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Discrete Mathematics with Ducks

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Table of Contents

THE BASICS Counting 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 Here Sets and Logic Introduction and Summary Sets Logic Try This! Problems on Sets and Logic Proof Techniques: Not! Try This! A Tricky Conundrum Where to Go from Here Bonus: Truth Tellers Graphs and Functions Introduction and Summary Function Introduction Try This! Play with Functions and Graphs Functions and Counting Graphs: Definitions and Examples Isomorphisms Graphs: Operations and Uses Try This! More Graph Problems Ramseyness Where to Go from Here Bonus: Party Tricks Bonus 2: Counting with the Characteristic Function Induction Introduction 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 Song Algorithms with Ciphers Introduction and Summary Algorithms Modular Arithmetic (and Equivalence Relations) Cryptography: Some Ciphers Try This! Encryptoequivalent Modulagorithmic Problems Where to Go from Here Bonus: Algorithms for Searching Graphs Bonus 2: Pigeons and Divisibility COMBINATORICS Binomial Coefficients and Pascal's Triangle Introduction 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: Mastermind Balls and Boxes and PIE-Counting Techniques Introduction and Summary Combinatorial Problem Types Try This! Let's Have Some PIE Combinatorial Problem Solutions and Strategies Let'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 Here Bonus: Linear and Integer Programming Recurrences Introduction 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 Stories Cutting 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 Gems GRAPH THEORY Trees Introduction and Summary Basic Facts about Trees Try This! Spanning Trees Spanning Tree Algorithms Binary Trees Try This! Binary Trees and Matchings Matchings Backtracking Where to Go from Here Bonus: The Branch-and-Bound Technique in Integer Programming Euler's Formula and Applications Introduction 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 Theory Graph Traversals Introduction 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 Theorems Graph Coloring Introduction and Summary Try This! Coloring Vertices and Edges Introduction to Coloring Try This! Let's Think about Coloring Coloring and Things (Graphs and Concepts) That Have Come Before Where to Go from Here Bonus: The Four-Color Theorem OTHER MATERIAL Probability and Expectation Introduction and Summary What Is Probability, Exactly? High Expectations You are Probably Expected to Try This! Conditional Probability and Independence Try This! . . . Probably, Under Certain Conditions Higher Expectations Where to Go from Here Bonus: Ramsey Numbers and the Probabilistic Method Fun with Cardinality Introduction and Summary Read This! Parasitology, The Play How Big Is Infinite? Try This! Investigating the Play How High Can We Count? Where to Go from Here Bonus: The Schroeder-Bernstein Theorem Additional Problems Solutions to Check Yourself Problems The Greek Alphabet and Some Uses for Some Letters List of Symbols Glossary Bibliography Problems and Instructor Notes appear at the end of each chapter.


This book can certainly be used to teach the standard course in discrete mathematics designed for computer science majors. The lighter touch of duck references does amuse you and does not ever overshadow or disguise the mathematical concepts. -Charles Ashbacher, MAA Reviews, August 2012 When I used Discrete Mathematics with Ducks in class, I assigned readings, and my students came to class full of questions and ideas. Every day we had a good time in one way or another; these classes were a highlight of my teaching career. I give a lot of credit to this book, thanks to the author's skill at blending rigor, great examples, casual humor, and precise writing. -David Perkins, author of Calculus and Its Origins I had a lot of fun teaching from Discrete Mathematics with Ducks! ... I think the discovery/exploratory/problem solving approach is ABSOLUTELY the way this course should be taught, and of all the discrete books I have looked at, this text does the best job of supporting that kind of approach to the subject while still giving enough of the material in writing to fill in the gaps. ... I found that the material provided and the instructor notes cut down on my prep time, and I definitely referred to them. Having the in-class activities included is a huge benefit to this book. -Dana Rowland, Associate Professor of Mathematics, Merrimack College ... an incredible book ... readable by students, useful for instructors, and constructed with style and flair. This book will make it much easier to teach an exciting, student-centered discrete mathematics course that will also serve as an excellent introduction to advanced critical thinking, problem solving, and proofs. And there are ducks! -Douglas Shaw, Professor of Mathematics, University of Northern Iowa

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