Merge branch 'master' of https://github.com/oscarlevin/discrete-book

oscarlevin · Jan 6, 2015 · c669b2c · c669b2c
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diff --git a/chapters/AdvancedCounting.pdf b/chapters/AdvancedCounting.pdf
diff --git a/chapters/AdvancedCounting.tex b/chapters/AdvancedCounting.tex
@@ -1,12 +1,11 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
+\usepackage{discrete}
 
-\usetikzlibrary{shapes}
+\def\thetitle{Advanced Counting} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{\thetitle} % will be put in the right header
 
-\usepackage{pgf}
-
-\heading{Math 228}{}{Advanced Counting Notes}
 
 
 

diff --git a/chapters/CombinatorialProofs.pdf b/chapters/CombinatorialProofs.pdf
diff --git a/chapters/CombinatorialProofs.tex b/chapters/CombinatorialProofs.tex
@@ -1,12 +1,12 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
+\usepackage{discrete}
 
-\usetikzlibrary{shapes}
+\def\thetitle{Combinatorial Proofs} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{\thetitle} % will be put in the right header
 
-\usepackage{pgf}
 
-\heading{Math 228}{}{Combinatorial Proofs Notes}
 
 
 

diff --git a/chapters/Counting.pdf b/chapters/Counting.pdf
diff --git a/chapters/Counting.tex b/chapters/Counting.tex
@@ -1,11 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-\usetikzlibrary{shapes,snakes}
-\usepackage{pgf}
-
-\heading{Math 228}{}{Counting Notes}
+\usepackage{discrete}
 
+\def\thetitle{Introduction to Counting} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{Counting} % will be put in the right header
 
 
 \begin{document}
@@ -119,7 +118,7 @@ \section{Counting with sets}
 From the example above, we see that in order to investigate the multiplicative principle carefully, we need to consider ordered pairs.  We should define this carefully.
 
 \begin{definition}
- Given sets $A$ and $B$, we can form the {\em set} $A \times B = \{(x,y) \st x \in A \and y \in B\}$ to be the set of all ordered pairs $(x,y)$ where $x$ is an element of $A$ and $y$ is an element of $B$.  We call $A \times B$ the {\em Cartesian product} of $A$ and $B$.
+ Given sets $A$ and $B$, we can form the {\em set} $A \times B = \{(x,y) \st x \in A \wedge y \in B\}$ to be the set of all ordered pairs $(x,y)$ where $x$ is an element of $A$ and $y$ is an element of $B$.  We call $A \times B$ the {\em Cartesian product} of $A$ and $B$.
 \end{definition}
 
 The question is, what is $|A \times B|$ - the cardinality of the Cartesian product?  To figure this out, let's write out $A \times B$.

diff --git a/chapters/GeneratingFunctions.pdf b/chapters/GeneratingFunctions.pdf
diff --git a/chapters/GeneratingFunctions.tex b/chapters/GeneratingFunctions.tex
@@ -1,10 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-
-\heading{Math 228}{}{Generating Functions Notes}
-
+\usepackage{discrete}
 
+\def\thetitle{Generating Functions} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{\thetitle} % will be put in the right header
 
 \begin{document}
 \section{Generating Functions}

diff --git a/chapters/GraphTheory.pdf b/chapters/GraphTheory.pdf
diff --git a/chapters/GraphTheory.tex b/chapters/GraphTheory.tex
@@ -1,12 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
+\usepackage{discrete}
 
-\usetikzlibrary{shapes}
-
-\usepackage{pgf}
-
-\heading{Math 228}{}{Graph Theory Notes}
+\def\thetitle{Introduction to Graph Theory} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{\thetitle} % will be put in the right header
 
 
 

diff --git a/chapters/Induction.pdf b/chapters/Induction.pdf
diff --git a/chapters/Induction.tex b/chapters/Induction.tex
@@ -1,9 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-
-\heading{Math 228}{}{Induction Notes}
+\usepackage{discrete}
 
+\def\thetitle{The Principle of Mathematical Induction} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{Induction} % will be put in the right header
 
 
 \begin{document}

diff --git a/chapters/Logic.pdf b/chapters/Logic.pdf
diff --git a/chapters/Logic.tex b/chapters/Logic.tex
@@ -1,9 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-
-\heading{Math 228}{}{Logic Notes}
+\usepackage{discrete}
 
+\def\thetitle{Mathematical Logic} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{Logic} % will be put in the right header
 
 
 \begin{document}

diff --git a/chapters/NumberTheory.pdf b/chapters/NumberTheory.pdf
diff --git a/chapters/NumberTheory.tex b/chapters/NumberTheory.tex
@@ -1,10 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-
-\heading{Math 228}{}{Number Theory Notes}
-
+\usepackage{discrete}
 
+\def\thetitle{Introduction to Number Theory} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{Number Theory} % will be put in the right header
 
 \begin{document}
 %\section*{Introduction to Number Theory}\setcounter{section}{1}

diff --git a/chapters/Quantifiers_Sets.pdf b/chapters/Quantifiers_Sets.pdf
diff --git a/chapters/Quantifiers_Sets.tex b/chapters/Quantifiers_Sets.tex
@@ -1,13 +1,13 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-
-\heading{Math 228}{}{Logic 2: Quantifiers and Sets Notes }
-
+\usepackage{discrete}
 
