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\pset{3}{February 23, 1995}{In recitation, March 3, 1995}

%	{\bf Reading:\ }


\bigskip
\begin{problems}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 1
\problem

Let $n$ be a natural number greater than $2$. Define relation $R$ on 
$\integers$ by
\[a \, R \, b \; \iff \;  n \mbox{ divides } a-b. \]
 
\bparts
\ppart Prove that $R$ is an equivalence relation.
\ppart Identify the total number of equivalence classes and describe them.
%	the equivalence classes.
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 2
\problem 

The following relation $\prec$ defines a partial order on 
$\{2,3,4, \cdots, 100 \}$:
\[
a \, \prec \, b \; \iff \; a \mbox{  divides  } b. 
\]
%\mbox{  iff }   where $a, b \in 

\bparts
\ppart Describe the minimal and maximal elements of this poset.
\ppart Find a longest chain.
\ppart Find a smallest antichain cover and prove that it is the smallest.
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 3
\problem
Prove that any partial ordering of a finite set 
can be extended to a total ordering of that set.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 4
\problem

Let $\prec$ be a partial ordering. Define $R$ by 
\[ b \, R \, a \;  \iff  \; a \prec b.\]
Show that $R$ is also a partial ordering.


\end{problems}

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