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\noindent {\large\bf Homework 3 [53 pts] \hfill 21-127 Concepts of Math \hfill due 07.10.2012}\\
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\noindent{\bf\large Notes:} This is due on {\bf Friday at 5:00 pm}. You may bring it to class in the morning, or you may bring it to my office any time during the day before 5:00 pm. If I am not there, slip it under my door. I will be there just before 5:00 to make sure you are there on time!

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\noindent{\large\bf Explain}:
\begin{enumerate}
\item{\bf World Traveler:} Define some appropriate sets and variable propositions to express the following claim using logical notation:
\begin{quote}
``Some student in this course has been to a capital city on every world continent (except Antarctica).''
\end{quote}
Then, use that notation to express the logical negation of the claim. Note: it is {\it not enough} to write something like: Let $C(x)$ be the statement ``$x$ has been to a capital city on every world continent.'' Break the statement into parts as much as possible, within reason!\\
{\bf[3 pts]}
\item {\bf Don't Not Forget To Not Do This Problem:} Let $U,V\subseteq\R$, let $P=\{x\in\R\mid x>0\}$, and let $f$ be a function from $\R$ to $\R$. For each of the following statements, write its logical negation {\it using logical symbols} (even if the original statement is written in a ``wordy'' style).
\begin{enumerate}
\item For all $x\in A$, there is a $y\in B$ such that $b>x$.
\item $\exs x\in A\st\; \fa b\in B\st\; b>x$.
\item For all $x,y\in\R$, $f(x)=f(y)\imp x=y$.
\item $\fa b\in\R\st\;\exs x\in\R\st\;f(x)=b$.
\item For all $x,y\in\R$ and for all $\ve\in P$, there is a $\delta\in P$ such that $|x-y|<\delta$ implies $|f(x)-f(y)|<\ve$,
\item $\fa\ve\in P\st\;\exs\delta\in P\st\;\fa x,y\in\R\st(|x-y|<\delta\imp |f(x)-f(y)|<\ve)$.
\end{enumerate}
{\bf[8 pts]}
\item {\bf Fue or Tralse:} Let $P,Q,R$ be logical statements. For each of the following propositions, say whether it is {\sf T} or {\sf F} and explain why.
\begin{enumerate}
\item $\ofb{P\imp (1=1)}\imp P$
\item $\ofb{P\imp (0=1)}\imp\neg P$
\item $\neg\ofb{(\text{Today is Saturday})\imp\pi=3}$
\item $(P\oor Q)\bimp P\oor(Q\oor R)$
\end{enumerate}
{\bf[8 pts]}
\item{\bf Working Together:} Find three non-mathematics-related logical statements $A,B,C$ such that $(A\oand B)\imp C$, but neither $A$ nor $B$ alone imply $C$. Be sure to explain why your selections satisfy the desired properties.\\
{\bf[3 pts]}

\end{enumerate}
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{\large\bf Apply}:
\begin{enumerate}[resume]
\item{\bf The Goods Are Odd:} Let $O(x)$ be the proposition ``$x$ is odd''. Prove the following claim:
\[ \fa a,b\in\Z\st\;[(O(a)\oand O(b))\iff O(a\cdot b)] \]
{\bf[6 pts]}
\item{\bf Stay Positive (Well, Non-negative):} Let $P=\{x\in\R\mid x>0\}$. Let $a\in\R$. Prove the following claim:
\[ \of{\fa p\in P\st\; a>-p} \imp a\geq 0 \]
{\bf[6 pts]}
\item{\bf Reciprocate:} Prove or disprove the following claim:
\[ \exs x\in \setof{y\in\R\mid y>0}\st\; x+\frac{1}{x}< 2 \]

\item{\bf :} Let $T,U,V$ be sets, with $T\neq\es$. Prove that
\[ (T\times U=T\times V) \iff U=V \]
Explain what happens to this claim when $T=\es$. Is it still true? Why or why not?\\
{\bf[6 pts]}
\end{enumerate}
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{\large\bf Explore}:
\begin{enumerate}[resume]
\item{\bf Can We Distribute the $-$?:} Prove or disprove the following claim:
\[ \fa A,B,C\text{ that are sets}\st\;A-(B-C)=(A-B)\cup C \]
{\bf[5 pts]}
\item{\bf Can We Un-distribute the $\PP$?:} Prove or disprove the following claim:
\[ \fa A,B\text{ that are sets}\st\; \PP(A\cup B)=\PP(A)\cup\PP(B) \]
{\bf[5 pts]}
\item{\bf Order Matters:} Find an example of a property $P(x,y)$ such that
\[ \fa x\in\N\st\exs y\in\N\st P(x,y) \]
is {\sf T}, but
\[ \exs x\in\N\st\fa y\in\N\st P(x,y) \]
is {\sf F}. Be sure to explain why your selection works.\\
{\bf[3 pts]}
\end{enumerate}
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