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\handout{diag}{Diagnostic Test}{September 7, 2000}
Due:  Friday, September 8, 2000, at the beginning of tutorial.
Instructions:  Please answer these questions on your own,
and without looking at any of the course material.  Allow at most 30 minutes. 
{\bf This test will not affect your grade in any way.}
It is intended to help us compare what you know before and after taking
the course.

\begin{enumerate}  %%[[[start numbering at 0]]]
\item Write your name:  
\item %(Prereq)
   \begin{enumerate}
   \item Find the derivative with respect to $x$ of $(x - 1) \ln x$.
   \item Find the indefinite integral of $\ln x$.
   \end{enumerate}

\item %(Diagnostic) 
What is the value modulo 5 of the product of 4,199,001 and 3,777,775?


\item 
For any set $A$, let $P(A)$ be its \emph{power set}, namely, the set
of \emph{all} its subsets; note that $A \in P(A)$.

   \begin{enumerate}
   \item %(Prereq) 
   List the elements in $P(\{1,2\})$.
   \item %(Prereq) 
   List the elements in $P(\{\emptyset, \{\emptyset\}\})$.
   \item %(Diagnostic) 
   How many elements are in $P(\{1,2,\ldots,8\})$?
   \end{enumerate}

\item %(Prereq) 
Describe a simple counterexample to the claim that if $A$
and $B$ are finite sets, the size of $A \cup B$ is the same as the size of
$A$ plus the size of $B$.


\item %(Diagnostic) 
Which of the following logical formulas are
\emph{tautologies}, i.e., they always hold:

   \begin{enumerate}
   \item $\mbox{not}[P \mbox{ or } Q] \equiv \mbox{not}[P] \mbox{ and } \mbox{not}[Q]$
   \item $[P \mbox{ implies } Q] \mbox{ or } [Q \mbox{ implies } P]$
   \item $\forall x [P(x) \mbox{ or } \mbox{not}[P(x)]]$
   \item $\forall x \exists y [P(x) \mbox{ or }  \mbox{not}[P(y)]]$
   \end{enumerate}

\item %(Diagnostic) 
Write a simple arithmetic expression for $\sum_{i=1}^n 2i$.

\item %(Diagnostic) 
Write a simple arithmetic expression for $\sum_{i=0}^n x^i$.

\item %(Diagnostic) 
If you flip three fair coins,
   \begin{enumerate}
      \item What is the probability that all three will land on Heads?
      \item What is the probability that at least one will land on Heads?
      \item What is the average number of Heads you can expect to flip?
   \end{enumerate}

\item %(Diagnostic) 
The \emph{Well-Ordering Principle} is an axiom of
Mathematics asserting that every nonempty set of \emph{nonnegative
integers} has a least member.

   \begin{enumerate}
        \item Explain why the \emph{Well-Ordering Principle} would not
   hold if the word ``nonnegative'' above were deleted.
        \item Explain why the \emph{Well-Ordering Principle} would not
   hold if the word ``integers'' above were replaced by ``rationals''.
        \item By this Principle, there must be a least nonnegative
   integer, $m$, that cannot be defined in fewer than fourteen words.  But
   ``The least nonnegative integer that cannot be defined in fewer than
   fourteen words,'' is a thirteen word definition of $m$.  Is there
   something wrong with the Well-Ordering Principle?  Comment please.
   \end{enumerate}
\end{enumerate}

\end{document}
