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%TCIDATA{Created=Sat Jan 10 07:33:43 2004}
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\begin{document}

\title{Problem Set 2}
\author{}
\maketitle

\begin{enumerate}
\item  Analyze the logical forms of the following statements.

\begin{enumerate}
\item  If anyone in the dorm has the measles, then everyone who has a friend
in the dorm will have to be quarantined.

\item  Jane saw a bear, and Roger saw one too.

\item  Every number that is larger than $x$ is larger than $y.$

\item  If there is a number $x$ such that $x^{2}+5x=w$ and there is a number 
$y$ such that $4-y^{2}=w,$ then $w$ is between $-10$ and $10.$

\item Anyone who has bought a Rolls Royce with cash must have a rich uncle.

\item If nobody failed then test, then everybod who got an A will tutor
someone who got a D.
\end{enumerate}


\item  Consider the following incorrect theorem:

\begin{theorem}
Suppose $n$ is a natural number larger than $2$ and $n$ is not a prime
number. Then $2n+13$ is not a prime number.
\end{theorem}

\begin{enumerate}
\item  What are the hyptheses and conclusion of this theorem?

\item  Show that the theorem is incorrect by finding a counterexample.
\end{enumerate}

\item  Suppose $A\backslash B\subseteq C\cap D$ and $x\in A.$ Prove that if $%
x\notin D$ then $x\in B.$


\item  Suppose $x$ is a real number and $x\neq 0.$ Prove that if $\frac{\sqrt%
[3]{x}+5}{x^{2}+6}=\frac{1}{x}$ then $x\neq 8.$

\item  Suppose $a,b,c,$ and $d$ are real numbers, $0<a<b$ and $d>0.$ Prove
that if $ac\geq bd$ then $c>d.$

\item  Suppose $x$ and $y$ are real numbers and $3x+2y\leq 5.$ Prove that if 
$x>1$ then $y<1.$

\item  Consider the following theorem.

\begin{theorem}
Suppose $x$ is a real number and $x\neq 4.$ If $\frac{2x-5}{x-4}=3$ then $%
x=7.$
\end{theorem}

\begin{enumerate}
\item  What wrong with the following proof of the theorem?

\begin{proof}
Suppose $x=7.$ Then $\frac{2x-5}{x-4}=\frac{2\left( 7\right) -5}{7-4}=\frac{9%
}{3}=3.$ Therefore if $\frac{2x-5}{x-4}=3$ then $x=7.$
\end{proof}

\item  Give a correct proof of the theorem.
\end{enumerate}

\item  Suppose $A\subseteq C$ and $B$ and $C$ are disjoint. Prove that if $%
x\in A$ then $x\notin B.$

\item  Use proof by contradiction to prove the following theorem.

\begin{theorem}
Suppose $A\cap C\subseteq B$ and $a\in C.$ Prove that $a\notin A\backslash
B. $
\end{theorem}

\item  Use proof by contradiction to prove the following theorem.

\begin{theorem}
Suppose $A\subseteq B,a\in A,$ and $a$ and $b$ are not both elements of $B.$
Prove that $b\notin B.$
\end{theorem}

\item  Consider the following incorrect theorem.

\begin{theorem}
Suppose $x$ and $y$ are real numbers and $x+y=10.$ Then $x\neq 3$ and $y\neq
8.$
\end{theorem}

\begin{enumerate}
\item  What is wrong with the following proof of the theorem?

\begin{proof}
Supose the conclusion of the theorem is false. Then $x=3$ and $y=8.$ But
then $x+y=11$, which contradicts the given information that $x+y=10.$
Therefore, the conclusion must be true.
\end{proof}

\item  Show that the theorem is incorrect by finding a counterexample.
\end{enumerate}

\item  Prove that if $A$ and $B\backslash C$ are disjoint, then $A\cap
B\subseteq C$

\item  Suppose $x$ is a real number.

\begin{enumerate}
\item  Prove that if $x\neq 1$ then there is a real number $y$ such that $%
\frac{y+1}{y-2}=x$

\item  Prove that if there is a real number $y$ such that $\frac{y+1}{y-2}=x$
then $x\neq 1.$
\end{enumerate}

\item  Prove that for every real number $x$, if $x>2$ then there is a real
number $y$ such that $y+\frac{1}{y}=x$

\item  Prove that if $A\subseteq B$ and $A\subseteq C$ then $A\subseteq
B\cap C$

\item  Suppose $A\subseteq B$. Prove that for every set $C,$ $C\backslash
B\subseteq C\backslash A.$

\item  Prove

\begin{enumerate}
\item  A sufficient condition for the demand for a good to incresae when its
price falls is that it is a normal good.

\item  A necessary but not sufficient condition for the demand for a good to
decrease when its price falls is that it is an inferior good.
\end{enumerate}
\end{enumerate}

\end{document}