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\title{General Mathematics and Computational Science I}
\author{Midterm I}
\date{September 29, 2005}
\maketitle

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\begin{enumerate}

\item Let $T_n$ denote the number of equally spaced points that fill
an equilateral triangle where each side is built of $n$ equally spaced
points:

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Find a general formula for $T_n$ and prove that your formula is
correct.
\hfill(8)


\item Show that $2^n > n^2$ for every natural number $n \geq 5$.
\hfill(8)


\item Are the following functions surjective?  Are they injective?
Prove or disprove!
\begin{enumerate}
\item $f \colon \{1,2,3\} \to \{1,2,3\}$ where $f(1)=2$, $f(2)=3$,
$f(3)=3$. 

\item Let $X$ be a nonempty set, and $P(X)$ the set of all subsets of
$X$, called the \emph{power set} of $X$.

Let $f \colon P(X) \to P(X)$ be
defined as $f(A) = X\setminus A$.

\end{enumerate}
\hfill(5+5)


\item Consider a map $G \colon \mathbb{N} \times \mathbb{N} \to
\mathbb{N}$ with the following properties:
\begin{itemize}
\item[(M1)] $G(a,1) = a$ for all $a \in \mathbb{N}$,
\item[(M2)] $G(a,s(b)) = G(a,b) + a$ 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 $G$ is commutative, i.e.
\begin{equation*}
  G(a, b) = G(b, a)
\end{equation*}
for all $a,b \in \mathbb{N}$.

\emph{Hint:} Consider the special case $b=1$ first.
\hfill(8)


\item Give an example of a relation on $\mathbb{N}$ which is
transitive, but is neither reflexive nor symmetric.

State explicitly why each of these properties holds respectively
fails.
\hfill(8)


\item Show that $\mathbb{N} \cong \mathbb{Z}$.
\hfill(8)


\end{enumerate}


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