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\begin{center}
{\bf Review Sheet for Exam 1}
\end{center}

\noindent{\bf Format:} the exam will most likely go as follows
\begin{itemize}
\item A page of short answer questions like ``State the well ordering
principle'' or ``State the definition of an injective function.''
\item A part that begins with instructions something like ``Following
are 8 assertions, 5 of which are false.''  Identify the false ones and
give examples showing that they are false.
\item Problems asking you to prove some assertion or other.  I expect
there to be about three of these.
\end{itemize}
I don't promise to keep to exactly this structure (for instance, I could
ask you compute the base 5 decimal expansion of some number or other---that
wouldn't fall into any of the above categories.)  However, I won't
deviate much from it.

\vspace*{.2in}
\noindent{\bf Things to know: }
\begin{description}
\item[definitions and statements.] function; injective, surjective, and
bijective functions; well-ordering principle; complement of a set;
cardinality; union, intersection and Cartesian product of two sets; 
prime number; `a divides b'; prime number; domain, range, and image of a 
function; 
base q decimal expansion of n; increasing, decreasing, and monotone function;
the division algorithm.
\item[proofs of specific theorems.] $\sqrt{2}$ is irrational; there are
infinitely many prime numbers.
\item[techniques of proof.] proof by (strong) induction; proof by 
contradiction;
proof by exhaustion; proving two sets are equal;  Proving a function is 
injective, surjective, or bijective;  Proving a function is increasing or 
decreasing.
\item[logic.] Negation of a statement; contrapositive of a statement;
quantifier symbols like `for every' and `there exists'.
\end{description}
I'm sure I've forgotten something in all this.  However, I think I've got
most things down.

\vspace*{.2in}
\noindent{\bf Advice for studying: }
Beyond remembering the specific things I mentioned above, make sure you review 
the homework problems.  Compare your solutions with my solutions.  Solve
old problems from scratch.  Solve some problems that look similar to
the ones I assigned.  Etc.  For one thing, some of the test will be taken
nearly verbatim from the homework.  For another, in my opinion, solving 
problems is the only way to really learn and understand math.


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