PART 1 Fundamentals of Discrete Mathematics
1 Fundamental Principles of Counting
1.1 The Rules of Sum and Product
1.2 Permutations
1.3 Combinations:The Binomial Theorem
1.4 Combinations with Repetition
1.5 The Catalan Numbers(Optional)
1.6 Summary and Historical Review
2 Fundamentals of Logic
2.1 Basic Connectives and Truth Tables
2.2 Logical Equivalence:The Laws of Logic
2.3 Logical Implication:Rules of Inference
2.4 The Use of Quantifiers
2.5 Quantifiers,Definitions,and the Proofs of Theorems
2.6 Summary and Historical Review
3 Set Theory
3.1 Sets and Subsets
3.2 Set Operations and the Laws of Set Theory
3.3 Counting and Venn Diagrams
3.4 A FirstWord on Probability
3.5 The Axioms of Probability(Optional)
3.6 Conditional Probability:Independence(Optional)
3.7 Discrete Random Variables(Optional)
3.8 Summary and Historical Review
4 Properties of the Integers:Mathematical Induction
4.1 TheWell-Ordering Principle:Mathematical Induction
4.2 Recursive Definitions
4.3 The Division Algorithm:Prime Numbers
4.4 The Greatest Common Divisor:The Euclidean Algorithm
4.5 The Fundamental Theorem of Arithmetic
4.6 Summary and Historical Review
5 Relations and Functions
5.1 Cartesian Products and Relations
5.2 Functions:Plain and One-to-One
5.3 Onto Functions:Stirling Numbers of the Second Kind
5.4 Special Functions
5.5 The Pigeonhole Principle
5.6 Function Composition and Inverse Functions
5.7 Computational Complexity
5.8 Analysis of Algorithms
5.9 Summary and Historical Review
6 Languages:Finite State Machines
6.1 Language:The Set Theory of Strings
6.2 Finite State Machines:A First Encounter
6.3 Finite State Machines:A Second Encounter
6.4 Summary and Historical Review
7 Relations:The Second Time Around
7.1 Relations Revisited:Properties of Relations
7.2 Computer Recognition:Zero-One Matrices and Directed Graphs
7.3 Partial Orders:Hasse Diagrams
7.4 Equivalence Relations and Partitions
7.5 Finite State Machines:The Minimization Process
7.6 Summary and Historical Review
PART 2 Further Topics in Enumeration
8 The Principle of Inclusion and Exclusion
8.1 The Principle of Inclusion and Exclusion
8.2 Generalizations of the Principle
8.3 Derangements:Nothing Is in Its Right Place
8.4 Rook Polynomials
8.5 Arrangements with Forbidden Positions
8.6 Summary and Historical Review
9 Generating Functions
9.1 Introductory Examples
9.2 Definition and Examples:Calculational Techniques
9.3 Partitions of Integers
9.4 The Exponential Generating Function
9.5 The Summation Operator
9.6 Summary and Historical Review
10 Recurrence Relations
10.1 The First-Order Linear Recurrence Relation
10.2 The Second-Order Linear Homogeneous Recurrence Relation with Constant
10.3 The Nonhomogeneous Recurrence Relation
10.4 The Method of Generating Functions
10.5 A Special Kind of Nonlinear Recurrence Relation(Optional)
10.6 Divide-and-Conquer Algorithms(Optional)
10.7 Summary and Historical Review
PART 3 Graph Theory and Applications
11 An Introduction to Graph Theory
11.1 Definitions and Examples
11.2 Subgraphs,Complements,and Graph Isomorphism
11.3 Vertex Degree:Euler Trails and Circuits
11.4 Planar Graphs
11.5 Hamilton Paths and Cycles
11.6 Graph Coloring and Chromatic Polynomials
11.7 Summary and Historical Review
12 Trees
12.1 Definitions,Properties,and Examples
12.2 Rooted Trees
12.3 Trees and Sorting
12.4 Weighted Trees and Prefix Codes
12.5 Biconnected Components and Articulation Points
12.6 Summary and Historical Review
13 Optimization and Matching
13.1 Dijkstra恠 Shortest-Path Algorithm
13.2 Minimal Spanning Trees:The Algorithms of Kruskal and Prim
13.3 Transport Networks:The Max-Flow Min-Cut Theorem
13.4 Matching Theory
13.5 Summary and Historical Review
PART 4 Modern Applied Algebra
14 Rings and Modular Arithmetic
14.1 The Ring Structure:Definition and Examples
14.2 Ring Properties and Substructures
14.3 The Integers Modulo n
14.4 Ring Homomorphisms and Isomorphisms
14.5 Summary and Historical Review
15 Boolean Algebra and Switching Functions
15.1 Switching Functions:Disjunctive and Conjunctive Normal Forms
15.2 Gating Networks:Minimal Sums of Products:Karnaugh Maps
15.3 Further Applications:Don恡-Care Conditions
15.4 The Structure of a Boolean Algebra(Optional)
15.5 Summary and Historical Review
16 Groups,Coding Theory,and Polya恠 Method of Enumeration
16.1 Definition,Examples,and Elementary Properties
16.2 Homomorphisms,Isomorphisms,and Cyclic Groups
16.3 Cosets and Lagrange恠 Theorem
16.4 The RSA Cryptosystem(Optional)
16.5 Elements of Coding Theory
16.6 The Hamming Metric
16.7 The Parity-Check and Generator Matrices
16.8 Group Codes:Decoding with Coset Leaders
16.9 Hamming Matrices
16.10 Counting and Equivalence:Burnside恠 Theorem
16.11 The Cycle Index
16.12 The Pattern Inventory:Polya恠 Method of Enumeration
16.13 Summary and Historical Review
17 Finite Fields and Combinatorial Designs
17.1 Polynomial Rings
17.2 Irreducible Polynomials:Finite Fields
17.3 Latin Squares
17.4 Finite Geometries and Affine Planes
17.5 Block Designs and Projective Planes
17.6 Summary and Historical Review
Appendix 1 Exponential and Logarithmic Functions A-1
Appendix 2 Matrices,Matrix Operations,and Determinants A-11
Appendix 3 Countable and Uncountable Sets A-23
Solutions S-1
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