\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\begin{document}

\title{General Mathematics and Computational Science I}
\author{Exercise 3}
\date{September 11, 2007}
\maketitle

\begin{enumerate}

\item (Repeat from Exercise Sheet 2.)

  Let $X$ be a nonempty set with an \emph{equivalence relation} $\sim$
  on it.

Recall that this means that
\begin{itemize}
\item[(i)] $x \sim x$ for every $x \in X$ (``reflexivity''),
\item[(ii)] $x \sim y$ implies that $y \sim x$ for $x,y\in X$
  (``symmetry'') 
\item[(iii)] $x \sim y$ and $y \sim z$ implies $x \sim z$ for $x,y,z
  \in X$ (``transitivity'').
\end{itemize}
Also recall that for any $x\in X$, we define the \emph{equivalence
  class} represented by $x$ as
\[
  [x] = \{ y \in X \colon y \sim x \} \,.
\]

Prove that for all $x,y \in X$, 
\begin{equation*}
  [x] = [y] \qquad \text{if and only if} \qquad x \sim y \,.
\end{equation*}
\item Check whether each of the following relations is an
equivalence relation, i.e.\ check whether it is reflexive,
symmetric, and transitive.  If a property holds, prove that it
does.  If a property does not hold, give a counter example.

\begin{enumerate}
\item On $\mathbb{Z}$, define $x \sim y$ if and only if $x-y$ is
divisible by $3$.

\emph{Note:} Any number $n \in \mathbb{Z}$ is divisible by $3$ if
there exists $k \in \mathbb{Z}$ such that $n = 3 \cdot k$.

\item Let $X$ be a nonempty set.  Define, for any two subsets $A,B
\subseteq X$, that $A \sim B$ if and only if $A \subseteq B$.
\item On $\mathbb{Z}\times \mathbb{Z}$, define $(a,b) \sim
(a',b')$ if and only if $ab'=ba'$.
\end{enumerate}

\item Recall from class that we studied a binary operation $F \colon
\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ with the following
properties:
\begin{itemize}
\item[(A1)] $F(a,1) = s(a)$ for all $a \in \mathbb{N}$,
\item[(A2)] $F(a,s(b)) = s(F(a,b))$ for all $a,b \in \mathbb{N}$,
\end{itemize}
where $s\colon \mathbb{N} \to \mathbb{N}$ is as in Peano's axioms.

Prove that if
\begin{equation*}
  F(a, c) = F(b, c)
\end{equation*}
for some $a,b,c \in \mathbb{N}$, then $a=b$.
\end{enumerate}



\end{document}
