\documentclass{article}   

\title{Discrete Mathematics\\ Drill-1} 
\author 
{Moshe Rosenfeld\\ 
\\ Hanoi 2011\\
{\tt moishe@u.washington.edu}}
\date{}
        

\begin{document}           

\maketitle                

\section{Drill – 1}
Mr. Nguyen runs a mail order expensive jewelry business. His mode of operation is to use boxes with personal coded locks to exchange money and jewelry. Once a customer initiates a transaction, Mr. Nguyen can send him a an open box where the customer can code a private number and lock the box. Every open box can be intercepted by intruders and its content will be stolen. A locked box will always reach its destination.\\
How can Dung buy safely an expensive diamond ring from Mr. Nguyen?
\vskip 10pt
Let p, q and s be the propositions:
\begin{enumerate}
 \item p: you drive your xe may in Hanoi over 80 km an hour.
\item q: the police stops you.
\item s: the police takes away your xe may.   
\end{enumerate}
Write a sentence that means: $ s \rightarrow  p \wedge q$
\vskip 5pt
Write the following propositions using the Boolean variables p, q, s and Boolean operators:
\begin{enumerate}
\item The police do not stop you.
\item You drive over 80 km an hour, the police stops you but they do not take away your xe may.
\item If you do not drive over 80 km an hour then the police will not stop you and they  will not take away you xe may.
\end{enumerate}
\vskip 10pt
Construct the truth table for the following compound propositions:
\begin{enumerate}
\item $(\neg p \leftrightarrow \neg q) \leftrightarrow (p \leftrightarrow q)$
\item $((p \rightarrow q) \rightarrow r) \rightarrow s$
\item $(p \leftrightarrow q)\leftrightarrow (r \leftrightarrow s)$
\end{enumerate}
\vskip 10pt
Show that the following pairs are logically equivalent:
\begin{enumerate}
\item $p\vee (q \wedge r)  ,  (p \vee q) \wedge (p \vee r)$
\item $(p \wedge q) \rightarrow (p \vee q)$ ,  {\bf TRUE}
\end{enumerate}


\end{document}