Discrete Mathematics and Its Applications with MathZone

PrefaceThe MathZone Companion WebsiteTo the Student1 The Foundations: Logic and Proofs1.1 Propositional Logic1.2 Propositional Equivalences1.3 Predicates and Quantifiers1.4 Nested Quantifiers1.5 Rules of Inference1.6 Introduction to Proofs1.7 Proof Methods and StrategyEnd-of-Chapter Material2 Basic Structures: Sets, Functions, Sequences and Sums2.1 Sets2.2 Set Operations2.3 Functions2.4 Sequences and SummationsEnd-of-Chapter Material3 The Fundamentals: Algorithms, the Integers, and Matrices3.1 Algorithms3.2 The Growth of Functions3.3 Complexity of Algorithms3.4 The Integers and Division3.5 Primes and Greatest Common Divisors3.6 Integers and Algorithms3.7 Applications of Number Theory3.8 MatricesEnd-of-Chapter Material4 Induction and Recursion4.1 Mathematical Induction4.2 Strong Induction and Well-Ordering4.3 Recursive Definitions and Structural Induction4.4 Recursive Algorithms4.5 Program CorrectnessEnd-of-Chapter Material5 Counting5.1 The Basics of Counting5.2 The Pigeonhole Principle5.3 Permutations and Combinations5.4 Binomial Coefficients5.5 Generalized Permutations and Combinations5.6 Generating Permutations and CombinationsEnd-of-Chapter Material6 Discrete Probability6.1 An Introduction to Discrete Probability6.2 Probability Theory6.3 Bayes' Theorem6.4 Expected Value and VarianceEnd-of-Chapter Material7 Advanced Counting Techniques7.1 Recurrence Relations7.2 Solving Linear Recurrence Relations7.3 Divide-and-Conquer Algorithms and Recurrence elations7.4 Generating Functions7.5 Inclusion-Exclusion7.6 Applications of Inclusion-ExclusionEnd-of-Chapter Material8 Relations8.1 Relations and Their Properties8.2 n-ary Relations and Their Applications8.3 Representing Relations8.4 Closures of Relations8.5 Equivalence Relations8.6 Partial OrderingsEnd-of-Chapter Material9 Graphs9.1 Graphs and Graph Models9.2 Graph Terminology and Special Types of Graphs9.3 Representing Graphs and Graph Isomorphism9.4 Connectivity9.5 Euler and Hamilton Paths9.6 Shortest-Path Problems9.7 Planar Graphs9.8 Graph ColoringEnd-of-Chapter Material10 Trees10.1 Introduction to Trees10.2 Applications of Trees10.3 Tree Traversal10.4 Spanning Trees10.5 Minimum Spanning TreesEnd-of-Chapter Material11 Boolean Algebra11.1 Boolean Functions11.2 Representing Boolean Functions11.3 Logic Gates11.4 Minimization of CircuitsEnd-of-Chapter Material12 Modeling Computation12.1 Languages and Grammars12.2 Finite-State Machines with Output12.3 Finite-State Machines with No Output12.4 Language Recognition12.5 Turing MachinesEnd-of-Chapter MaterialAppendixesA.1 Axioms for the Real Numbers and the Positive IntegersA.2 Exponential and Logarithmic FunctionsA.3 PseudocodeSuggested ReadingsAnswers to Odd-Numbered ExercisesPhoto CreditsIndex of BiographiesIndex