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\noindent
\hspace*{\fill} {\bf Page 1 of 2} \\
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\noindent
{\bf Eric Eaton} \hfill {\bf 2 Feb 2005} \\
{\bf CMSC 203 -- Discrete Structures} \hfill {\bf Section 0401, Spring 2005} \\ 
{\bf Example Set 1} \\
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\noindent
1.1.7. What are the contrapositive, the converse,a nd the inverse of the implication \\

\indent
``The home team wins whenever it is raining.''? \\

\noindent
{\it Solution}: First, it is best to rewrite the implication for clarity. \\

\indent
If it is raining, then the home team wins. \\

\indent
{\bf Converse}: If the home team wins, then it is raining. \\

\indent
{\bf Contrapositive}: If the home team does not win, then it is not raining.\\ 

\indent
{\bf Inverse}: If it is not raining, then the home team does not win. \\

\vspace{0.2in}
\noindent
1.1.10. How can this English sentence be translated into a logical expression? \\

\indent ``You cannot ride the roller coaster if you are under four feet tall unless you are older than 16 years old.'' \\

\noindent
{\it Solution}: The first thing that must be done is to use propositional variables to represent each of the sentence parts.  Let 
\begin{itemize}
\item $q$ be ``You cannot ride the roller coaster,''
\item $r$ be ``You are under four feet tall,'' and
\item $s$ be ``You are older than 16 years old.''
\end{itemize}
\noindent
Then the sentence can be translated into
$$(r \wedge \neg s) \rightarrow \neg q.$$ 

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\noindent
\hspace*{\fill} {\bf Page 2 of 2} \\
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\vspace{0.1in}
\noindent
{\bf Eric Eaton} \hfill {\bf 2 Feb 2005} \\
{\bf CMSC 203 -- Discrete Structures} \hfill {\bf Section 0401, Spring 2005} \\ 
{\bf Example Set 1} \\
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\vspace{0.2in}
\noindent
1.1.12. Determine whether the following system specifications are
        consistent: \\

\indent
``The diagnostic message is stored in the buffer or it is retransmitted.''

\indent
``The diagnostic message is not stored in the buffer.''

\indent
``If the diagnostic message is stored in the buffer, then it is retranmitted.'' \\

\noindent
{\it Solution}: First, translate the sentence parts into propositional variables:
\begin{itemize}
\item $p$ is ``The diagnostic message is stored in the buffer'' and
\item $q$ is ``The diagnostic message is retransmitted.''
\end{itemize}
\noindent
Then the specification can be rewritten as
$$p \vee q,$$
$$\neg p,$$
$$p \rightarrow q.$$
\noindent
Now, for the system specification to be consistent, we must be able to assign truth values to the variables to make all specifications true.  So, $p$ must be false to make $\neg p$ true.  Since $p \vee q$ must be true, but $p$ is false, then $q$ must be true.  Then $p \rightarrow q$ is also true as $p$ is false and $q$ is true.  Therefore, the system specification is consistent.

\vspace{0.2in}
\noindent
1.2.6. Show that $(p \wedge q) \rightarrow (p \vee q)$ is a tautology. \\

\noindent
{\it Solution}: It is necessary to use logical equivalences to show that the statement above is a tautology.
\begin{eqnarray*}
(p \wedge q) \rightarrow (p \vee q) & \equiv & \neg (p \wedge q) \vee (p \vee q) \qquad \qquad \mbox{(by implication)}  \\
 & \equiv & (\neg p \vee \neg q) \vee (p \vee q) \qquad \mbox{  (by De Morgan)}  \\
 & \equiv & (\neg p \vee p) \vee (\neg q \vee q) \qquad \qquad \mbox{(by associative and commutative)}  \\
 & \equiv & \mbox{\bf T} \vee \mbox{\bf T} \qquad \qquad \qquad \qquad \qquad \mbox{(by negation)}  \\
 & \equiv & \mbox{\bf T} \\ 
\end{eqnarray*}

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