\documentclass[11pt]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb,amscd}
\usepackage{graphicx, epsfig}
\usepackage{multicol}
\usepackage{boxedminipage}
\usepackage{subfigure}
\usepackage{euler,palatino,array}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref}
\usepackage{wrapfig}
\usepackage{dsfont}



\setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in} 
\setlength{\parindent}{0in}
\newcounter{prob}
\setcounter{prob}{0}
\newcommand{\problem}{\stepcounter{prob}\medskip{\bf Presentation problem \theprob.}\ }
\newcommand{\answer}{\medskip{\bf Answer to Question \theprob.}\ }
\newcommand{\solution}{\medskip{\bf Solution to Question \theprob.}\ }



% ***** Some useful abreviations *****
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\ra}{\rightarrow}
\newcommand{\ds}{\displaystyle}
\newcommand{\bnum}{\begin{enumerate}} %Use with \item to create numbered lists
\newcommand{\enum}{\end{enumerate}}
\newcommand{\babc}{\bnum\renewcommand{\labelenumi}{(\alph{enumi})}}%Use with \item for abc lists
\newcommand{\eabc}{\end{enumerate}}
\newcommand{\bijk}{\bnum\renewcommand{\labelenmi}{\roman{enumi}.}}%Roman numeral lists
\newcommand{\eijk}{\end{enumerate}}

\renewcommand{\ss}{\smallskip}
\newcommand{\ms}{\medskip}
\newcommand{\bs}{\bigskip}




\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}}
\end{minipage}
\hfill
\begin{minipage}[t]{1.8in}

{\large Math 4320

\vspace{.15in}

Homework 1

\vspace{.15in}

Due 1/28/15}
\end{minipage}
\hfill
\begin{minipage}[t]{3.4in}

{\large Name: \hrulefill

\vspace{.15in}

Collaborators: \hrulefill

\vspace{.15in}

 \hrulefill 

}
\end{minipage}


\begin{flushright}
\begin{boxedminipage}[t]{3in}
{\footnotesize{
\medskip $\phantom{o}$GRADE \\
\rule{0mm}{8mm} $\phantom{o}$   \hrulefill $\phantom{o}$ \Large/  \phantom{ 5000} \ \\ 
\smallskip

\footnotesize{\hfill (For instructors's use only)}
}}
 \end{boxedminipage}
\end{flushright}

\vspace*{-33mm}


\parbox{82mm}{\footnotesize{
{\em 
\hskip 0.15in Please print out this page.  
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.
As always, please write neatly and legibly. 

 

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

}
} }


\vskip 12mm

%% END PREAMBLE




\noindent {\bf Exercises.} 

\begin{enumerate}
\item Define a relation $\sim$ on $\R^2$ by $(x_1, y_1) \sim (x_2, y_2)$ if and only if there exists $r\neq 0$ such that
$(x_1, y_1) =(r\cdot x_2, r\cdot y_2)$. Prove that $\sim$ is an equivalence relation.

Alternatively, do the following exercise which is a generalization of the above. Let $V$
be a vector space and define $\sim$ on $V$ by $x \sim y$ if and only if there exists a nonzero scalar $r$ such
that $x = r\cdot y$. Prove that $\sim$ is an equivalence relation.

\item  Show that the following two definitions of injective are the same:

(a) A function $f : X \to Y$ is injective if $f(x) = f(y)$ implies $x = y$ for all $x$ and $y$.

(b) A function $f : X \to Y$ is injective if there exists $\ell : Y \to X$ with $\ell \circ f = \mathds{1}_X$.

\item From the book, p.\ 103, $\# 2.1$ (v), (vi), (vii) and (ix).


\item From the book, p.\ 106, $\# 2.18$.

\item From the book, p.\ 123, $\# 2.21$ (ii), (iv) and (x).

\end{enumerate}



\end{document}
