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Full Description
Containing exercises and materials that engage students at all levels, Discrete Mathematics with Ducks presents a gentle introduction for students who find the proofs and abstractions of mathematics challenging. This classroom-tested text uses discrete mathematics as the context for introducing proofwriting. Facilitating effective and active learning, each chapter contains a mixture of discovery activities, expository text, in-class exercises, and homework problems.Elementary exercises at the end of each expository section prompt students to review the materialTry This! sections encourage students to construct fundamental components of the concepts, theorems, and proofs discussed.Sets of discovery problems and illustrative examples reinforce learning.Bonus sections can be used for take-home exams, projects, or further studyInstructor Notes sections offer suggestions on how to use the material in each chapterDiscrete Mathematics with Ducks offers students a diverse introduction to the field and a solid foundation for further study in discrete mathematics and complies with SIGCSE guidelines. The book shows how combinatorics and graph theory are used in both computer science and mathematics.
Contents
THE BASICSCounting and Proofs Introduction and Summary Try This! Let's Count The Sum and Product Principles Preliminaries on Proofs and Disproofs Pigeons and Correspondences Where to Go from HereSets and LogicIntroduction and Summary SetsLogicTry This! Problems on Sets and Logic Proof Techniques: Not! Try This! A Tricky Conundrum Where to Go from Here Bonus: Truth TellersGraphs and FunctionsIntroduction and Summary Function Introduction Try This! Play with Functions and GraphsFunctions and Counting Graphs: Definitions and Examples Isomorphisms Graphs: Operations and UsesTry This! More Graph Problems RamseynessWhere to Go from Here Bonus: Party Tricks Bonus 2: Counting with the Characteristic FunctionInductionIntroduction and Summary Induction Try This! Induction More Examples The Best Inducktion Proof Ever Try This! More Problems about Induction Are They or Aren't They? Resolving Grey Ducks Where to Go from Here Bonus: Small Crooks Bonus 2: An Induction SongAlgorithms with CiphersIntroduction and Summary AlgorithmsModular Arithmetic (and Equivalence Relations) Cryptography: Some CiphersTry This! Encryptoequivalent Modulagorithmic Problems Where to Go from Here Bonus: Algorithms for Searching GraphsBonus 2: Pigeons and DivisibilityCOMBINATORICSBinomial Coefficients and Pascal's TriangleIntroduction and Summary You Have a Choice Try This! Investigate a Triangle Pascal's Triangle Overcounting Carefully and Reordering at Will Try This! Play with Powers and Permutations Binomial Basics Combinatorial Proof Try This! Pancakes and Proofs Where to Go from Here Bonus: Sorting Bubbles in Order of Size Bonus 2: MastermindBalls and Boxes and PIE-Counting TechniquesIntroduction and Summary Combinatorial Problem Types Try This! Let's Have Some PIECombinatorial Problem Solutions and StrategiesLet's Explain Our PIE! Try This! What Are the Balls and What Are the Boxes? And Do You Want Some PIE? Where to Go from HereBonus: Linear and Integer ProgrammingRecurrencesIntroduction and Summary Fibonacci Numbers and Identities Recurrences and Integer Sequences and Induction Try This! Sequences and Fibonacci Identities Naive Techniques for Finding Closed Forms and Recurrences Arithmetic Sequences and Finite Differences Try This! Recurrence Exercises Geometric Sequences and the Characteristic Equation Try This! Find Closed Forms for These Recurrence Relations! Where to Go from Here Bonus: Recurring StoriesCutting up Food (Counting and Geometry)Introduction and Summary Try This! Slice Pizza (and a Yam) Pizza Numbers Try This! Spaghetti, Yams, and More Yam, Spaghetti and Pizza Numbers Where to Go from Here Bonus: Geometric GemsGRAPH THEORYTreesIntroduction and Summary Basic Facts about Trees Try This! Spanning TreesSpanning Tree AlgorithmsBinary Trees Try This! Binary Trees and Matchings Matchings Backtracking Where to Go from Here Bonus: The Branch-and-Bound Technique in Integer ProgrammingEuler's Formula and ApplicationsIntroduction and Summary Try This! Planarity Explorations Planarity A Lovely Story Or, Are Emus Full?: A Theorem and a Proof Applications of Euler's Formula Try This! Applications of Euler's Formula Where to Go from Here Bonus: Topological Graph TheoryGraph TraversalsIntroduction and Summary Try This! Euler Traversals Euler Paths and Circuits Hamilton Circuits, the Traveling Salesperson Problem, and Dijkstra's Algorithm Try This!-Do This!-Try This! Where to Go from Here Bonus: Digraphs, Euler Traversals, and RNA Chains Bonus 2: Network Flows Bonus 3: Two Hamiltonian TheoremsGraph ColoringIntroduction and Summary Try This! Coloring Vertices and EdgesIntroduction to ColoringTry This! Let's Think about Coloring Coloring and Things (Graphs and Concepts) That Have Come BeforeWhere to Go from Here Bonus: The Four-Color TheoremOTHER MATERIALProbability and ExpectationIntroduction and Summary What Is Probability, Exactly? High Expectations You are Probably Expected to Try This! Conditional Probability and IndependenceTry This! . . . Probably, Under Certain Conditions Higher ExpectationsWhere to Go from Here Bonus: Ramsey Numbers and the Probabilistic MethodFun with CardinalityIntroduction and Summary Read This! Parasitology, The PlayHow Big Is Infinite? Try This! Investigating the PlayHow High Can We Count?Where to Go from Here Bonus: The Schroeder-Bernstein TheoremAdditional Problems Solutions to Check Yourself Problems The Greek Alphabet and Some Uses for Some Letters List of Symbols Glossary BibliographyProblems and Instructor Notes appear at the end of each chapter.
