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Discrete Mathematics and its Applications
by Rosen, Kenneth H.Edition: 5th
Format: Hardcover
Pub. Date: 2002-09-09
Publisher(s): McGraw-Hill Science/Engineering/Math
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Summary
Discrete Mathematics and its Applications is a focused introduction to the primary themes in a discrete mathematics course, as introduced through extensive applications, expansive discussion, and detailed exercise sets. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students. The Fifth Edition includes a more thorough and linear presentation of logic, proof types and proof writing, and mathematical reasoning. This enhanced coverage will provide students with a solid understanding of the material as it relates to their immediate field of study and other relevant subjects. The inclusion of applications and examples to key topics has been significantly addressed to add clarity to every subject.
Table of Contents
Discrete Mathematics and Its Applications, Fifth Edition1 The Foundations: Logic and Proof, Sets, and Functions1.1 Logic1.2 Propositional Equivalences1.3 Predicates and Quantifiers1.4 Nested Quantifiers1.5 Methods of Proof1.6 Sets1.7 Set Operations1.8 Functions2 The Fundamentals: Algorithms, the Integers, and Matrices2.1 Algorithms2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
1.1 Logic1.2 Propositional Equivalences1.3 Predicates and Quantifiers1.4 Nested Quantifiers1.5 Methods of Proof1.6 Sets1.7 Set Operations1.8 Functions2 The Fundamentals: Algorithms, the Integers, and Matrices2.1 Algorithms2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
1.3 Predicates and Quantifiers1.4 Nested Quantifiers1.5 Methods of Proof1.6 Sets1.7 Set Operations1.8 Functions2 The Fundamentals: Algorithms, the Integers, and Matrices2.1 Algorithms2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
1.5 Methods of Proof1.6 Sets1.7 Set Operations1.8 Functions2 The Fundamentals: Algorithms, the Integers, and Matrices2.1 Algorithms2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
1.7 Set Operations1.8 Functions2 The Fundamentals: Algorithms, the Integers, and Matrices2.1 Algorithms2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
2 The Fundamentals: Algorithms, the Integers, and Matrices2.1 Algorithms2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
2.2 The Growth of Functions2.3 Complexity of Algorithms2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
2.4 The Integers and Division2.5 Applications of Number Theory2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
2.6 Matrices3 Mathematical Reasoning, Induction, and Recursion3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
3.1 Proof Strategy3.2 Sequences and Summations3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
3.3 Mathematical Induction3.4 Recursive Definitions and Structural Induction3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
3.5 Recursive Algorithms3.6 Program Correctness4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
4 Counting4.1 The Basics of Counting4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
4.2 The Pigeonhole Principle4.3 Permutations and Combinations4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
4.4 Binomial Coefficients4.5 Generalized Permutations and Combinations4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
4.6 Generating Permutations and Combinations5 Discrete Probability5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
5.1 An Introduction to Discrete Probability5.2 Probability Theory5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
5.3 Expected Value and Variance6 Advanced Counting Techniques6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
6.1 Recurrence Relations6.2 Solving Recurrence Relations6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
6.3 Divide-and-Conquer Algorithms and Recurrence Relations6.4 Generating Functions6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
6.5 Inclusion-Exclusion6.6 Applications of Inclusion-Exclusion7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
7 Relations7.1 Relations and Their Properties7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
7.2 n-ary Relations and Their Applications7.3 Representing Relations7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
7.4 Closures of Relations7.5 Equivalence Relations7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
7.6 Partial Orderings8 Graphs8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
8.1 Introduction to Graphs8.2 Graph Terminology8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
8.3 Representing Graphs and Graph Isomorphism8.4 Connectivity8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
8.5 Euler and Hamilton Paths8.6 Shortest-Path Problems8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
8.7 Planar Graphs8.8 Graph Coloring9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
9 Trees9.1 Introduction to Trees9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
9.2 Applications of Trees9.3 Tree Traversal9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
9.4 Spanning Trees9.5 Minimum Spanning Trees10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
10 Boolean Algebra10.1 Boolean Functions10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
10.2 Representing Boolean Functions10.3 Logic Gates10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
10.4 Minimization of Circuits11 Modeling Computation11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
11.1 Languages and Grammars11.2 Finite-State Machines with Output11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
11.3 Finite-State Machines with No Output11.4 Language Recognition11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
11.5 Turing MachinesAppendixesA.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
A.1 Exponential and Logarithmic FunctionsA.2 Pseudocode
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