%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{amsart}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   ENVIRONMENTS
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%% \theoremstyle{plain} %% This is the default
\newtheorem{th}{Theorem}[section]
\newtheorem{cor}[th]{Corollary}
\newtheorem{lm}[th]{Lemma}
\newtheorem{prp}[th]{Proposition}
\newtheorem{clm}[th]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[th]{Definition}
\newtheorem{exmp}[th]{Example}
\newtheorem{prb}[th]{Problem}
%
\theoremstyle{remark}
\newtheorem{rem}[th]{Remark}
%
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Math macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Logical symbols
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\satf}{\models}
\newcommand{\id}{{\text{\sl id}}}       % The identity map
\newcommand{\miff}{\Longleftrightarrow} % math `if and only if'
\newcommand{\mimplies}{\Longrightarrow} % math `implies'
\newcommand{\mimplied}{\Longleftarrow}  % math `implied by'
\newcommand{\miffdef}{\stackrel{\text{\normalshape
   def}}{\miff}}                        % defining iff
\newcommand{\eqdef}{\stackrel{\text{\normalshape
   def}}{=}}                            % defining equation
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
%
%\newcommand{\neareq}{\stackrel{\ae}{=}}
%\newcommand{\aeqv}{\,\stackrel{\text{\normalshape ae}}{\equiv}\,}
%\newcommand{\aeq}{\,\stackrel{\text{\normalshape ae}}{=}\,}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%  Classical math
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\tmod}[1]{\text{\normalshape mod~$#1$}}

\newcommand{\BZ}{\Bbb Z} % integers
\newcommand{\BQ}{\Bbb Q} % rationals
\newcommand{\BR}{\Bbb R} % reals
\newcommand{\BC}{\Bbb C} % complex numbers
\newcommand{\re}{\text{\normalshape Re}\,}    % real part
\newcommand{\im}{\text{\normalshape Im}\,}    % imaginary part
%
\newcommand{\meet}{\wedge}
\newcommand{\join}{\vee}
\newcommand{\union}{\cup}
\newcommand{\inter}{\cap}  % intersection
\newcommand{\iso}{\cong}   % isomorphic
\newcommand{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\newcommand{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\bold{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\Ker}{\text{\normalshape Ker}\,}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%  Operators on algebras
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%  \m is used to produce models (algebras) as follows:
%%%  typing `\m a'  produces in bold face, A.
%%%  \vr does the same for varieties
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\m}[1]{\mbox{${\bold{\uppercase{#1}}}$}}
\newcommand{\sm}[1]{\mbox{{\scriptsize\bf{\uppercase{#1}}}}}
\newcommand{\vr}[1]{{\cal{\uppercase {#1}}}}
%
\newcommand{\Var}{{\bold{V}}}
\newcommand{\Ho}{{\bold{H}}}
\newcommand{\Su}{{\bold{S}}}
\newcommand{\Pd}{{\bold{P}}}
\newcommand{\Iso}{{\bold{I}}}
\newcommand{\Ps}{{\bold{P}}_{\text{\normalshape s}}}
\newcommand{\fin}[1]{{\bold{#1}}_{\text{\normalshape fin}}}
%
\newcommand{\Pol}{{\text{\normalshape Pol}\,}}
\newcommand{\Clo}{{\text{\normalshape Clo}\,}}
\newcommand{\Con}{{\text{\normalshape Con}\,}}
\newcommand{\Conb}{{\bold{Con}\,}}       % Con(A) as a lattice
%\newcommand{\Cg}[1]{\text{\normalshape Cg}^{\m{#1}}}
%\newcommand{\Sg}[1]{\text{\normalshape Sg}^{\m{#1}}}
\newcommand{\Cg}[1]{\text{\normalshape Cg}^{\sm{#1}}}
\newcommand{\Sg}[1]{\text{\normalshape Sg}^{\sm{#1}}}
\newcommand{\cg}{\text{\normalshape Cg}\,}
\newcommand{\sg}{\text{\normalshape Sg}\,}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  macros for tame congruence theory.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\M}[3]{\text{\normalshape
   M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\normalshape
   Min}_{\m{#1}}}
%
\newcommand{\typ}{{\text{\normalshape typ}\,}}
\newcommand{\ann}{{\text{\normalshape ann}\,}}
\newcommand{\Sep}{{\text{\normalshape Sep}\,}}
\newcommand{\ssb}{{\text{\normalshape ssb}\,}}
\newcommand{\sst}{{\text{\normalshape sst}\,}}
%
\newcommand{\utyp}{{\bold{1}}\,} % unary
\newcommand{\atyp}{{\bold{2}}\,} % affine
\newcommand{\btyp}{{\bold{3}}\,} % boolean
\newcommand{\ltyp}{{\bold{4}}\,} % lattice
\newcommand{\styp}{{\bold{5}}\,} % semilattice
\newcommand{\ityp}{{\bold{i}}\,}
\newcommand{\jtyp}{{\bold{j}}\,}
%
\newcommand{\solv}{\stackrel{s}{\sim}}
\newcommand{\ssolv}{\stackrel{ss}{\sim}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Abbreviations
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\s}[1]{\underset{\mbox{\scriptsize\sl #1}}{\textstyle\bold \sim}}
%\newcommand{\s}[1]{\underset{#1}{\textstyle\bold \sim}}

\begin{document}

\noindent
{\large In-Class Exercise \#2.}

\bigskip

Prove the lemma.

\bigskip

\noindent
{\bf Lemma.} Suppose that $L$ and $R$ are permuting 
equivalence relations on a set $X$.
Suppose that each of the relations 
$L$ and $R$ has exactly $m$
equivalence classes, all of which have size $n$.
(So $|X| = mn$.)
Prove that there is a set 
$\{x_1,\ldots,x_m\}\subseteq X$ which contains exactly
one element from every $L$-class and exactly one element 
from every $R$-class.

\bigskip

\vfill

Prove P. Hall's Theorem.

\bigskip

\noindent
{\bf Theorem.} If $H<G$ are finite groups, then
there is a common system of representatives for the left
cosets and the right cosets of $H$ in $G$.

\vfill



\end{document}

