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\title{General Mathematics and Computational Science I}
\author{Exercise 3}
\date{September 13, 2005}
\maketitle

\begin{enumerate}

\item (From Daepp/Gorkin, p.~113) 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 $\mathbb{Z}$, define $x \sim y$ if and only if $\lvert x
\rvert = \lvert y \rvert$.
\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}
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