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\title{General Mathematics and Computational Science I}
\author{Exercise 6}
\date{September 20, 2007}
\maketitle

\begin{enumerate}

\item (Bernoulli inequality.)  Use induction to prove that for every
$x \in \mathbb{Q}$ with $x \geq -1$ and every $n \in \mathbb{N}$,
\begin{equation*}
  (1+x)^n \geq 1 + n \, x \,.
\end{equation*}

\item Recall from class that two sets $A$ and $B$ are said to be
  \emph{isomorphic}, written $A \cong B$, if there exists a bijective
  map $\phi \colon A \to B$.

Show that the relation ``$\cong$'' is in fact an equivalence
relation. 

\item We say a field $\mathbb{F}$ is \emph{ordered} if there exists a
relation $>$ which respects addition and multiplication in the
following way.  For any $x,y,z\in\mathbb{F}$,
\begin{itemize}
\item[(O1)] if $x>y$, then $x+z>y+z$;
\item[(O2)] if $x>y$ and $z>0$, then $xz>yz$.
\end{itemize}

Recall from class that the set of rational numbers $\mathbb{Q}$ are
equivalence classes $[a/b]$ with respect to the equivalence relation
$(a,b) \sim (a',b')$ if $a\cdot b'=a'\cdot b$.  Recall that addition,
multiplication, and ordering of  two rationals $x=[a/b]\in \mathbb{Q}$
and $y=[c/d]\in \mathbb{Q}$ are defined as follows.
\begin{enumerate}
\item $x+y=[(ad+bc)/bd]$ 
\item $x\cdot y = [ac/bd]$
\item $x$ is \emph{positive} if $ab>0$ and \emph{negative} if $ab<0$;
  $x>y$ if $x-y$ is positive.
\end{enumerate}

Show that this order relation satisfies (O1) and (O2).

\end{enumerate}
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