\include{preamble}

\newcommand{\Exercise}[2]{
\begin{center}
\fbox{
\begin{minipage}{5in}
{\bf See exercise #1:} #2 
\end{minipage}}
\end{center}

}


\makeatletter
\newcommand\binomialCoefficient[2]{%
    % Store values 
    \c@pgf@counta=#1% n
    \c@pgf@countb=#2% k
    %
    % Take advantage of symmetry if k > n - k
    \c@pgf@countc=\c@pgf@counta%
    \advance\c@pgf@countc by-\c@pgf@countb%
    \ifnum\c@pgf@countb>\c@pgf@countc%
        \c@pgf@countb=\c@pgf@countc%
    \fi%
    %
    % Recursively compute the coefficients
    \c@pgf@countc=1% will hold the result
    \c@pgf@countd=0% counter
    \pgfmathloop% c -> c*(n-i)/(i+1) for i=0,...,k-1
        \ifnum\c@pgf@countd<\c@pgf@countb%
        \multiply\c@pgf@countc by\c@pgf@counta%
        \advance\c@pgf@counta by-1%
        \advance\c@pgf@countd by1%
        \divide\c@pgf@countc by\c@pgf@countd%
    \repeatpgfmathloop%
    \the\c@pgf@countc%
}
\makeatother

\title{}
\author{Zajj Daugherty}
\date{\today}

\usepackage{multicol}
\setlength{\columnsep}{1.5cm}
\setlength{\columnseprule}{0.2pt}

\begin{document}
\centerline{\bf Welcome to Math 365!}

\noindent Course website (including syllabus): https://zdaugherty.ccnysites.cuny.edu/teaching/m365s19/\\
Professor: Zajj Daugherty, zdaugherty@gmail.com\\
Textbook: Discrete Mathematics and Its Applications (7th edition), by Kenneth Rosen.


%\begin{warmup}\qquad
%\begin{enumerate}
%	\item How many ways can you choose 2 things from a set of 4?\\
%		(Example, pick a committee of 2 people from a group of 4. 
%		This is different from the number of ways to choose a president 
%		and a vice president from a group of 4 people.)
%	\item How many ways can you choose 3 things from a set of 5?
%	\item Explain why there are exactly the same number of ways to choose 1 thing from a set of 5 as there are ways to choose 4 things from a set of 5. 
%	\item How many ways are there to choose 3 things from a set of 3? 4 things from a set of 4? 5 things from a set of 5?
%	\item How many ways are there to choose 0 things from a set of 3? 0 things from a set of 4? 0 things from a set of 5?
%	\item Expand:
%	\vspace{-.2in}
%	\begin{align*}
%		(1+x)^2 &= 1 + 2x + x^2\\
%		(1 + x)^3 &= \\
%		(1 + x)^4 &= 
%	\end{align*} 
%\end{enumerate}
%\end{warmup}
\medskip
\hrule
\medskip


\noindent {\bf Homework 0:} due Monday 2/4 by email. \quad (See course website.) 


\medskip
\hrule
\medskip

\noindent {\bf Attach at the end of Homework 1:} \\
Before writing up your homework, read handouts ``Communicating Mathematics through Homework and Exams'' and ``Some Guidelines for Good Mathematical Writing''. Then, later, at the end of your write-up, include the following, labeling this as \textbf{``Writing exercise''}. 
\begin{enumerate}[(a)]
\item List three things you learned or thought about more carefully after reading these documents. 
\item Mark up your finished homework assignment, showing where you followed or failed to follow the mechanical and stylistic issues outlined in the handout \emph{Communicating Mathematics\dots}. This means \textbf{treat your write-up as a second-to-last draft}, and go point-by-point through the handout and address instances where you followed or did not follow each direction in your writing. Use a different-colored pen if you have one, and hand in this marked up draft. You do not need to rewrite the result.

How might you improve in the future? 

