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Intro to Mathematical Reasoning (Math 300) -- Homework 2 
\footnote{Version 1/19/04}
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\begin{enumerate}

\item
This problem
is only  for those of you
who {\em did not} get the correct minimum number of rounds in
the hat switching problem.
Since your solution was
not the minimum, this means that {\em there was something wrong
with your reasoning}.  Analyze your own written argument and
find {\em the exact point in your argument} where you made
an untrue statement.  Write a brief and specific explanation
of what is wrong in your argument. (In most case you
should be able to find a particular sentence where you went
wrong in your reasoning.  It is not enough to say
``My argument went wrong because I said the answer was 6 rounds.''
You must identify the key place in your argument where you (unintentionally)
lied.

Be sure to attach your paper with your solution to the original
problem with this homework.


\item
Let $S$ denote the set $\{2,10,11,18,19,27\}$.
For each of the following statements,
determine whether the statement is
true or false.  Give a careful explanation for each answer. (Note:
``careful'' does not mean ``long''.)

Recall that when we say
that an integer $a$ is a multiple of $b$ we mean that there is
an integer $k$ such that $bk=a$.
\begin{enumerate}
\item For all elements $x$ belonging to $S$ there is a $y$ belonging
to  $S$ such that $x+y$ is a multiple
of 5.
\item There exists
an element $x$ belonging to $S$ such that for all elements
$y$ belonging to $S$,
$x+y$ is a multiple of 5.
\item For all $x$ belonging to $S$ there is a $y$ belonging to $S$
such that $x+y$ is a multiple of 7.
\end{enumerate}

\item
%(({\em Comment to instructor}: This problem makes reference to material
%in handout 1.))
Here are some universal statements.
For each one, (i) prove that the statement is incorrect,
and (ii)if possible,
find a simple restriction of the hypothesis of the statement
that makes it true.  (This is not possible for all of them.)
In choosing your restriction, try to add a condition
that is least restrictive.


\begin{enumerate}
\item
For any real numbers $a,b,c$, if $ab=ac$ then $b=c$.

\item For any positive integer $a$, positive integer $b$ and
positive integer $c$, if $bc$ is a multiple of $a$ then
$b$ is a multiple of $a$ or $c$ is a multiple of $a$.

\item For any four positive real
numbers $r,s,t,u$ if  $r \leq s$ and $t \leq u$ then $rs \leq tu$.



\item For all positive integers $n$, $n^4 \geq 2^{n-1}$.

\item For any two positive real numbers $x,y$, $\frac{1}{x}+\frac{1}{y} \leq \frac{4}{x+y}$.

\item For any two real numbers $y_1,y_2$ belonging to the interval
$[-\pi,\pi]$, if $\cos(y_1)=\cos(y_2)$ then $y_1=y_2$.

\end{enumerate}

\item In section 8.1 of the book on page 180 are listed
five ``bullets'' which state some properties of the real numbers.
Bullets 1,2 and 5 each give a single universal statement,
bullet 3 has two existential statements, and bullet 4
has two universal statements, for a total of 7 statements.

\begin{enumerate}
\item Suppose you replace the words ``real number'' by ``integer''
in each of the statements.  Which of the seven statements
is now false?  Explain.
\item Suppose you replace the words ``real number'' by ``positive
integer'' in each of the statements.  Which of the seven statements
is now false?  Explain.
\end{enumerate}


\item  Consider the statement:
``For all pairs $x,y$ of integers, if there is an integer $d$ that is greater
than 1 such that $x$ is a multiple of $d$
and $y$ is a multiple of $d$, then there is an integer $m$ that is less than $xy$
such that $m$ is a multiple of $x$ and $m$ is a multiple of $y$.
\begin{enumerate}
\item Draw the parse tree for the above sentence.
\item Write an English sentence that is equivalent to the
negation of the above sentence and is obtained by pushing negations
through.
\end{enumerate}

\item Here is a predicate with free variables $x$, which stands
for a real number and $S$, which
stands for a set of real numbers.
``$x$ is not
a member of $S$ and for all real numbers $\varepsilon>0$,
there exists  a member $y$ of $S$ such that $|x-y| \leq \varepsilon$''.
\end{enumerate}
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