1. Prove that p E F- [p] E F$. (Hint: the forward direction is immediate. For the converse, consider using the empty string for x in the definition of ~.)

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1. Prove that p EF- [p] € F$. (Hint: the forward
direction is immediate. For the converse, consider
using the empty string for x in the definition of ~.)
Transcribed Image Text:1. Prove that p EF- [p] € F$. (Hint: the forward direction is immediate. For the converse, consider using the empty string for x in the definition of ~.)
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\usepackage{enumerate}
\usepackage{ifthen}

\usepackage{amssymb}

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\setlength{\textwidth}{7.5in}

\newcommand{\comp}{ \,{\scriptstyle \stackrel{\circ}{}}\, } 
\newcommand{\nullset}{\mathrm{O}\!\!\!\!\big/\,}
\newcommand{\divides}{\,\Big|\,}
\newcommand{\myspace}{\mbox{\parbox[c][0.35in]{0.35in}{\hfill}}}
\newcommand{\myspacea}{\mbox{\parbox[c][0.35in]{0.65in}{\hfill}}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}

\newcommand{\force}{{$\mbox{\;}$}}

%\newcommand{\bigtimes}[1]{ \mbox{\raisebox{-5pt}{$\stackrel{\mbox{\resizebox{0.2in}{!}{$\times$}}}{\mbox{\resizebox{0.15in}{!}{$#1$}}}$}}}

\begin{document}

%\pagestyle{empty}

\noindent
\parbox{1.5in}{\bf Math 4710/5710} 
\hfill {\Large \bf  Set Test --- In Class} \hfill
\parbox{1.5in}{\bf \hfill September $15^{\mathrm{th}}$, 2014}

\vspace{0.1in}

\noindent {\large\bf Name:} \underline{\Large\color{red}\sc \quad Answer Key \quad} \hfill {\bf Be sure to show your work!}

\vspace{0.1in}

\noindent {\bf\large 1. (20 points)} Memorization

\begin{enumerate}[(a)]
   \item Let $T$ be a (non-empty) set with relation $<$. What properties must $(T,<)$ possess so that it is {\it totally ordered} (i.e. simply ordered)? Don't just state these properties. Define the properties themselves -- give details!

\vspace{0.1in}

\begin{itemize}
\item {\bf Irreflexive:} For all $a \in T$ we have $a \not< a$.
\item {\bf Transitive:} For all $a,b,c \in T$ such that $a<b$ and $b<c$ we have $a<c$.
\item {\bf Comparability:} For all $a,b \in T$  exactly one of the following holds: $a<b$, $b<a$, or $a=b$.
\end{itemize}
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