Discrete and Combinatorial Mathematics (Classic Version)

I. FUNDAMENTALS OF DISCRETE MATHEMATICS

1. Fundamental Principles of Counting

The Rules of Sum and Product
Permutations
Combinations: The Binomial Theorem
Combinations with Repetition
The Catalan Numbers (Optional)
Summary and Historical Review


2. Fundamentals of Logic

Basic Connectives and Truth Tables
Logical Equivalence: The Laws of Logic
Logical Implication: Rules of Inference
The Use of Quantifiers
Quantifiers, Definitions, and the Proofs of Theorems
Summary and Historical Review


3. Set Theory

Sets and Subsets
Set Operations and the Laws of Set Theory
Counting and Venn Diagrams
A First Word on Probability
The Axioms of Probability (Optional)
Conditional Probability: Independence (Optional)
Discrete Random Variables (Optional)
Summary and Historical Review


4. Properties of the Integers: Mathematical Induction

The Well-Ordering Principle: Mathematical Induction
Recursive Definitions
The Division Algorithm: Prime Numbers
The Greatest Common Divisor: The Euclidean Algorithm
The Fundamental Theorem of Arithmetic
Summary and Historical Review


5. Relations and Functions

Cartesian Products and Relations
Functions: Plain and One-to-One
Onto Functions: Stirling Numbers of the Second Kind
Special Functions
The Pigeonhole Principle
Function Composition and Inverse Functions
Computational Complexity
Analysis of Algorithms
Summary and Historical Review


6. Languages: Finite State Machines

Language: The Set Theory of Strings
Finite State Machines: A First Encounter
Finite State Machines: A Second Encounter
Summary and Historical Review


7. Relations: The Second Time Around

Relations Revisited: Properties of Relations
Computer Recognition: Zero-One Matrices and Directed Graphs
Partial Orders: Hasse Diagrams
Equivalence Relations and Partitions
Finite State Machines: The Minimization Process
Summary and Historical Review



II. FURTHER TOPICS IN ENUMERATION

8. The Principle of Inclusion and Exclusion

Derangements: Nothing Is in Its Right Place
Rook Polynomials
Arrangements with Forbidden Positions
Summary and Historical Review


9. Generating Functions

Introductory Examples
Definition and Examples: Calculational Techniques
Partitions of Integers
The Exponential Generating Functions
The Summation Operator
Summary and Historical Review


10. Recurrence Relations

The First-Order Linear Recurrence Relation
The Second-Order Linear Homogeneous Recurrence Relation with Constant Coefficients
The Nonhomogeneous Recurrence Relation
The Method of Generating Functions
A Special Kind of Nonlinear Recurrence Relation (Optional)
Divide and Conquer Algorithms
Summary and Historical Review



III. GRAPH THEORY AND APPLICATIONS

11. An Introduction to Graph Theory

Definitions and Examples
Subgraphs, Complements, and Graph Isomorphism
Vertex Degree: Euler Trails and Circuits
Planar Graphs
Hamilton Paths and Cycles
Graph Coloring and Chromatic Polynomials
Summary and Historical Review


12. Trees

Definitions, Properties, and Examples
Rooted Trees
Trees and Sorting
Weighted Trees and Prefix Codes
Biconnected Components and Articulation Points
Summary and Historical Review


13. Optimization and Matching

Dijkstra's Shortest Path Algorithm
Minimal Spanning Trees: The Algorithms of Kruskal and Prim
Transport Networks: The Max-Flow Min-Cut Theorem
Matching Theory
Summary and Historical Review



IV. MODERN APPLIED ALGEBRA

14. Rings and Modular Arithmetic

The Ring Structure: Definition and Examples
Ring Properties and Substructures
The Integers Modulo n. Cryptology
Ring Homomorphisms and Isomorphisms: The Chinese Remainder Theorem
Summary and Historical Review


15. Boolean Algebra and Switching Functions

Switching Functions: Disjunctive and Conjunctive Normal Forms
Gating Networks: Minimal Sums of Products: Karnaugh Maps
Further Applications: Don't-Care Conditions
The Structure of a Boolean Algebra (Optional)
Summary and Historical Review.


16. Groups, Coding Theory, and Polya's Theory of Enumeration

Definition, Examples, and Elementary Properties
Homomorphisms, Isomorphisms, and Cyclic Groups
Cosets and Lagrange's Theorem
The RSA Cipher (Optional)
Elements of Coding Theory
The Hamming Metric
The Parity-Check and Generator Matrices
Group Codes: Decoding with Coset Leaders
Hamming Matrices
Counting and Equivalence: Burnside's Theorem
The Cycle Index
The Pattern Inventory: Polya's Method of Enumeration
Summary and Historical Review


17. Finite Fields and Combinatorial Designs

Polynomial Rings
Irreducible Polynomials: Finite Fields
Latin Squares
Finite Geometries and Affine Planes
Block Designs and Projective Planes
Summary and Historical Review



Appendices

Exponential and Logarithmic Functions
Matrices, Matrix Operations, and Determinants
Countable and Uncountable Sets
Solutions
Index