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\title{CS200 - Problem Set 4\\\small{Due: Monday, Mar. 12 to Canvas before class}}
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Please read the sections of the syllabus on problem sets and honor code before starting this homework.
\begin{enumerate}
\item \textbf{[6 points]} What can you deduce from the following true statements? Please write the simplest new true statement possible. (Partial credit will be given for more complex statements.)
\begin{align}
P&\rightarrow R\nonumber\\
Q&\rightarrow R\nonumber\\
P&\vee Q\nonumber\\
\rule{1cm}{0.15mm}&\rule{1.5cm}{0.15mm}\nonumber\\
\therefore\qquad&
\end{align}
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\item \textbf{[0 points]}~\\
(Challenge problem) You meet two spiders on the road. Everyone knows that a spider either always tells the truth, or always lies. The first spider says, ``If we are brothers, then we are both liars.'' The second spider says, ``We are cousins or we are both liars.'' Are both spiders telling the truth? (Hint, create a truth table for their statements and consider the possible cases of each spider lying or telling the truth, and use deduction to see if there is a contradiction. Also, can brothers be cousins?)
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\item Read section 5.3 of \href{http://www.people.vcu.edu/~rhammack/BookOfProof/Contrapositive.pdf}{Proof} by Richard Hammack. Then read the following poorly written proof of the statement: If $n$ is even, then $n^2$ is even.
\textbf{Proof:}
\begin{enumerate}[label=\arabic*.]
\item Let $n=$ an integer.
\item Suppose $n$ is even.
\item Then $n=2k$.
\item $n^2=(2k)^2$, $(2k)^2=4k^2$, so $4k^2=2(2k^2)$
\item Since $(2k^2)$ is an integer, I've shown it is even.
\end{enumerate}
(The sentences in the proof are numbered to make it easier to reference specific lines in your answer.)
\begin{enumerate}
\item \textbf{[1 point per guideline violation found]} Identify sentences that violate Hammack's mathematical writing guidelines and explain why. (A sentence can violate multiple guidelines, and so can be included multiple times.)
\item \textbf{[6 points]} Rewrite the proof so that it follows Hammack's mathematical writing guidelines.
\end{enumerate}
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\item\textit{Pigeon Hole Principle} \textbf{[11 point each]}
% (4 points)
The pigeonhole principle is an extremely important tool in computer science
(see \href{https://math.stackexchange.com/questions/97812/pigeonhole-
practical-applications-in-computer-science}{this StackExchange post} for just
some of its many diverse applications). It states: If you put $n+1$ pigeons in
$n$ cubbies, there must be a cubby with more than one pigeon in it. Create two proofs of this fact, one that uses proof by induction and one that is a contrapositve proof. Each proof should be graded using the
standard $11$-point scale.
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\item \textbf{[11 points]} Prove $\forall n\in \mathbb{Z}$, $n$ is even if and only if $5n+3$ is odd. Prove one direction using a direct proof, and one direction using a contrapositive proof.
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\item \textbf{[11 points]} Prove that you can make any postage greater than or equal to $14$ cents using $3$-cent stamps and $7$-cent stamps. (Hint - combine proof by cases and induction.)
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\item \textbf{[3 points]} When we use a direct
proof to prove $P\rightarrow Q$ is true, we start by assuming $P$ is true. Why do we
not also consider the case that $P$ is false?
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\item How long did you spend on this homework?
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