\item List three or more ways that you succeeded or failed at following the advice in \emph{Some Guidelines\dots}. How might you improve in the future?  
\end{enumerate}

\noindent \textbf{To receive any credit for homework 1, you must do this writing exercise.}

\medskip
\hrule



\begin{try}\quad
\begin{enumerate}[(a)]
\item Use set-builder notation, i.e.\ $\{$ elements $|$ conditions $\}$, to write the following sets.
\begin{enumerate}[(i)]
\item The set of positive integers that are multiples of $5$.
\item The set of real numbers that are not integers.
\item The set of rational number that are between $-3$ and $19$, inclusive.
\end{enumerate}





\item Let $U = \{-3,-2,-1,0,1,2,3\}$ and $V = U \times U$. List the elements in the following sets\vspace{-.25in}
\begin{multicols}{2}
\begin{enumerate}[(i)]
	\item $A = \{x \in \ZZ ~|~  -2 \le x <2\}$ \smallskip
	\item $B = \{x \in \ZZ ~|~ 0 < x <3\}$ \smallskip
	\item $\cP(B)$
	\item $A \cup B$ \smallskip
	\item $A \cap B$ \smallskip
	\item $A - B$\smallskip
	\item $\overline{A}$, where $U$ is the universal set. \columnbreak
~\\
	\item $C = \{ (x,y) \in A \times B ~|~ x \ne y\}$. \smallskip
	\item $\overline{C}$, where $V$ is the universal set.
	\item $A \times A$ \smallskip
	\item $(A \times A) \cap C$\smallskip
	\item $(A \times A) \cup C$  \smallskip
	\item $\overline{A \times A} \cap C$, where $V$ is the universal set.
\end{enumerate}\end{multicols}

\item Pick a set $S$ and two universal sets $U_1$ and $U_2$ that illustrate that $\bar{A}$ depends on the choice of universal set.

\smallskip


\item Let $A$ and $B$ be sets contained in a universal set $U$. Decide whether the following identities are {\bf true or false}. If false, give an example where the identity doesn't hold. If true, explain why (in complete sentences). \vspace{-.25in}
\begin{multicols}{2}
\begin{enumerate}[(i)]
\item $A \cap B = B \cap A$ \smallskip
\item $A \cup B = B \cup A$ \smallskip
\item $A - B = B - A$ \smallskip
\item $A \times B = B \times A$ 
\columnbreak
~\\
\item $|A - B| = |A| - |B|$ \smallskip
\item If $A$ is finite, then so is $\cP(A)$ \smallskip
\item $\overline{A} \cap \overline{B} = \overline{(A \cup B)}$
\end{enumerate}\end{multicols}
\end{enumerate}

\end{try}

\begin{try}
\begin{enumerate}[(a)]

\item Let $X$ be the set of students who live within one mile of school and let $Y$ be the set of students who walk home after school. Describe the students in each of these sets. 
\begin{enumerate}[(i)]
\item $X \cap Y$

\item $X \cup Y$

\item $X - Y$ 

\item $Y - X$
\end{enumerate}

\medskip

\item How many elements does each of these sets have (where $a$ and $b$ are distinct elements)?
\begin{enumerate}[(i)]
\item $P(\{a,b,\{a,b\}\})$
\item $P(\{\emptyset,a,\{a\},\{\{a\}\}\}) $
\item $P(P(\emptyset))$
\end{enumerate}

\medskip

\item Suppose that $A\times B =\emptyset$, where $A$ and $B$ are sets. What can you conclude?

\medskip

\item Look up Pascal's triangle online. Give three interesting facts about it (not including any covered in class). 
\end{enumerate}
\end{try}


\vfill

\noindent {\color{black}\fbox{ \begin{minipage}{6.5in}
{\bf Due Wednesday 2/6:} 
Handout Exercises 1--7; and the Writing Exercise.\\
{\bf For next time:} read sections 2.1 and 2.2.\end{minipage}}}
\thispagestyle{empty}




\vfill

\noindent Some shorthands you'll see in the book:\\
\centerline{
\begin{tabular}{|rl|l|}
\hline
$\in$ & means 	``in'', ``contained in'' & Ex: $x \in \RR$ means $x$ is a real number.  $\phantom{\Big|}$\\\hline
$\forall$ & means ``for all'' & Ex: $A \subseteq B$ if $\forall a \in A$, we have $a \in B$. $\phantom{\Big|}$\\\hline
$\wedge$ & means ``and" (both) & Ex: $A \cap B = \{x \in U ~|~ ( x \in A ) \wedge (x \in B)\}$. $\phantom{\Big|}$\\ \hline
$\vee$ & means ``or" (one or the other or both) & Ex: $A \cup B = \{x \in U ~|~ (x \in A) \vee (x \in B)\}$. $\phantom{\Big|}$\\\hline
$\neg$ & means ``not'' & Ex: $\overline{A} = \{x \in U ~|~ \neg(x \in A)\}$.  $\phantom{\Big|}$\\
\hline
\end{tabular}
}

\end{document}