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\noindent {\large\bf Homework 2 [50 pts] \hfill 21-127 Concepts of Math \hfill due 07.10.2012}\\
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\noindent{\bf\large Notes:} This assignment is designed to give you practice working with sets. Reviewing the definitions and notation we have seen in class, and in Chapter 3 of the textbook, might be helpful. It looks long, perhaps, but many of these questions have short answers, and there are plenty of opportunities for easy points!

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{\large\bf Explain}:

\begin{enumerate}
\item{\bf Setting Up Sets:} Rewrite the following sentences using the ``set-builder notation'' to define a set. Then, if possible, write out all the elements of the set; if not possible, explain why not and write out three example elements of the set.\\
	\begin{enumerate}
	\item Let $A$ be the set of all natural numbers whose squares are less than 44
	\item Let $B$ be the set of all real numbers that are roots of the equation $x^2+7x-8=0$
	\item Let $C$ be the set of pairs of integers whose sum is positive.
	\item Let $D$ be the set of pairs of integers whose first coordinate is positive and whose second coordinate is negative and whose sum is nonnegative.
	\item Fix an integer $a$. Let $E_{a}$ be the set of all integers $z$ with the property that $a\cdot k = z$, for some integer $k$. (For writing out specific elements, use the case where $a=7$).
	\end{enumerate}
{\bf [8 pts]}
\item{\bf Plane and Simple:} Let $(a,b)$ be a point on the Cartesian plane $\R\times\R$ and let $\ve$ be a nonnegative real number. Let $C_{(a,b),\ve}$ be the set of real numbers that are ``close'' to $(a,b)$; that is, define
\[ C_{(a,b),\ve} = \setof{(x,y)\in\R\times\R\mid \sqrt{(x-a)^2+(y-b)^2}<\ve} \]
\begin{enumerate}
\item Come up with a geometric description of the set $C_\ve$. What happens to the set as we change $r$? What happens if we change $\ve$?
\item What is $C_{(0,0),1}\cap C_{(0,0),2}$?
\item What is $C_{(0,0),1}\cup C_{(0,0),2}$?
\item What is $C_{(0,0),1}\cap C_{(2,2),1}$?
\end{enumerate}
{\bf [5 pts]}
\item{\bf I've Got The Power Sets:} Let $A$ and $B$ be sets.
\begin{enumerate}
\item Assume that $A\subseteq B$. Explain why $\PP(A)\subseteq\PP(B)$.
\item Does this work the other way? If we assume $\PP(A)\subseteq\PP(B)$, can we conclude that $A\subseteq B$? Why or why not?
\end{enumerate}
{\bf [3 pts]}
\item{\bf Intellectual Properties:} Let $X$ be the set of students in this course. Identify a property $P(x)$ such that $A:=\{x\in X\mid P(x)\}$ is a proper subset of $X$ and $A\neq\es$. Then, identify a property $Q(x)$ such that $B:=\{x\in X\mid Q(x)\}$ is a proper subset of $A$ and $B\neq\es$.\\
{\bf [2 pts]}
\end{enumerate}
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{\large\bf Apply}:
\begin{enumerate}[resume]

\item{\bf Contain Yourself:} Let $A,B,C$ be sets. Prove that $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ via a ``double-containment'' argument.\\
{\bf [4 pts]}

\item{\bf Let's Get Strict:} Let $A$, $B$, and $C$ be sets with $A\subseteq C$ and $B\subseteq C$.
\begin{enumerate}
\item Draw a Venn diagram for the sets $\overline{A}\cap\overline{B}$ and $\overline{(A\cap B)}$.
\item Prove that $\overline{A}\cap\overline{B}\subseteq\overline{(A\cap B)}$.
\item Define specific sets $A,B,C$ such that the inequality is {\it strict}, i.e. $\overline{A}\cap\overline{B}\subset\overline{(A\cap B)}$
\item Define specific sets $A,B,C$ such that $\overline{A}\cap\overline{B}=\overline{(A\cap B)}$.
\end{enumerate}
{\bf [4 pts]}

\item{\bf Brace Yourself:} Let $A=\{\es,\star\}$. Write out all the elements of $\PP(A)$. Then, write out all the elements of $\PP(\PP(A))$.\\
{\bf [3 pts]}

\item{\bf Are All Sets Created Equal?:} Let $A=[3]\times[4]$ (Remember that $[n]=\setof{1,2,3,\dots,n}$). Let $B=\setof{(x,y)\in\Z\times\Z\mid 0\leq 3x-y+1\leq 9}$. Show that $A\subseteq B$. Is it true that $A=B$? Why or why not?\\
{\bf [3 pts]}

\item{\bf What am I?:} In each of the following, the defined set is a commonly-known set in disguise. Explain what that set is and why.
\begin{enumerate}
\item What is the following set?
\[ F = \setof{n\in\N\mid \text{if }x\text{ is a factor of }n\text{ then }x=1\text{ or }x=n} \]
\item Using the same $F$ as the line above, what is the following set?
\[ G = \bigcup_{f\in F} \setof{fu\mid u\in\Z} \]
\end{enumerate}
{\bf [3 pts]}

\end{enumerate}
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{\large\bf Explore}:
\begin{enumerate}[resume]
\item{\bf See Dese Sets:} Can you identify specific sets $C$ and $D$ such that $C-D=D-C$? Can you make a general claim when this equality holds true?\\
{\bf [2 pts]}

\item{\bf The Power of Induction:} Provide an inductive argument that explains why $\PP([n])$ has $2^n$ elements. (Remember that $[n]=\{1,2,3,\dots,n\}$).\\
{\bf [5 pts]}

\item{\bf What's $\Z$ point?:} In this problem, we are going to ``prove'' the existence of the negative integers! I say ``prove'' because we won't really understand what we've done until later but, trust me, it's what we're doing. Because of this goal, you cannot {\it assume} any integers strictly less than 0 exist, so your algebraic steps (especially in part \ref{transitive}{\bf -d}) should not involve any terms that might be negative. For example, $a$ and $b$ are just natural numbers, so we don't yet ``know'' (in this context) that $a-b$ exists. However, something like $a-a=0$ is fine.

Let $P=\N\times\N$. Define the set $R$ by
\[ R = \setof{((a,b),(c,d))\in P\times P\mid a+d=b+c} \]
\begin{enumerate}
\label{transitive}
\item Find three different pairs $(c,d)$ such that $((1,3),(c,d))\in R$.
\item Let $(a,b)\in P$. Is it true that $((a,b),(a,b))\in R$? Why or why not?
\item Let $((a,b),(c,d))\in R$. Is it also true that $((c,d),(a,b))\in R$? Why or why not?
\item Assume $((a,b),(c,d))\in R$ and $((c,d),(e,f))\in R$. Prove that $((a,b),(e,f))\in R$, too.
\end{enumerate}
{\bf [8 pts]}
\end{enumerate}
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