+\def\thetitle{Quantifiers and Set Theory} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{Quantifiers and Sets} % will be put in the right header
 
 \begin{document}
-\section*{Quantifiers and Sets}
+%\section*{Quantifiers and Sets}
 
 So far we have seen how statements can be combined with logical symbols.  This is helpful when trying to understand a complicated mathematical statement - you can determine under which conditions the complicated statement is true.  Additionally, we have been able to analyze the logical form of arguments to decide which arguments are valid and which are not.  However, the types of statements we have been able to make so far has be sorely limited.  For example, consider a classic argument:
 

diff --git a/chapters/Sequences.pdf b/chapters/Sequences.pdf
diff --git a/chapters/Sequences.tex b/chapters/Sequences.tex
@@ -1,10 +1,10 @@
 \documentclass[12pt]{article}
 
-\usepackage{../discrete}
-
-\heading{Math 228}{}{Sequences Notes}
-
+\usepackage{discrete}
 
+\def\thetitle{Sequences} % will be put in the center header on the first page only.
+\def\lefthead{Math 228 Notes} % will be put in the left header
+\def\righthead{\thetitle} % will be put in the right header
 
 \begin{document}
 As the legend goes, there is a monastery in Hanoi with a great hall containing 3 tall pillars.  Resting on the first pillar are 64 giant disks (or washers) - all different sizes, stacked from largest to smallest.  The monks are charged with the following task: they must move the entire stack of disks to the third pillar.  However, due to the size of the disks, the monks cannot move more than one at a time.  Each disk must be placed on one of the pillars before the next disk is moved.  And because the disks are so heavy and fragile, the monks may never place a larger disk on top of a smaller disk.

diff --git a/chapters/discrete.sty b/chapters/discrete.sty
@@ -0,0 +1,80 @@
+%%%  This is discrete.sty
+%%%  It is an attempt at designing a package to be used in my discrete course notes.  It will load amsmath, amsthm, etc and all the definitions I like.
+
+\ProvidesPackage{discrete}
+
+\usepackage{amssymb, amsmath, amsthm, mathrsfs, multicol, graphicx} 
+\usepackage[top=1in, bottom=1in, left=1in, right=1in]{geometry}
+\usepackage{changepage} % for adding extra indents in environments
+\usepackage{fancyhdr}
+\usepackage{tikz}
+\usepackage{framed}
+\usepackage{textcomp}
+\usepackage{booktabs}
+\usepackage{ifthen}
+\usepackage{everypage}
+
+\pagestyle{fancy}
+
+
+\AddEverypageHook{%
+  \ifthenelse{\value{page}=1}%
+    {\renewcommand{\headrulewidth}{0.0pt}}%
+    {\renewcommand{\headrulewidth}{0.4pt}}%
+}
+%\makeatletter\let\thetitle\@title\makeatother
+%headers and footers:
+\renewcommand{\headrulewidth}{0.1pt}
+%\renewcommand{\chaptermark}[1]{\markboth{#1}{}}
+\renewcommand{\sectionmark}[1]{\markright{#1}{}}
+\lhead{\ifthenelse{\value{page}=1}{}{\footnotesize \lefthead}}
+\chead{\ifthenelse{\value{page}=1}{\textbf{\Large \thetitle}}{}}
+\rhead{\ifthenelse{\value{page}=1}{}{\footnotesize \righthead}}
+\newcommand{\heading}[3]{\clearpage{\lhead{\footnotesize #1} \chead{\bf #2} \rhead{\footnotesize #3}}}
+%\chead{\title}
+%\newcommand{\cleanpage}{\cfoot{}}
+
+
+
+%new list environments
+\newcounter{qcounter}
+\newcounter{pcounter}
+\newenvironment{questions}{\begin{list}{\arabic{qcounter}.~}{\usecounter{qcounter}}}{\end{list}}
+\newcommand{\question}[1]{\item #1}
+\newenvironment{parts}{\begin{list}{(\alph{pcounter})~}{\usecounter{pcounter}}}{\end{list}}
+\renewcommand{\part}[1]{\item #1}
+
+
+\theoremstyle{plain}
+\newtheorem*{theorem}{Theorem}
+\newtheorem{problem}{Problem}
+\newtheorem*{conjecture}{Conjecture}
+%\newtheorem{answer}{Answer}
+
+\theoremstyle{definition}
+\newtheorem{definition}{Definition}
+%\newtheorem{example}{Example}
+\newtheorem{probfact}{Probability Fact}
+
+\theoremstyle{remark}
+\newtheorem*{remark}{Remark}
+%\newtheorem*{solution}{Solution}
+
+%Create indented ``example'' environment:
+\newenvironment{example}{\vskip 2em \begin{adjustwidth}{3em}{3em}{\bf Example}: }{\end{adjustwidth}\vskip 2em}
+%Create ``solution'' environment:
+\newenvironment{solution}{\vskip 1ex \noindent {\em Solution}:}{}
+%Create ``defbox'' environment to highlight important definitions
+\newenvironment{defbox}[1]{\vskip 1em \begin{framed}\noindent{\bf #1\\}}{\end{framed}}
+
+\newcommand{\ex}{\noindent{\bf Ex: }}
+
+
+\renewcommand{\qedsymbol}{\small QED}
+
+
+\input{../macros-discrete}
+
+\endinput
+
+%%% End of File.
diff --git a/chapters/images/7bridgescolor.png b/chapters/images/7bridgescolor.png