@CRC Press. Discrete Mathematics. with Ducks. sarah-marie belcastro. let this be your watchword. serious mathematics treated with levity
|
|
- Emil Peters
- 5 years ago
- Views:
Transcription
1 Discrete Mathematics with Ducks sarah-marie belcastro serious mathematics treated with levity let this be your Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an Informa business AN A K PETERS BOOK
2 * Contents Preface for Instructors and Other Teachers xvii 1 About This Book xvii 2 How to Use This Book xx 21 A Start on Discovery-Based Learning xxi 22 Details of Conducting Group Work xxiii 3 Chapter and Bonus-Section Dependencies xxvi Preface for Students and Other Learners 1 About This Book (and about Learning Mathematics) xxix xxix 2 How to Use This Book xxx 21 How to Use This Book in a Class xxxii 22 How to Use This Book for Self-Study xxxii 3 Tips for Reading Mathematics xxxiii 4 Problem-Solving Prompts xxxiv 5 Tips for Writing Mathematics xxxv Acknowledgments xxxix I Theme: The Basics 1 1 Counting and Proofs 3 11 Introduction and Summary 3 12 Try This! Let's Count 3 13 The Sum and Product Principles 6 14 Prehminaries on Proofs and Disproofs 9 15 Pigeons and Correspondences Where to Go from Here Problems That Use Counting or Proofs Instructor Notes 22 vii
3 viii Contents 2 Sets and Logic Introduction and Summary Sets Making New Sets from Scratch Finding Sets inside Other Sets Proof Technique: Double-Inclusion Making New Sets from Old Looking at Sets Logic Combining Statements Restriction of Variables via Quantifiers Negation Interactions Try This! Problems on Sets and Logic Proof Techniques: Not! Try This! A Tricky Conundrum Where to Go from Here Bonus: Truth Tellers Problems about Sets and Logic Instructor Notes 55 3 Graphs and Functions Introduction and Summary Function Introdunction Try This! Play with Functions and Graphs Play with Functions Play with Graphs A Dot Game Functions and Counting Graphs: Definitions and Examples Isomorphisms Graphs: Operations and Uses Sets and Graphs Have Some Things in Common How Are Graphs Useful? Try This! More Graph Problems Ramseyness Where to Go from Here 81
4 Contents ix 311 Bonus: Party Tricks Bonus 2: Counting with the Characteristic Function Problems about Graphs and Functions Instructor Notes 88 4 induction Introduction and Summary Induction Summation Notation Induction Types and Styles 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 Problems That Use Induction Instructor Notes Ill 4121 Potential Practice Proof Problems Algorithms with Ciphers Introduction and Summary Algorithms Conditionals and Loops Efficiency Mgorithms and Existence Proofs Modular Arithmetic (and Equivalence Relations) Cryptography: Some Ciphers Shift Ciphers Atbash Ciphers The Vigenere Cipher Decryption and the Real World Try This! Encryptoequivalent Modulalgorithmic Problems Where to Go from Here 138
5 X Contents 57 Bonus: Algorithms for Searching Graphs Bonus 2: Pigeons and Divisibility Problems about Algorithms, Modular Arithmetic, and Ciphers Instructor Notes 148 II Theme: 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 One Strategy for Playing Mini-Project Problems Binomially Combinatorial in Nature Instructor Notes 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 Strategy: Slots Strategy: Stars and Bars Solutions to Problem Types Denouement: Bijective Counting, Again Let's Explain Our PIE! 204
6 Contents xi 76 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 Problems about Balls, Boxes, and PIEs Instructor Notes 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 Recurring Problems Instructor Notes 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 Let's Go for It! Hyperbeet Numbers Where to Go from Here Bonus: Geometric Gems Problems That Combine Combinatorial Topics Instructor Notes 272
