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{\bf Math 165: Honors Calculus I}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Quiz 1}\quad {\sl September 2, 1999}

\bigskip

\begin{center}
\fbox{\begin{minipage}{4.5in}
\centerline{\textbf{Axioms for the Real Numbers}}

A set $\R$, called the set of real numbers, is assumed to exist
satisfying the ten axioms below.

\emph{The Field Axioms.}
Two operations on $\R$, \emph{addition} and \emph{multiplication}, are
assumed to be defined, so that for each pair, $x$, $y\in \R$, there is
a uniquely determined \emph{sum}, $x+y\in\R$, and a uniquely determined
\emph{product}, $x\cdot y\in\R$, satisfying the following axioms.
  \begin{itemize}
  \item \textsc{Axiom 1.} $x + y = y + x$, $xy = yx$
  \item \textsc{Axiom 2.} $x + (y + z) = (x + y) + z$, $x(yz) = (xy)z$
  \item \textsc{Axiom 3.} $x(y + z) = xy + xz$
  \item \textsc{Axiom 4.}  There exists distinct numbers $0$ and $1$ such
  that for every $x\in\R$, $x+0 = x$ and $1\cdot x = x$.
  \item \textsc{Axiom 5.}  For every $x\in\R$, there is a $y\in\R$ such that
  $x+y = 0$.
  \item \textsc{Axiom 6.}  For every $x\in\R$, $x\ne 0$, there is a
  $y\in\R$ such that $xy = 1$.
  \end{itemize}
\emph{The Order axioms} A subset $\R^+\subset \R$, called the positive
numbers, is assumed to exist satisfying the following axioms.
  \begin{itemize}
  \item \textsc{Axiom 7.} If $x$ and $y$ are in $\R^+$, so are $x+y$ and $xy$.
  \item \textsc{Axiom 8.} For every real $x\ne0$, either $x\in\R^+$ or
  $-x\in\R^+$, but not both.
  \item \textsc{Axiom 9.} $0\not\in\R^+$.
  \end{itemize}
\emph{The Completeness Axiom}
  \begin{itemize}
  \item \textsc{Axiom 10.}   For every non-empty subset $S\subset\R$
  that is bounded above there is a $B\in \R$ that is  the supremum of~$S$,
  $B = \sup S$.
  \end{itemize}
\end{minipage}}
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\begin{enumerate}

\item Using only the axioms for the real numbers, prove the following
statement:
Given any real numbers $a$, $b$, there is a \emph{unique} real number
$x$ such that $a + x = b$.

\answerspace
\answerspace

\newpage
\item Define an inductive set, $S$.
\answerspace
\answerspace

\problemspace
\item Use the Principle of Mathematical Induction to prove
$2^n > n$ for any positive integer~$n$.
\answerspace\answerspace

\end{enumerate}

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