 
\documentclass {article}
\usepackage{amsfonts}
\usepackage{fancyhdr}
\usepackage{graphicx}


\setlength{\textwidth}{6.5in}
\setlength{\textheight}{8.5in}

\setlength{\oddsidemargin}{-.3in}
\setlength{\evensidemargin}{-.3in}

\setlength{\parindent}{0pt}
\setlength{\parskip}{12pt}


\def\dt        {{\Delta t}}
\def\dx        {{\Delta x}}

\def\eq        {\, = \,}
\def\gt        {\, > \,}
\def\implies   {{\, \Longrightarrow \,}}
\def\norm      {{| \! |}}


\begin{document}

\pagestyle{fancy}

\lhead{\bf HOMEWORK 1}
\chead{\bf  DUE:  Fri., Mar. 30}

\rhead{\bf  NAME: \hspace{80} }
\noindent

\cfoot{\thepage}
 


{\bf DIRECTIONS:}  

\begin{itemize}                 
         \item Turn in your homework as {\bf SINGLE-SIDED} typed or handwritten pages.  
         \item {\bf STAPLE} your homework together.  Do not use paper clips, folds, etc.
	\item {\bf STAPLE} this page to the front of your homework.
         \item {Be sure to write your name on your homework.}
	\item Show all work, {\bf clearly and in order}.
\end{itemize}

{\bf You will lose point 0.5 points for each instruction not followed.}

\bigskip\bigskip
\centerline{\begin{tabular}{|c||c|c|}\hline
& & \\
Questions & \quad Points \quad & \qquad Score \qquad\,\\  \\ \hline\hline 
& & \\
1 & 0.5 & \\ \hline
& & \\
2& 0.5 & \\ \hline
& & \\
3& 1 & \\ \hline
& & \\
4& 1 & \\ \hline
& & \\
5& 6 & \\ \hline
& & \\
6& 1 & \\ \hline
\hline
& & \\
Total& 10 & \\
& & \\ \hline
\end{tabular}}

\newpage

{\bf Problem 1:}  (0.5 points) Read ``A Guide to Writing Mathematics'' by K. Lee.  List and describe five items pertaining to writing math of which you were previously unaware.  {\it (Use full sentences.)}

{\bf Problem 2:}  (0.5 points) Sections I and II of ``Writing Proofs'' by T. Hsu.  List and describe five items pertaining to writing mathematical proofs of which you were previously unaware.   {\it (Use full sentences.)}

{\bf Problem 3:} (1 point)  Let $A = \{ b,f,p,c,a\}$.  

{\bf (a)} (0.5 points) List all the subsets of $A$.

{\bf (b)} (0.5 points) List all the proper subsets of $A$.

{\bf Problem 4:} (1 point)  In each of the following, form a set whose elements are the symbols of the given word or phrase (remember, a `space' is a symbol too).

{\bf (a)} (0.25 points) BUBBLE

{\bf (b)} (0.25 points) MATHEMATICS IS FUN!

{\bf (c)} (0.25 points) MADMAN

{\bf (d)} (0.25 points) HUH?

{\bf Problem 5:} (6 points)  Let $A$, $B$, $C$ be subsets of some universal set $X$.  Label the following true or false.  If true, prove.  If false, provide a counter example.

{\bf (a)} (1 point)  $A \cap (B \cap C) = (A \cap B) \cap C$.

{\bf (b)} (1 point) $A\setminus B = B \setminus A \iff B= \emptyset$.

{\bf (c)} (1 point) $\emptyset \subset \emptyset$.

{\bf (d)} (1 point) $A \cap B = X \iff A=X$ and $B=X$.

{\bf (e)} (1 point) $(A\cap B)^c = A^c \cup B^c$.

{\bf (f)} (1 point) $A\cap(B \bigtriangleup C) = (A \cap B) \bigtriangleup (A \cap C)$

{\bf Problem 6:}  (1 point)  In 1895, George Cantor gave the following definition:  ``By a set we shall understand any collection into a whole of definite distinguishable objects of our intuition or thought.  The objects will be called members of the collection.''   In 1902, the philosopher and mathematician Bertrand Russel constructed and object of his intuition satisfying Cantor's definition of a set, but that lead to a logical contradiction (Russell's Paradox):
\begin{quote}
{\it 
Consider a village in which there lives one barber.  The barber does not shave people who shave themselves.  Moreover, he shaves all those people who do not shave themselves.}
\end{quote}
Analyze this situation.  Why does it result in a paradox? 




\end{document}