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\lhead{MATH 311-02}
\chead{Problem Set \#1\answer{ solutions}}
\rhead{\answer{YOUR NAME HERE}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due January 26, 2017}}
\begin{document}
\begin{enumerate}
\item \question{The following questions build towards a future proof
technique.}
\begin{enumerate}
\item \question{Write a complete proof of the following statement:
if $n$ is an even integer, then $3n^2+n-5$ is odd.}
\answer{
}
\item \question{Write a complete proof of the following statement:
if $n$ is an odd integer, then $3n^2+n-5$ is odd.}
\answer{
YOUR ANSWER HERE
}
\item \question{What true statement would adequately combine the two
results above?}
\answer{
YOUR ANSWER HERE
}
\end{enumerate}
\item \question{In preparation for the next few questions, recall that
the logical implication operator can be defined entirely in terms
of disjunctions and negations: $P\rightarrow Q\equiv\neg P\vee
Q$. In fact, \emph{any} logical expression you might wish can be
built using conjunctions, disjunctions, and negations. For
instance, consider the mysterious function of P, Q, and R with the truth
table given below:
\begin{center}
\begin{tabular}{|c|c|c||c|}\hline
$P$&$Q$&$R$&$f(P,Q,R)$\\\hline\hline
F&F&F&F\\\hline
F&F&T&F\\\hline
F&T&F&T\\\hline
F&T&T&F\\\hline
T&F&F&T\\\hline
T&F&T&T\\\hline
T&T&F&F\\\hline
T&T&T&F\\\hline
\end{tabular}
\end{center}
We could write that simply as a disjunction of 3 conjunctions
corresponding to each of the places where it's true, so
\[f(P,Q,R)\equiv(\neg P\wedge Q\wedge \neg R)\vee(P\wedge\neg
Q\wedge\neg R)\vee(P\wedge\neg Q\wedge R)\] This is not the most
efficient way to do this, but it's an easy way to demonstrate that
conjunctions, disjunctions, and negations are all you will ever
truly need to write a logical expression. The following questions
address the possibility that you might not even need all three of
these operations.}
\begin{enumerate}
\item \question{Is it possible to write the conjunction operation
($P\wedge Q$) entirely in terms of disjunctions and negations?
Either show how or explain why not.}
\answer{
YOUR ANSWER HERE
}
\item \question{Is it possible to write the disjunction operation
($P\vee Q$) entirely in terms of conjunctions and negations?
Either show how or explain why not.}
\answer{
YOUR ANSWER HERE
}
\item \question{Is it possible to write the negation operation
($\neg P$) entirely in terms of conjunctions and disjunctions?
Either show how or explain why not.}
\answer{
YOUR ANSWER HERE
}
\item \question{\textbf{BONUS:} It is impossible to write every
single possible logical statement in terms of only \emph{one} of
the standard operations (negations, conjunctions, disjunctions,
or implications). However, if we consider nonstandard binary
operations, there are 16 different possible ways to fill in a
truth table, and maybe one of them will suffice. Note that these
16 ways include some pretty silly operations like ``always
result in true'' or ``negate the left operand and ignore the
right'', but maybe some of the others will work. Below, show a
truth table for some nonstandard operation $\otimes$, and show
how conjunction, disjunction, and negation could all be
rewritten in terms of $\otimes$.}
\answer{
% In the truth table below, replace each "?" with a "T" or "F"
% to describe whatever novel operation you choose.
\begin{tabular}{|c|c||c|}\hline
$P$&$Q$&$P\otimes Q$\\\hline\hline
F&F&?\\\hline
F&T&?\\\hline
T&F&?\\\hline
T&T&?\\\hline
\end{tabular}
YOUR ANSWER HERE
}
\end{enumerate}
\end{enumerate}
\end{document}