bonus)} \emph{The idyllic village of Salemo
% (population 150) has inhabitants who are happy, healthy, and
% expert logicians. They believe (oddly enough) that their
% prosperity and intelligence derives from a specific sort of
% ignorance: none of the villagers knows their own eye color. As
% none of the inhabitants wishes to ruin the others' success, anyone
% who does find out their own eye color during the day will leave
% town quietly at night (although everyone will learn that they left
% by the following morning). To avoid such a misfortune, they never
% speak of eye color and have no mirrors. However, one fine day, a
% stranger indiscreetly made reference to a blue-eyed villager (but
% not by name, or in any other way which identified a specific
% person). As it so happens, 50 of the villagers have blue eyes and
% the remaining 100 have brown eyes, and every villager knows the
% eye color of every other villager. What, if anything, will result
% from the stranger's indiscretion?}
% \onehalfspacing
\end{enumerate}
\end{document}