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\pset{1}{September 9, 1999}{2:35pm in tutorial, September 16, 1999}

\textbf{Reading}: The
Rosen text, \S 1.1--1.5 and 3.1--2; Notes for Lectures 1 \& 2; Notes for
Spring '99 Tutorial 1.

Most of this reading reviews elementary mathematical topics which should
be familiar to you; you can skim these.  Do look more closely at any
topics which seem unfamiliar, and if there are many such topics, we
encourage you to ask about them in tutorial.

\medskip

Do the following problems from the Rosen textbook:

\begin{problems}

\problem \textbf{Logic}:
\bparts
\ppart \S 1.1, Exercise 24(b).
\ppart \S 1.2, Exercise 6.
\ppart \S 1.2, Exercise 8(a).
\ppart \S 1.3, Exercise 14 (d), (h), (i), (o).

\ppart For each of the logical formulas in \S 1.3, Exercise 22(a), (c),
(d), (g), and (j), indicate whether or not it is true when the domain of
discourse is $\naturals$ (the natural numbers 0, 1, 2, \dots), $\integers$
(the integers), $\rationals$ (the rationals), $\reals$ (the real numbers), and
$\complex$ (the complex numbers).

\ppart \S 1.3, Exercise 42.
\eparts

\problem  \textbf{Sets}
\bparts
\ppart \S 1.4, Exercise 14.  Prove it or give a counterexample.

\ppart \S 1.5, Exercise 12(e).

\ppart \S 1.5, Exercise 14.

\ppart \S 1.5, Exercise 36.
\eparts

\problem \textbf{Proofs}  %From Fall 95 ps1, prob 7

Determine whether each of the following arguments is valid.  If an
argument is correct, what rule of inference (from Rosen, \S 3.1) is being
used?  If it is not, what fallacy (again from Rosen \S 3.1) occurs?

\begin{problemparts}
\problempart
{\em Theorem:} if $n^2>1$, then $n>1$.  {\em Proof:}
If $n$ is a real number such that $n>1$, then $n^2>1$.  Suppose that
$n^2>1$, then $n>1$.

\problempart
{\em Theorem:} The number $\log_2 3$ is irrational. {\em Proof:} A number is irrational if it is not the ratio of two integers.
Therefore, since $\log_2 3$ cannot be written in the form $a/b$ where
$a$ and $b$ are integers, it is irrational.

\problempart
{\em Theorem:} If $n^2 \leq 9$ then, $n \leq 3$.  {\em Proof:} Because
if $n$ is a real number with $n>3$, then $n^2 >9$.

\problempart
{\em Theorem} If $n\leq 2$, then $n^2\leq 4$. {\em Proof:} Because if $n$ is a real number with $n>2$, then $n^2>4$.

\end{problemparts}

\problem \textbf{Simple Induction}
\bparts

\ppart \S 3.2, Exercise 18.

\ppart \S 3.2, Exercise 32.

\ppart \S 3.2, Exercise 34.
\eparts


\end{problems}

Remember to include a \emph{collaboration statement} on the front page of
your solutions (see Handout 2, \emph{Course Information}).

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