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\renewcommand{\coursecopyrightnames}{Prof.~Albert~R.~Meyer}
%\renewcommand{\coursecopyrightyear}{2002}

\begin{document}
\paper{diag}{}{Diagnostic Questionnaire}

This is an entry survey/diagnostic questionnaire that will be used to
assess quality of teaching in 6.042.  It is intended to help us judge how
much background in the course material you already have.  You score, which
will count toward your grade, will be the \emph{number of questions} you
answer.  Your \textbf{actual answers will not affect your grade in any
way}; in fact, they will not be seen by the course staff until the term is
over.

\textbf{Instructions}: Please answer these questions on your own, without
looking at any references.  This should take about 30 minutes.


\begin{enumerate}  %%[[[start numbering at 0]]]

\item[0.] 
\textbf{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 3,777,778?

\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 %(Prereq)

   \begin{enumerate}
   \item How many of the following mathematical terms have you seen already?
   \item For how many could you write definitions without looking them up
   first?
   \end{enumerate}

   \begin{itemize}
     \item gcd, 
     \item prime number,
     \item proof by contradiction,
     \item one-to-one (injective) function,
     \item lub,
     \item equivalence relation.
  \end{itemize}


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

   \begin{enumerate}
   \item $\mbox{not}[P \mbox{ or } Q] \iff \mbox{not}[P] \mbox{ and } \mbox{not}[Q]$
   \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 More than one of the following expressions describes the number of
\emph{size three subsets} of the set $\set{1,2,\dots,6}$.  Which ones?
\begin{enumerate}
    \item $3 \times 6$
    \item $\binom{6}{3}$
    \item $6 \times 5 \times 4$
    \item $5 \times 4$
    \item $\dfrac{6!}{3!}$
    \item $3! \times 3!$
    \item $\dfrac{6!}{3!3!}$
\end{enumerate}

\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''.
        \remove{\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}
