\documentclass{article}
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\usepackage{amsmath,amssymb,amsthm}
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\usepackage{tikz}
\usetikzlibrary{arrows,shapes,positioning}
\newcommand{\XXX}{{\color{red} \textsf{\textbf{XXX}}}}
\begin{document}
% To run this file, you'll need to have LaTeX installed. All of the
% lab computers have it. You can download a complete LaTeX
% distribution from
% https://www.tug.org/texlive/acquire-netinstall.html.
% Once you have LaTeX, you can either build your PDF on the
% command-line by running 'pdflatex hwXX.tex' to generate hwXX.pdf, or
% you can use an editor like LyX, TeXShop, or ShareLaTeX which
% automates the building of your PDF.
% Please print out your solution double-sided (a/k/a duplex) and bring
% it to class on the Wednesday it's due.
\noindent
% fill in the XXXs below
{\Large CS055 HW11 \qquad Name: \XXX \qquad CAS ID: \XXX} \\[.5em]
% I encourage you to collaborate, but please list any other students
% you talked to about the homework. If you worked alone, please just
% remove the XXXs.
Collaborators: \XXX
% Okay! Solve the problems below. please don't delete the problem
% statement or the ``enumerate'' bracketing which provides the
% numbering.
\begin{enumerate}
\item \label{rnsubr} If $R \subseteq A \times A$ is transitive, prove
that $R^n \subseteq R$ for all $n \ge 1$. (Hint: be careful with
your base cases!)
\textbf{Proof:} \XXX
\item If $R \subseteq A \times A$ is transitive, prove that $R^n$ is
transitive for all $n \ge 1$.
\textbf{Proof:} \XXX
\item Consider the set $\textbf{2} = \{ \top, \bot \}$. How many
different partial orders are there on this set? List all of them,
but no need to prove that they are partial orders.
\textbf{Proof:} \XXX
\item Suppose $\langle A, R \rangle$ is a poset. Prove that $\langle
A, R^{-1} \rangle$ is also a poset. (Hint: not sure where to start?
Look up the definition of $R^{-1}$!)
\textbf{Proof:} \XXX
\item Consider the poset $\langle \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \},
\mathord{\mid} \rangle$, where $a \mid b$ when $a$ evenly divides
$b$. So $2 \mid 4$, but $2 \nmid 3$.
\begin{enumerate}
\item What are the maximal elements?
\textbf{Answer:} \XXX
\item What are the minimal elements?
\textbf{Answer:} \XXX
\end{enumerate}
\item Prove that every totally ordered set is a lattice.
\textbf{Proof:} \XXX
\end{enumerate}
\end{document}