7 xii Contents II! Theme: Graph Theory Trees Introduction and Summary Basic Facts about Trees Try This! Spanning Trees Spanning Tree Algorithms Greedy 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 Tree Problems Instructor Notes 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 117 Try This! Applications of Euler's Formula 320 of Euler's Formula Where to Go from Here Bonus: Topological Graph Theory Problems about Planar Graphs Instructor Notes Graph Traversals Introduction and Summary Try This! Euler Traversals Euler Paths and Circuits 335
8 Contents xiii 124 Dijkstra's Algorithm, with sides of Hamilton Circuits and the Traveling Salesperson Problem 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 Problems with Traversing Instructor Notes Graph Coloring Introduction and Summary Try This! Coloring Vertices and Edges Vertex Coloring ; Edge Coloring More on Vertex Coloring More on Edge Coloring Introduction to Coloring Coloring Bounds Applications of Vertex Coloring Try This! Let's Think about Coloring Coloring and Things (Graphs and Concepts) That Have Come Before Let's Color the Edges of Complete Graphs Let's Color Bipartite Graphs Add a Condition, Get a Different Bound Greedy Matchings Where to Go from Here Bonus: The Four-Color Theorem Colorful Problems Instructor Notes 390 IV Other Material Probability and Expectation Introduction and Summary What Is Probability, Exactly? 396
9 xiv Contents High Expectations You Are Probably Expected to Try This! Conditional Probability and Independence The Helpfulness of PIE in the Real World of Probability Independence versus Exclusivity Try This! Probably, Under Certain Conditions 147 Higher Expectations That's Wild! (A Hint at the Probabilistic Method) 148 Where to Go from Here Bonus: Ramsey Numbers and the Probabilistic Method Expect Problems, Probably Instructor Notes Fun with Cardinality Introduction and Summary Read This! Parasitology, the Play Scene 1: The Storage Coordinator Scene 2: The Taxonomist Scene 3: The Cafe Scene 4: Cataloguing How Big Is Infinite? Try This: Investigating the Play Questions about Sample Storage More Questions about Sample Storage Questions about Cafe Conversations Indiscrete Questions How High Can We Count? The Continuum Hypothesis Where to Go from Here Bonus: The Schroder-Bernstein Theorem Infinitely Large Problems Instructor Notes 457 A Additional Problems 459 B Solutions to Check Yourself Problems 487
10 Contents w C The Greek Alphabet and Some Uses for Some Letters 517 D List of Symbols 519 Glossary 523 Bibliography 537
Preface for Instructors and Other Teachers 1 About This Book... xvii
Preface for Instructors and Other Teachers xvii 1 About This Book.... xvii 2 How tousethis Book...................... xx 2.1 A Start on Discovery-Based Learning..... xxi 2.2 Details of Conducting Group
More informationRevised Curriculum for Bachelor of Computer Science & Engineering, 2011
Revised Curriculum for Bachelor of Computer Science & Engineering, 2011 FIRST YEAR FIRST SEMESTER al I Hum/ T / 111A Humanities 4 100 3 II Ph /CSE/T/ 112A Physics - I III Math /CSE/ T/ Mathematics - I
More informationCONTENTS GRAPH THEORY
CONTENTS i GRAPH THEORY GRAPH THEORY By Udit Agarwal M.Sc. (Maths), M.C.A. Sr. Lecturer, Rakshpal Bahadur Management Institute, Bareilly Umeshpal Singh (MCA) Director, Rotary Institute of Management and
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION MH1301 DISCRETE MATHEMATICS. Time Allowed: 2 hours
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER II EXAMINATION 206-207 DISCRETE MATHEMATICS May 207 Time Allowed: 2 hours INSTRUCTIONS TO CANDIDATES. This examination paper contains FOUR (4) questions and comprises
More informationCounting integral solutions
Thought exercise 2.2 20 Counting integral solutions Question: How many non-negative integer solutions are there of x 1 +x 2 +x 3 +x 4 = 10? Thought exercise 2.2 20 Counting integral solutions Question:
More informationHonors Precalculus Chapter 9 Summary Basic Combinatorics
Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each
More informationMath 3012 Applied Combinatorics Lecture 2
August 20, 2015 Math 3012 Applied Combinatorics Lecture 2 William T. Trotter trotter@math.gatech.edu The Road Ahead Alert The next two to three lectures will be an integrated approach to material from
