\documentstyle[12pt,twoside,amssymb]{article}
\input{macros-course}

\let\comp\circ
\let\union\cup
\let\intersect\cap
\newcommand{\Cal}[1]{{\cal #1}}

\begin{document}

% NOTE: I used book's notation for composition, i.e. $S\comp R = R;S$.
 
%\problem

Let $A$ be a set. Answer the following questions about relations on
$A$. If your answer is {\em no}, give a counterexample if $A$ is the
three element set $\{1,2,3\}$.

\bparts

\ppart
Let $R$ and $S$ be transitive relations on $A$, and $S\comp R$ the
composition of $R$ with $S$ (as defined in the book). Is $S\comp R$
transitive?

% transitiviry fails
% let R = (1,3), (2,2)
% let S = (2,3), (2,2)
% then (1,2) and (2,3) are in R;S, but (1,3) is not

\ppart
Let $R$ and $S$ be partial orders on $A$. Is $R\intersect S$ a partial
order?

% yes

\ppart
Let $E$ be an equivalence relation and let $R$ be a partial order on
$A$. Is the relation $E\intersect R$ a partial order? Is it an
equivalence relation?

% yes, it is the identity relation 
% it may be a better question to ask them to describe the relation

\ppart
Let $R$ be a relation on $\Cal{P}(A)\times\Cal{P}(A)$ defined by
\[
(s_1,s_2)\ R\ (t_1,t_2)\text{ iff }s_1\oplus s_2\subseteq t_1\oplus t_2{.}
\]
Is $R$ a partial order?

\noindent
Note: $\Cal{P}(A)$ denotes the power set of $A$, and $s\oplus t$
denotes the set of all elements in either $s$ or $t$ but not in both.

% no antisymerty fails (reflexivity and transitivity hold)
% ({1,2}, {2,3}) R ({1}, {3}) and ({1}, {3}) R ({1,2}, {2,3}), but

\ppart
Let $R$ be an arbitrary relation on $A$, and let $S$ be reflexive and
transitive relation on $A$. Is the relation $S\comp(R\intersect S)$
transitive?

% yes

\ppart
Let $R$ be a total order on $A$, and let $E$ be an equivalence
relation on $A$. Is $\overline{R}\union(E\intersect R)$ a total order
on $A$?

% yes

\ppart
Let $R$ be a partial order on $A$, and let $I$ be identity relation on
$A$. Is $\overline{R}\union I$ an partial order on $A$?

% no, let R = (2,1), (3,1), (1,1), (2,2), (3,3)
% then (2,3) and (3,2) are in the complement
 
\eparts

\end{document}
