\documentclass[11pt,pages]{exam}
\usepackage{amsmath,amsfonts,times, graphicx,multicol,setspace}

\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{(\theenumi)}

\newcommand{\ds}{\displaystyle}
\newcommand{\ben}{\renewcommand{\theenumi}{\alph{enumi}}\renewcommand{\labelenumi}{(\theenumi)}\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}

\newcommand{\ba}{\begin{longanswer}}
\newcommand{\ea}{\end{longanswer}}


\newcommand{\bp}{\addtocounter{problemnum}{1}\begin{problem}}
\newcommand{\ep}{\end{problem}}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
 \newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Inn}{\operatorname{Inn}} 

\newcommand{\ra}{\rightarrow} 
\newcommand{\id}{\operatorname{id}} 


\newcommand{\<}{\langle}
\renewcommand{\>}{\rangle}


\title{Exam 1}
\target{Math 157}

\begin{document}


\begin{exam}{Thursday, October 3, 2013}

\bp Let $S=\{\heartsuit, \diamondsuit, \clubsuit,\spadesuit\}$
and define two binary operations $+$ and $\times$ as follows:

{%\Large
\begin{center}
\begin{tabular}{|l||c|c|c|c|}
\hline 
$+$ & $\heartsuit$ & $\diamondsuit$ & $\clubsuit$  & $\spadesuit$  \\
\hline \hline
$\heartsuit$ & $\heartsuit$ & $\diamondsuit$ & $\clubsuit$ & $\spadesuit$ \\
\hline
$\diamondsuit$ &$\diamondsuit$  & $\heartsuit$ & $\spadesuit$ & $\clubsuit$ \\
\hline
$\clubsuit$ & $\clubsuit$ & $\spadesuit$ & $\heartsuit$ & $\diamondsuit$ \\
\hline
$\spadesuit$ & $\spadesuit$ & $\clubsuit$ & $\diamondsuit$ & $\heartsuit$ \\
\hline
\end{tabular}
$\qquad$
\begin{tabular}{|l||c|c|c|c|}
\hline 
$\times$ & $\heartsuit$ & $\diamondsuit$ & $\clubsuit$  & $\spadesuit$  \\
\hline \hline
$\heartsuit$ & $\heartsuit$ & $\heartsuit$ & $\heartsuit$ & $\heartsuit$\\
\hline
$\diamondsuit$ & $\heartsuit$  & $\diamondsuit$ & $\heartsuit$ & $\diamondsuit$ \\
\hline
$\clubsuit$ & $\heartsuit$ & $\heartsuit$ & $\clubsuit$&$\clubsuit$\\
\hline
$\spadesuit$ & $\heartsuit$ & $\diamondsuit$ & $\clubsuit$  & $\spadesuit$  \\
\hline
\end{tabular}
\end{center}
} 
\ben
\addtolength{\itemsep}{2in}
\item Which element of $S$ is an identity for the operation $+$?

\item Solve the equation $\diamondsuit x+ \clubsuit =
  \spadesuit$ for $x$.

\item Only one of the field axioms is not satisfied by $S$ with
  $+$ and $\times$.  Which one?

\een

\ep


% \bp Let $A,B$ and $C$ be sets.  
% \ben
% \item One of the following
% conditions is sufficient for $A\setminus B = A
% \setminus C.$  Which one? 
% \begin{itemize}
% \item $A \subset (B\cup C)$
% \item $(B\cup C) \subset A$
% \item $A\cap B\cap C =\emptyset$
% \item $(B\cap C) \subset A$
% \item $A\cap C=A \cap B$
% \end{itemize}
% \item Prove that the condition you identified implies that $A\setminus
%   B = A
% \setminus C$.
% \vspace{4in}
% \item Give an example to show that the identified condition is not
%   \emph{necessary} for $A\setminus B = A
% \setminus C$.
% \een
% \ep


\bp Let $A,B$ and $C$ be sets.  
\ben
\item One of the following
conditions is sufficient for $(A\setminus B)\setminus C = A
\setminus (B\setminus C).$  Which one? 
\begin{itemize}
\item $A \subset (B\cup C)$
\item $(B\cup C) \subset A$
\item $A\cap B\cap C =\emptyset$
\item $C \subset (B\setminus A)$
\item $A\cap B=C \cap B$
\end{itemize}
\item Prove that the condition you identified implies that  $(A\setminus B)\setminus C = A
\setminus (B\setminus C).$
\vspace{5in}

\item Give an example to show that the identified condition is not
  \emph{necessary} for $A\setminus
  B = A
\setminus C$.

\een
\ep

\bp True or False.  Give brief, but conclusive evidence, to support
your answer.
\ben
\addtolength{\itemsep}{2in}
\item For all sets $S$ and for all $A\subseteq S$ there exists a
  unique set
  $B\subseteq S$ with $A\cup B =S$.

\item For all sets $A\subseteq \R$, either $A$ or $\R\setminus A$ is bounded
  above.

\item For all $x,y\in \R$, if $x^2<y^2$ then either $x<y$ or $-x<y$.

\item For all $x\in \R$ and for all $n\in \N\left
    (|2x-6|<\dfrac{1}{n}\Rightarrow x=3\right).$

\een
\ep

\bp 
\ben
\item Use mathematical induction to prove that 
$\ds \sum_{k=1}^n k^3 =\frac{(n)^2(n+1)^2}{4}$ for all $n\in \N$.
\een
\ep
\addtocounter{problemnum}{-1}
\bp
\ben
\addtocounter{enumi}{1}
\item Use this result to compute the area of the region pictured below 
(the
vertical distance between the point $b$ units from $0$ is
$\frac{1}{2}b^3$).
\bigskip

\begin{center}
\includegraphics[scale=0.7]{exam1_pic.pdf}
\end{center}
\een
\ep

\pagebreak



\end{exam}
\end{document}

