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\title{General Mathematics and Computational Science I}
\author{Practice Midterm I -- Not for Credit}
\date{September 22, 2005}
\maketitle

\begin{enumerate}

\item (Re)do Exercise 2 Question 3.

\item Show that $2^n \leq n!$ for all natural numbers $n \geq 5$. 

\item Are the following functions surjective?  Are they injective?
Prove or disprove! 
\begin{enumerate}
\item Define $f \colon \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}
\times \mathbb{Z}$ by $f(x,y) = (y,x)$.

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

Fix a proper subset $B \subset X$, and let $f \colon P(X) \to P(X)$ be
defined as $f(A) = A \cap B$.

\item Define $f \colon P(\mathbb{N}) \to \mathbb{N}$ by $f(A) = \min
A$, the minimum element of the set $A$. 

\end{enumerate}

\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 if
\begin{equation*}
  G(a, c) = G(b, c)
\end{equation*}
for some $a,b,c \in \mathbb{N}$, then $a=b$.

\item For functions $p,q \colon \mathbb{Z} \to \mathbb{Z}$, define the
relation $p \sim q$ if and only if $p(0) = q(0)$.

Is this an equivalence relation?  Prove or disprove!

\item Let $I_n = \{ k \in \mathbb{N} \colon k \leq n\}$.  Show that
\begin{equation*}
  I_m \times I_n \cong I_{mn} \,.
\end{equation*}

\end{enumerate}
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