More informationFacilitator Guide. Unit 2
Facilitator Guide Unit 2 UNIT 02 Facilitator Guide ACTIVITIES NOTE: At many points in the activities for Mathematics Illuminated, workshop participants will be asked to explain, either verbally or in
More informationMaths Revision Booklet. Year 6
Maths Revision Booklet Year 6 Name: Class: 1 Million 1 000 000 six zeros Maths Revision Place Value 750 000 ¾ million 500 000 ½ million 250 000 ¼ million 1.0 = 1 = 0.75 = ¾ = 0.50 = ½ = 0.25 = ¼ = 100
More informationCounting. Chapter 6. With Question/Answer Animations
. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Counting Chapter
More informationWhyTry Elementary Game Plan Journal
WhyTry Elementary Game Plan Journal I can promise you that if you will do the things in this journal, develop a Game Plan for your life, and stick to it, you will get opportunity, freedom, and self respect;
More informationStaircase Rook Polynomials and Cayley s Game of Mousetrap
Staircase Rook Polynomials and Cayley s Game of Mousetrap Michael Z. Spivey Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington 98416-1043 USA mspivey@ups.edu Phone:
More informationLecture 18 - Counting
Lecture 18 - Counting 6.0 - April, 003 One of the most common mathematical problems in computer science is counting the number of elements in a set. This is often the core difficulty in determining a program
More informationReplacement, Maintenance, and Reliability. Theory and Applications SECOND EDITION. (cfc. Andrew K.S. Jardine Albert H.C. Tsang.
Maintenance, Replacement, and Reliability Theory and Applications SECOND EDITION Andrew K.S. Jardine Albert H.C. Tsang (cfc CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint
More informationDIGITAL SIGNAL PROCESSING LABORATORY
DIGITAL SIGNAL PROCESSING LABORATORY SECOND EDITION В. Preetham Kumar CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
More informationTHE TAYLOR EXPANSIONS OF tan x AND sec x
THE TAYLOR EXPANSIONS OF tan x AND sec x TAM PHAM AND RYAN CROMPTON Abstract. The report clarifies the relationships among the completely ordered leveled binary trees, the coefficients of the Taylor expansion
More informationCS1802 Week 3: Counting Next Week : QUIZ 1 (30 min)
CS1802 Discrete Structures Recitation Fall 2018 September 25-26, 2018 CS1802 Week 3: Counting Next Week : QUIZ 1 (30 min) Permutations and Combinations i. Evaluate the following expressions. 1. P(10, 4)
More informationTopics to be covered
Basic Counting 1 Topics to be covered Sum rule, product rule, generalized product rule Permutations, combinations Binomial coefficients, combinatorial proof Inclusion-exclusion principle Pigeon Hole Principle
More informationSequences. like 1, 2, 3, 4 while you are doing a dance or movement? Have you ever group things into
Math of the universe Paper 1 Sequences Kelly Tong 2017/07/17 Sequences Introduction Have you ever stamped your foot while listening to music? Have you ever counted like 1, 2, 3, 4 while you are doing a
More informationCounting in Algorithms
Counting Counting in Algorithms How many comparisons are needed to sort n numbers? How many steps to compute the GCD of two numbers? How many steps to factor an integer? Counting in Games How many different
More informationSolutions to Problem Set 7
Massachusetts Institute of Technology 6.4J/8.6J, Fall 5: Mathematics for Computer Science November 9 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised November 3, 5, 3 minutes Solutions to Problem
More informationOn uniquely k-determined permutations
On uniquely k-determined permutations Sergey Avgustinovich and Sergey Kitaev 16th March 2007 Abstract Motivated by a new point of view to study occurrences of consecutive patterns in permutations, we introduce
More informationPrinciple of Inclusion-Exclusion Notes
Principle of Inclusion-Exclusion Notes The Principle of Inclusion-Exclusion (often abbreviated PIE is the following general formula used for finding the cardinality of a union of finite sets. Theorem 0.1.
