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Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors. Digital is becoming increasingly important and gaining popularity, crowning Connect as the digital leader for this discipline. McGraw-Hill Education's Connect, available as an optional, add on item. Connect is the only integrated learning system that empowers students by continuously adapting to deliver precisely what they need, when they need it, how they need it, so that class time is more effective. Connect allows the professor to assign homework, quizzes, and tests easily and automatically grades and records the scores of the student's work. Problems are randomized to prevent sharing of answers and may also have a "multi-step solution" which helps move the students' learning along if they experience difficulty. Cover 1 Title Page 2 Copyright Page 3 Contents 4 About the Author 7 Preface 8 Acknowledgments 14 Online Resources 17 To the Student 20 1 The Foundations: Logic and Proofs 24 1.1 Propositional Logic (C) 24 1.2 Applications of Propositional Logic (C) 40 1.3 Propositional Equivalences (C) 49 1.4 Predicates and Quantifiers (C) 63 1.5 Nested Quantifiers (C) 83 1.6 Rules of Inference (C) 96 1.7 Introduction to Proofs (C) 107 1.8 Proof Methods and Strategy (C) 119 End-of-Chapter Material 138 2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices 144 2.1 Sets (C) 144 2.2 Set Operations (C) 156 2.3 Functions (C) 170 2.4 Sequences and Summations (C) 188 2.5 Cardinality of Sets (M) 202 2.6 Matrices (C) 211 End-of-Chapter Material 218 3 Algorithms 224 3.1 Algorithms (S) 224 3.2 The Growth of Functions (S) 239 3.3 Complexity of Algorithms (S) 254 End-of-Chapter Material 267 4 Number Theory and Cryptography 274 4.1 Divisibility and Modular Arithmetic (C) 274 4.2 Integer Representations and Algorithms (C) 283 4.3 Primes and Greatest Common Divisors (C) 294 4.4 Solving Congruences (C) 313 4.5 Applications of Congruences (S) 326 4.6 Cryptography (S) 333 End-of-Chapter Material 347 5 Induction and Recursion 354 5.1 Mathematical Induction (C) 354 5.2 Strong Induction and Well-Ordering (C) 377 5.3 Recursive Definitions and Structural Induction (C) 388 5.4 Recursive Algorithms (S) 404 5.5 Program Correctness (S) 416 End-of-Chapter Material 421 6 Counting 428 6.1 The Basics of Counting (C) 428 6.2 The Pigeonhole Principle (C) 443 6.3 Permutations and Combinations (C) 451 6.4 Binomial Coefficients and Identities (M) 460 6.5 Generalized Permutations and Combinations (M) 468 6.6 Generating Permutations and Combinations (S) 480 End-of-Chapter Material 484 7 Discrete Probability 492 7.1 An Introduction to Discrete Probability (C) 492 7.2 Probability Theory (M) 500 7.3 Bayes’ Theorem (M) 517 7.4 Expected Value and Variance (S) 526 End-of-Chapter Material 543 8 Advanced Counting Techniques 550 8.1 Applications of Recurrence Relations (C) 550 8.2 Solving Linear Recurrence Relations (M) 563 8.3 Divide-and-Conquer Algorithms and Recurrence Relations (S) 576 8.4 Generating Functions (M) 586 8.5 Inclusion–Exclusion (C) 602 8.6 Applications of Inclusion–Exclusion (M) 608 End-of-Chapter Material 615 9 Relations 622 9.1 Relations and Their Properties (C) 622 9.2 n-ary Relations and Their Applications (S) 634 9.3 Representing Relations (C) 644 9.4 Closures of Relations (M) 651 9.5 Equivalence Relations (C) 661 9.6 Partial Orderings (M) 673 End-of-Chapter Material 688 10 Graphs 696 10.1 Graphs and Graph Models (C) 696 10.2 Graph Terminology and Special Types of Graphs (C) 708 10.3 Representing Graphs and Graph Isomorphism (C) 726 10.4 Connectivity (C) 737 10.5 Euler and Hamilton Paths (C) 751 10.6 Shortest-Path Problems (M) 766 10.7 Planar Graphs (M) 776 10.8 Graph Coloring (M) 785 End-of-Chapter Material 794 11 Trees 804 11.1 Introduction to Trees (C) 804 11.2 Applications of Trees (S) 816 11.3 Tree Traversal (S) 831 11.4 Spanning Trees (M) 844 11.5 Minimum Spanning Trees (M) 858 End-of-Chapter Material 864 12 Boolean Algebra 870 12.1 Boolean Functions (S) 870 12.2 Representing Boolean Functions (S) 878 12.3 Logic Gates (S) 881 12.4 Minimization of Circuits (S) 887 End-of-Chapter Material 902 13 Modeling Computation 908 13.1 Languages and Grammars (S) 908 13.2 Finite-State Machines with Output (S) 920 13.3 Finite-State Machines with No Output (S) 927 13.4 Language Recognition (S) 940 13.5 Turing Machines (S) 950 End-of-Chapter Material 961 Appendices 966 1 Axioms for the Real Numbers and the Positive Integers 966 2 Exponential and Logarithmic Functions 972 3 Pseudocode 976 Suggested Readings 982 Answers to Odd-Numbered Exercises 990 Index of Biographies 1088 Index 1089 Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors. Digital is becoming increasingly important and gaining popularity, crowning Connect as the digital leader for this discipline.McGraw-Hill Education's Connect, available as an optional add on item, is the only integrated learning system that empowers students by continuously adapting to deliver precisely what they need, when they need it and how they need it - ensuring class time is more effective. Connect allows the professor to assign homework, quizzes, and tests easily and automatically grades and records the scores of the student's work. Problems are randomized to prevent sharing of answers and may also have a'multi-step solution'which helps move the students'learning along if they experience difficulty.