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Discrete Math for Shockers
John Hammond
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\(\def\ds{\displaystyle} \def\d{\displaystyle} \def\N{\mathbb N} \def\B{\mathbf{B}} \def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\R{\mathbb R} \def\C{\mathbb C} \def\F{\mathbb F} \def\pow{\mathcal P} \def\inv{^{-1}} \def\iff{\leftrightarrow} \def\Iff{\Leftrightarrow} \def\land{\wedge} \def\And{\bigwedge} \def\entry{\entry} \def\AAnd{\d\bigwedge\mkern-18mu\bigwedge} \def\Vee{\bigvee} \def\VVee{\d\Vee\mkern-18mu\Vee} \def\imp{\rightarrow} \def\Imp{\Rightarrow} \def\Fi{\Leftarrow} \def\var{\mbox{var}} \def\Th{\mbox{Th}} \def\entry{\entry} \def\sat{\mbox{Sat}} \def\con{\mbox{Con}} \def\iffmodels{\bmodels\models} \def\dbland{\bigwedge \!\!\bigwedge} \def\dom{\mbox{dom}} \def\rng{\mbox{range}} \def\isom{\cong} \def\st{\mid} \def\divides{\mid} \def\and{\text{ and }} \def\lcm{\text{lcm}} \def\modulus{\mathbin{\%}} \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} \newcommand{\va}[1]{\vtx{above}{#1}} \newcommand{\vb}[1]{\vtx{below}{#1}} \newcommand{\vr}[1]{\vtx{right}{#1}} \newcommand{\vl}[1]{\vtx{left}{#1}} \renewcommand{\v}{\vtx{above}{}} \def\circleA{(-.5,0) circle (1)} \def\circleAlabel{(-1.5,.6) node[above]{$A$}} \def\circleB{(.5,0) circle (1)} \def\circleBlabel{(1.5,.6) node[above]{$B$}} \def\circleC{(0,-1) circle (1)} \def\circleClabel{(.5,-2) node[right]{$C$}} \def\twosetbox{(-2,-1.4) rectangle (2,1.4)} \def\threesetbox{(-2.5,-2.4) rectangle (2.5,1.4)} \def\ansfilename{practice-answers} \def\shadowprops{{fill=black!50,shadow xshift=0.5ex,shadow yshift=0.5ex,path fading={circle with fuzzy edge 10 percent}}} \newcommand{\hexbox}[3]{ \def\x{-cos{30}*\r*#1+cos{30}*#2*\r*2} \def\y{-\r*#1-sin{30}*\r*#1} \draw (\x,\y) node{#3}; } \renewcommand{\bar}{\overline} \newcommand{\card}[1]{\left| #1 \right|} \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} \newcommand{\fixspacing}{\vspace{0pt plus 1filll}\mbox{}} \usepackage{cancel} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Dedication
Acknowledgements
Preface
1
Basic Objects and Symbols
1.1
Propositional Logic
The Basics
Truth Tables for Logical Connectives
Computer Corner
Exercises
1.2
Sets
Basic terminology
Operations on sets
Verifying Set Identities
Exercises
1.3
Relations and Functions
Relations
Functions
Computer Corner
What kind of object? Binary operations vs.
\(a\mathrel{R} b\)
Exercises
2
Symbolic Logic and Proofs
2.1
Logical Equivalences
Exercises
2.2
Application: Set Equivalences
Exercises
2.3
Propositional Functions and Quantifiers
Predicates
Logical Quantifiers
Exercises
2.4
Logical Arguments
Exercises
2.5
An introduction to proofs
Basic concepts of proof
Direct Proof
Indirect Proofs
Proof by Contraposition
Proof by Contradiction
Proofs of equivalence
Proof by cases
Existence Proofs
Uniqueness proofs
Vacuous and Trivial Proofs
Counterexamples Revisited
Exercises
2.6
Chapter Review
2.6
Exercises
3
Some Classic Number Theory
3.1
Divisibility and Congruences
The Divides Relation
The Congruence Relation
Exercises
3.2
Prime Numbers
Computer Corner - Implementing the Sieve
Exercises
3.3
GCDs and The Euclidean Algorithm
Exercises
3.4
Multiplicative Inverses
Exercises
3.5
Modular exponentiation
Exercises
3.6
Application: Encryption
Basic Cipher Examples
Exercises
Number Theory Using Sage
The RSA Activity
4
Sequences, Recurrence, and Induction
4.1
Sequences and Series
Computer Corner
Exercises
4.2
Solving Recurrence Relations
Motivation, or, why do I care?
Solving recurrence relations
Solving particularly nice recurrence relations
Computer Corner
Exercises
4.3
Mathematical Induction
Computer Corner
Exercises
5
Counting Techniques
5.1
The Multiplicative and Additive Principles
Exercises
5.2
Permutations and Combinations
Exercises
5.3
Combinatorial Proofs
Binomial Coefficients
Combinatorial Proofs
Exercises
5.4
Counting Fibonacci numbers with tiles
Exercises
Back Matter
A
Solutions to the exercises
Dedication
Dedication
To Dad. It’s not an accounting book, but I did use a spreadsheet!