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\pset{1}{September 7, 2000}{2:35pm in lecture, September 12, 2000}

\textbf{Reading:} 
Vellman, \textit{How to Prove It}, Chapter 1; Sections 3.1, 3.2. \\
Rosen, \textit{Discrete Mathematics and Its Applications}, Sections 1.1, 1.2, 
1.4, 1.5, 2.3, 3.1 \\
Some of this reviews elementary mathematical topics that should be familiar 
to you; you can skim these.  
Do look closely at any topics that seem unfamiliar, and
if there are many such topics, we encourage you to discuss this with
your TA so he can help you plan a catch-up strategy.

\medskip
\begin{problems}

\problem \textbf{Propositional logic}
\bparts
\ppart Rosen \S 1.1, Exercise 24 (f).
\ppart Rosen \S 1.2, Exercise 8 (c),(d).
\ppart Velleman \S 1.5, Exercise 6:\\
\noindent\hspace*{40pt}\texttt{Which of the following formulas are equivalent?} \\
\noindent\hspace*{40pt}\texttt{(a) $P \rightarrow (Q \rightarrow R)$} \\
\noindent\hspace*{40pt}\texttt{(b) $Q \rightarrow (P \rightarrow R)$} \\
\noindent\hspace*{40pt}\texttt{(c) $(P \rightarrow Q) \land (P \rightarrow R)$} \\
\noindent\hspace*{40pt}\texttt{(d) $(P \land Q) \rightarrow R$} \\
\noindent\hspace*{40pt}\texttt{(e) $P \rightarrow (Q \land R)$}

\eparts

\bigskip

\problem  \textbf{Set theory}
\bparts
\ppart Rosen \S 1.4, Exercise 16 (a),(b),(c),(d).  
\ppart Rosen \S 1.5, Exercise 12(c).
\ppart Rosen \S 1.5, Exercise 18 (a),(b),(c).
\ppart Rosen \S 1.5, Exercise 36.

\eparts

\problem \textbf{Proofs or fallacies?}  %variant of fall99

Determine whether each of the following arguments is valid.
If it is not valid, then explain what the error is. (You may want to use the rules of inference and fallacies from Rosen in your solutions.)

\begin{problemparts}

\problempart
{\em Theorem:} if $n^3>1$, then $n>1$.  {\em Proof:}
If $n$ is a real number such that $n>1$, then $n^3>1$.  Suppose that
$n^3>1$, 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 $n^2 > 9$ then, $n > 3$.  {\em Proof:} Because
if $n$ is a real number with $n>3$, then $n^2 >9$.
\end{problemparts}

\problem \textbf{Proofs}
Velleman \S 3.1, Exercise 6:\\
\texttt{Suppose x is a real number and $x \ne 0$. Prove that if $\frac{\sqrt[3]{x} + 5}{x^2 + 6} = \frac{1}{x}$ then $x \ne 8$.}

\problem \textbf{And more proofs}
Rosen \S 3.1, Exercise 46.
\end{problems}

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