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\title{General Mathematics and Computational Science I}
\author{Exercise 1}
\date{September 14, 2006}
\maketitle

\begin{enumerate}

\item (From Ivanov, p.~2)  Into how many pieces do
\begin{enumerate}
\item $n$ points subdivide a line;
\item $n$ straight lines subdivide the plane, if no two of the lines
are parallel and no three meet in a single point?
\end{enumerate}

\item Use induction to show that for $n \in \mathbb{N} \cup \{0\}$ and
any (real) number $q \neq 1$,
\begin{equation*}
  \sum_{k=0}^n q^k = \frac{1-q^{n+1}}{1-q} \,.
\end{equation*}

\textbf{Remark:} This sum is called a \emph{geometric progression}.

\item A polygon is called \emph{convex} if any straight line
connecting two points of the polygon lies entirely within; see figure
below. 

Use induction to show that, for a convex $n$-gon (a polygon with $n$
vertices), the sum of the interior angles $S_n$ is
\begin{equation*}
  S_n = 180 \cdot (n-2)
\end{equation*}
\begin{center}
  \includegraphics[width=5cm,angle=270]{polygon.pdf}
\end{center}

\item (From Daepp/Gorkin, p.~114) Let $X$ be a nonempty set with an
equivalence relation $\sim$ on it.  Prove that for all $x,y \in X$, 
\begin{equation*}
  [x] = [y] \qquad \text{if and only if} \qquad x \sim y \,.
\end{equation*}
\end{enumerate}


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