% If you are working from this template to just write up your own
% solutions, I'm trying to keep it simple for you and there are only a
% few places you need to edit. On line 33 there's a place where the
% phrase "YOUR NAME HERE" appears; change it to your actual
% name. Several places below there are places where the phrase "YOUR
% ANSWER HERE" appears, and directly below (or replacing) those lines
% are good places to put your answers.
%
% One important thing you need to do, however (due to the complicated
% way I use a single form to produce both the question sheets and
% solution writeups) is to make sure that you keep the word 'solution'
% in your filename, or else your answers won't be printed at all.
\documentclass[12pt]{article}
\usepackage[margin=1in,dvips]{geometry}
\usepackage{fancyhdr,lastpage}
\usepackage{amsmath,amsfonts,amsthm,amssymb}
\usepackage{tikz}
\usepackage{substr}
\IfSubStringInString{\detokenize{solution}}{\jobname}{%
% Solutions
\newcommand\qfill{}
\newcommand\qpage{}
\newcommand\qonly[1]{}
\newcommand\answer[1]{#1}
\newcommand\question[1]{{\em #1}}
\newcommand\rubric[1]{}
}
{%
% Question sheets
\newcommand\qfill{\vfill}
\newcommand\qpage{\newpage}
\newcommand\qonly[1]{#1}
\newcommand\answer[1]{}
\newcommand\question[1]{#1}
\newcommand\rubric[1]{}
}
\newtheorem{lem}{Lemma}
\newtheorem{prop}{Proposition}
\pagestyle{fancy}
\lhead{MATH 311-01}
\chead{Problem Set \#10\answer{ solutions}}
\rhead{}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due April 14, 2016}}
\newcommand{\logicnot}{{\sim}}
\begin{document}
\begin{enumerate}
\item \question{Let $R$ be an equivalence relation on a set $S$. For
$A\subseteq S$, we define $R_A$ to be the \emph{restriction} of
$R$ to elements of the set $A$, i.e., $R_A$ is a relation on $A$
such that for any $a,b\in A$, the statement $a\mathrel{R_A}b$ is
true if and only if $a\mathrel{R} b$ is true. Alternatively, one
can define $R_A$ setwise as equal to $R\cap(A\times A)$.}
\begin{enumerate}
\item \question{Prove that for any equivalence relation $R$ on a set
$S$ and subset $A$ of $S$, the restriction $R_A$ is an
equivalence relation on $A$.}
\answer{
\begin{proof}
\end{proof}
}
\item \question{Briefly explain why $R_A$ might not be an
equivalence relation on $S$.}
\answer{}
\item \question{Prove that for any $R$, $A$, and $S$ as above and
$x\in A$, the equivalence class $[x]_{R_A}$ over the restricted
relation is equal to the intersection of $A$ and the equivalence
class $[x]_R$ over the unrestricted relation.}
\answer{
\begin{proof}
\end{proof}
}
\end{enumerate}
\item \textbf{Bonus question:} \question{Let $R_1$ and $R_2$ be
equivalence relations on a set $S$. Their intersection, as
described below, is on the relations considered as sets of ordered
pairs.}
\begin{enumerate}
\item \question{Prove that $R_1\cap R_2$ is also an equivalence
relation on $S$.}
\answer{
\begin{proof}
\end{proof}
}
\item \question{With proof, identify the equivalence classes of the
relation $R_1\cap R_2$.}
\answer{}
\end{enumerate}
\end{enumerate}
\end{document}