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\pagestyle{myheadings} \markright{Math 1110 (Fall 2011) \hspace{2in}
HW1}

\begin{document}


%% PREAMBLE

\thispagestyle{empty}

\begin{minipage}[t]{1.1in}
{\vspace{-0.2in}
\includegraphics[width=1.1in]{cornell-seal.pdf}}
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\hfill
\begin{minipage}[t]{1.8in}
%\vspace{.05in}
{\large Math 4310

\vspace{.15in}

Homework 1

\vspace{.15in}

Due 8/29/12}
\end{minipage}
\hfill
\begin{minipage}[t]{3.4in}
%\vspace{.05in}
{\large Name: \hrulefill

\vspace{.15in}

Collaborators: \hrulefill

\vspace{.15in}

 \hrulefill 

}
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%\vskip 0.1in

\begin{flushright}
\begin{boxedminipage}[t]{3in}
{\footnotesize{
\medskip $\phantom{o}$GRADES \\
\rule{0mm}{8mm} $\phantom{o}$Exercises   \hrulefill $\phantom{o}$ /   50 \ \\ 
\rule{0mm}{8mm} $\phantom{o}$Extended Glossary \ \\ 

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\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Component & Correct? & Well-written? \\ \hline
Definition & $\phantom{5}/6$ & $\phantom{5}/6$ \\ \hline
Example & $\phantom{5}/4$ & $\phantom{5}/4$ \\ \hline
Non-example & $\phantom{5}/4$  & $\phantom{5}/4$ \\ \hline
Theorem & $\phantom{5}/5$  & $\phantom{5}/5$  \\ \hline
Proof & $\phantom{5}/6$  & $\phantom{5}/6$  \\ \hline\hline
Total & $\phantom{5}/25$  & $\phantom{5}/25$  \\ \hline
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\vspace*{-66mm}


\parbox{82mm}{\footnotesize{
{\em 
\hskip 0.15in Please print out these pages.  
I encourage you to work with your classmates on this homework.
Please list your collaborators on this cover sheet.  (Your grade
will not be affected.)
Even if you work in a group, you should write up 
your  solutions yourself! 
You should include all computational details, and proofs should be carefully 
written with full details.
%These problems will  be assessed for completeness.
As always, please write neatly and legibly. 

\hskip 0.15in Please follow the instructions for the  ``extended glossary''
on separate paper (\LaTeX\ it if you can!)
Hand in your final draft,
including full explanations and write your glossary in complete, 
mathematically and grammatically correct sentences.  
Your answers will be  assessed for style and accuracy.  

\hskip 0.15in Please {\bf staple} this cover sheet,
your exercise solutions, and your glossary together, in that order,
and hand in your homework in class.  

}
} }


\vskip 12mm

%% END PREAMBLE




\noindent {\bf Exercises.} 

\begin{enumerate}
\item Use induction to verify the following statements.
\babc
\item For any integer $n\geq 1$, $\ds{1^{2}+2^2 +\cdots + n^2 = \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}}$.

\item Recall that the {\bf Fibonacci numbers} are defined by $f_1=f_2=1$, 
and  $f_n=f_{n-1}+f_{n-2}$ for values of $n\geq 3$.  So the Fibonacci sequence is
$
1,1,2,3,5,8,13,21,34,55,89,144,233,\dots .
$
For any integer $n\geq 1$, we have  $\ds{\sum_{k=1}^n (f_k)^2 = f_n\cdot f_{n+1}}$.

\item For any integer $n\geq 1$, the Fibonacci numbers satisfy 
$
f_1+f_3+\cdots +f_{2n-1} = f_{2n}.
$
\eabc

\item ({\bf Curtis, p.\ 15 \#4}) Let $\F_2$ denote the set consisting of two elements
$\{ 0,1\}$, with the operations defined by the tables

\begin{center}
\begin{tabular}{c|cc}
$+$ & 0 & 1 \\ \hline
0 & 0 & 1 \\
1 & 1 & 0
\end{tabular}
\hskip 0.5in
\begin{tabular}{c|cc}
$\cdot$ & 0 & 1 \\ \hline
0 & 0 & 0 \\
1 & 0 & 1
\end{tabular}
\end{center}

\noindent Show that $\F_2$ is a field, with the additional property that 
$2\alpha = \alpha + \alpha =0$ for all $\alpha\in\F_2$.

\item Is the set $\big\{\ a+b\sqrt{7}\ \ \big| \ \ a,b\in \Q\ \big\}$ a subfield of $\R$?
If so, please provide a proof.  If not, explain why not.

\end{enumerate}



\vfill\pagebreak


\noindent {\bf Extended Glossary.}
Please give a definition of a {\bf perfect square}.  
Then give an example of a perfect square, an example of 
a number that is not a perfect square, and state and prove a theorem 
about perfect squares.

\vskip 0.1in

You may work in groups, but please write up 
your solutions {\bf yourself}. If you do work together,
your group should come up with at least 
two examples, two non-examples, and two theorems.
Each one (example/non-example/theorem) should be included in some group member's 
extended glossary.
Your solutions should be written formally,
so that we could cut and paste them into a textbook.


\end{document}