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\begin{document}
\centerline{\bf \Large ASSIGNMENT 1}

\medskip

\section{Warm Up (do before recitation lab)}

\begin{enumerate}[(1)]
\item \math{p} and \math{q} are two propositions that are either \math{T} or 
\math{F}. How many rows are there in the truth table of the
compound proposition  \math{\neg (p\OR q)\AND\neg p}.
\item Give the truth table for the compound proposition above.
\item {\bf Theorem:} If integer \math{n\in\N} is even then \math{n^2} is even''.
What is wrong with the following proof:
{\sf\small
\begin{enumerate}[1.]
\item Suppose \math{n^2} is even.
\item Let \math{n} have a prime factorization \math{p_1^{q_1}p_2^{q_2}\cdots}.
\item So in the prime factorization of \math{n^2}, each prime appears 
an even number of times, \math{n^2=p_1^{2q_1}p_2^{2q_2}\cdots}. 
\item Since \math{n^2} is even, 2 is a prime factor of 
\math{n^2}, so \math{n^2=2^{2q}\cdots}, with 
\math{q>0}.
\item So \math{n=2^{q}\cdots}, with \math{q>0} and so \math{n} has 
\math{2} as a factor and is therefore even as stated in the theorem.
\end{enumerate}
}
What does the above proof actually prove?

\item Is \math{n^2+n+41} prime for 
\math{n=1,2,3,4,\ldots,10}. Does that mean that \math{n^2+n+41} is 
prime for all \math{n\in\N}?
\item Let \math{F(x)} denote `\math{x} is a freshman' and
\math{M(x)} denote `\math{x} is a math major'.
Translate into sensible english:
\begin{enumerate}[(a)]
\item \math{\forall x(M(x)\imp\neg F(x))}
\item \math{\neg\exists x(M(x)\AND\neg F(x))}
\end{enumerate}


\end{enumerate}


\section{Recitation Lab (TA will work these out in lab)}

\begin{enumerate}[(1)]
\item
Using a truth table, determine whether these two compound propositions are
logically equivalent:
\mand{(p\imp q)\imp r\qquad\qquad p\imp(q\imp r).}

\item What is the negation of ``Jan is rich and happy''
\item{\bf [DNF]} Using only 
\math{\neg,\AND,\OR}, give a compound proposition which has the following 
truth table:
\mand{
\begin{array}{ccc|c}
p&q&r&\\\hline
T&T&T&F\\
T&T&F&T\\
T&F&T&F\\
T&F&F&F\\
F&T&T&T\\
F&T&F&T\\
F&F&T&F\\
F&F&F&F\\
\end{array}
}
\item What is the set \math{\Z\cap\overline\N\cap \cl{S}}, where the
set \math{S} is the set of integers which are perfect squares.
\item Prove that there are infinitely many primes.

\end{enumerate}

\section{Problems (hand these in)}

\begin{enumerate}[(1)]
\item Use a direct proof to show that the product of two odd numbers is odd.
\item Using only 
\math{\neg,\AND,\OR}, give a compound proposition which has the following 
truth table:
\mand{
\begin{array}{cc|c}
q&r&\\\hline
T&T&F\\
T&F&T\\
F&T&F\\
F&F&F\\
\end{array}
}

\item Compute the \emph{number} of positive 
divisors of the following integers:
\mand{
\begin{array}{c}
6,8,12,18,30,\\
4,9,16,25,36
\end{array}
}
Formulate a conjecture that relates a property of the
number of divisors
of \math{n} to a property of \math{n}. Your conjecture should
at the very least agree with your data. State your conjecture 
as a theorem. You do not have
to prove your theorem but be {\bf precise} in your statement of the theorem.
(You may want to define some convenient notation, for example
let \math{\phi(n)} be the number of positive divisors of \math{n}).

\item For the Ebola spreading model, a square gets infected if
at least two (non-diagonal) squares are infected. Show the final state of the
grid (who is infected) when the shaded in squares are
initially infected.
\begin{center}
FIGURE NOT INCLUDED
\end{center}
Can you find an arrangement of 5 initial infections that can eventually
infect the whole \math{6\times 6} square. What about with 6 initial infections?
Try the same game with a \math{4\times 4} square and a 
\math{5\times 5} square.

Suppose the square is \math{n\times n} where \math{n\in\N}. Formulate a
conjecture that asserts the minimum number of initial
infections required to eventually infect the whole square.

State your conjecture as a theorem.
You do not have
to prove your theorem but be {\bf precise} in your statement of the theorem.



\item Prove or Disprove each of the following theorems:

{\bf Theorem.} If \math{x} and \math{y} are both \emph{irrational}, then 
\math{x^y} is irrational. 
\newline\hfill\emph{[Hint: We proved in class that
\math{\sqrt{2}} is irrational.]}

{\bf Theorem.} If \math{n} is an integer and \math{n^2} is divisible by 
3, then \math{n} is divisible by 3.

\end{enumerate}

\end{document}






