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\begin{document} \pagestyle{empty} \begin{flushleft}

\scs{Name(s):} 

\scs{Math 301 \hfill Logic \hfill January 28, 2015}


\prob{
Determine if the following statements are True or False.
\begin{itemize}
\item[a) ] Either horses have 4 legs or 17 is not prime.
\item[b) ] Neither do 3 quarters add up to one dollar, nor do horses have 4 legs.
\item[c) ] If ducks have webbed feet, then Canada lies south of the equator.
\item[d) ] If Canada lies south of the equator, then ducks are mammals. 
\end{itemize}
}

\dfn{
A statement is a {\em contradiction} if its only possible truth value is ``false''.
A statement is a {\em tautology} if its only possible truth value is ``true''.
}

\prob{
Make a truth table for $P \lor (\neg P )$ and $P \land (\neg P)$. 
}

\vfill

\prob{
Fill in the blank with a statement (written in English) that makes the entire statement true.
\begin{itemize}
\item[a) ] $\left[ P \land (\neg P) \right] \implies$ \underline{\hspace{3in}}

\vspace{.25in}

\item[b) ] \underline{\hspace{3in}} $\implies \left[ P \lor (\neg P) \right]$

\vspace{.25in}

\end{itemize}
}

\prob{
Make a truth table for $P \implies \left(P \lor Q\right)$.
}

\vfill

\prob{
Make a truth table for $\left[ (P \implies Q) \land P \right] \implies Q$ (this is known as Modus Ponens and is one of the most important rules of deductive logic).
}

\vfill

\vspace{.5in}

\newpage

\prob{
Last semester Prof.\@ Axon gave an A to any student who got a perfect score on the final exam.
\begin{itemize}
\item[a) ] 
Katie got a perfect score on the final exam.
What can you conclude about Katie's grade in the class?
(You're almost certainly using Modus Ponens to make this deduction).

\vspace{1in}

\item[b) ]
Mike didn't get an A in Prof. Axon's class.
What can you conclude about Mike's score on the final exam?

\vspace{1in}

\item[b) ]
You probably just used Modus Tollens, another very important tautology.
Apply the same reasoning to fill in the blanks below to get a useful rule of logic.
Modus Tollens in English: 
\begin{quote}
If we know that P implies Q and we know that Q is not true, then \underline{\hspace{2in}}
\end{quote}

Modus Tollens in symbols: \[ 
\left[ (P \implies Q) \land (\neg Q) \right] \implies \text{\underline{\hspace{1in}}}
\]

\vspace{.25in}

\end{itemize}
}

\dfn{
The {\em converse} of $P \implies Q$ is $Q \implies P$.
The {\em contrapositive} of $P \implies Q$ is $(\neg Q) \implies (\neg P)$.
}

\prob{
\begin{itemize}
\item[a) ] Write a statement that has a true contrapositive.

\vspace{.75in}

\item[b) ] Try to think of a statement that has a false contrapositive (if you can't do this, use a truth table to show that $P \implies Q$ is logically equivalent to its contrapositive).

\vfill

\end{itemize}
}

\prob{
We have seen that $P \implies Q$ is logically equivalent to $(\neg P) \lor Q$.
One of DeMorgan's laws states that $\neg (P \lor Q) \equiv (\neg P) \land (\neg Q)$.
Use this to find an alternative expression for $\neg (P \implies Q)$.
}

\vspace{1in}

\end{flushleft} \end{document}