Discrete Mathematics and Its Applications

Summary: Discrete Mathematics and Its Applications, Fifth Edition 1 The Foundations: Logic and Proof, Sets, and Functions 1.1 Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Methods of Proof 1.6 Sets 1.7 Set Operations 1.8 Functions 2 The Fundamentals: Algorithms, the Integers, and Matrices 2.1 Algorithms 2.2 The Growth of Functions 2.3 Complexity of Algorithms 2.4 The Integers a...nd Division 2.5 Applications of Number Theory 2.6 Matrices 3 Mathematical Reasoning, Induction, and Recursion 3.1 Proof Strategy 3.2 Sequences and Summations 3.3 Mathematical Induction 3.4 Recursive Definitions and Structural Induction 3.5 Recursive Algorithms 3.6 Program Correctness 4 Counting 4.1 The Basics of Counting 4.2 The Pigeonhole Principle 4.3 Permutations and Combinations 4.4 Binomial Coefficients 4.5 Generalized Permutations and Combinations 4.6 Generating Permutations and Combinations 5 Discrete Probability 5.1 An Introduction to Discrete Probability 5.2 Probability Theory 5.3 Expected Value and Variance 6 Advanced Counting Techniques 6.1 Recurrence Relations 6.2 Solving Recurrence Relations 6.3 Divide-and-Conquer Algorithms and Recurrence Relations 6.4 Generating Functions 6.5 Inclusion-Exclusion 6.6 Applications of Inclusion-Exclusion 7 Relations 7.1 Relations and Their Properties 7.2 n-ary Relations and Their Applications 7.3 Representing Relations 7.4 Closures of Relations 7.5 Equivalence Relations 7.6 Partial Orderings 8 Graphs 8.1 Introduction to Graphs 8.2 Graph Terminology 8.3 Representing Graphs and Graph Isomorphism 8.4 Connectivity 8.5 Euler and Hamilton Paths 8.6 Shortest-Path Problems 8.7 Planar Graphs 8.8 Graph Coloring 9 Trees 9.1 Introduction to Trees 9.2 Applications of Trees 9.3 Tree Traversal 9.4 Spanning Trees 9.5 Minimum Spanning Trees 10 Boolean Algebra 10.1 Boolean Functions 10.2 Representing Boolean Functions 10.3 Logic Gates 10.4 Minimization of Circuits 11 Modeling Computation 11.1 Languages and Grammars 11.2 Finite-State Machines with Output 11.3 Finite-State Machines with No Output 11.4 Language Recognition 11.5 Turing Machines Appendixes A.1 Exponential and Logarithmic Functions A.2 Pseudocode [read more]