More informationReading 14 : Counting
CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 14 : Counting In this reading we discuss counting. Often, we are interested in the cardinality
More informationRethinking the Licensing of New Attorneys - An Exploration of Alternatives to the Bar Exam: Introduction
Georgia State University Law Review Volume 20 Issue 4 Summer 2004 Article 3 9-1-2003 Rethinking the Licensing of New Attorneys - An Exploration of Alternatives to the Bar Exam: Introduction Clark D. Cunningham
More informationIdeas beyond Number. Activity worksheets
Ideas beyond Number Activity sheet 1 Task 1 Some students started to solve this equation in different ways: For each statement tick True or False: = = = = Task 2: Counter-examples The exception disproves
More informationPOLYA'S FOUR STEP PROBLEM SOLVING PROCESS Understand. Devise a Plan. Carry out Plan. Look Back. PROBLEM SOLVING STRATEGIES (exmples) Making a Drawlnq
1.1 KEY IDEAS POLYA'S FOUR STEP PROBLEM SOLVING PROCESS Understand Devise a Plan Carry out Plan Look Back PROBLEM SOLVING STRATEGIES (exmples) Making a Drawlnq Guesslnc and Checking Making a Table UsinQ
More informationEuropean Journal of Combinatorics. Staircase rook polynomials and Cayley s game of Mousetrap
European Journal of Combinatorics 30 (2009) 532 539 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Staircase rook polynomials
More informationCSE 312: Foundations of Computing II Quiz Section #2: Combinations, Counting Tricks (solutions)
CSE 312: Foundations of Computing II Quiz Section #2: Combinations, Counting Tricks (solutions Review: Main Theorems and Concepts Combinations (number of ways to choose k objects out of n distinct objects,
More informationPARTICIPANT Guide. Unit 2
PARTICIPANT Guide Unit 2 UNIT 02 participant Guide ACTIVITIES NOTE: At many points in the activities for Mathematics Illuminated, workshop participants will be asked to explain, either verbally or in
More informationFormalising Event Reconstruction in Digital Investigations
Formalising Event Reconstruction in Digital Investigations Pavel Gladyshev The thesis is submitted to University College Dublin for the degree of PhD in the Faculty of Science August 2004 Department of
More informationChapter 6.1. Cycles in Permutations
Chapter 6.1. Cycles in Permutations Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 6.1. Cycles in Permutations Math 184A / Fall 2017 1 / 27 Notations for permutations Consider a permutation in 1-line
More informationMA/CSSE 473 Day 13. Student Questions. Permutation Generation. HW 6 due Monday, HW 7 next Thursday, Tuesday s exam. Permutation generation
MA/CSSE 473 Day 13 Permutation Generation MA/CSSE 473 Day 13 HW 6 due Monday, HW 7 next Thursday, Student Questions Tuesday s exam Permutation generation 1 Exam 1 If you want additional practice problems
More informationMATH 1112 FINAL EXAM REVIEW e. None of these. d. 1 e. None of these. d. 1 e. None of these. e. None of these. e. None of these.
I. State the equation of the unit circle. MATH 111 FINAL EXAM REVIEW x y y = 1 x+ y = 1 x = 1 x + y = 1 II. III. If 1 tan x =, find sin x for x in Quadrant IV. 1 1 1 Give the exact value of each expression.
More informationContents. List of Figures List of Tables. Structure of the Book How to Use this Book Online Resources Acknowledgements
Contents List of Figures List of Tables Preface Notation Structure of the Book How to Use this Book Online Resources Acknowledgements Notational Conventions Notational Conventions for Probabilities xiii
More informationCS1800 Discrete Structures Fall 2016 Profs. Aslam, Gold, Ossowski, Pavlu, & Sprague 7 November, CS1800 Discrete Structures Midterm Version C
CS1800 Discrete Structures Fall 2016 Profs. Aslam, Gold, Ossowski, Pavlu, & Sprague 7 November, 2016 CS1800 Discrete Structures Midterm Version C Instructions: 1. The exam is closed book and closed notes.
More informationUnit Nine Precalculus Practice Test Probability & Statistics. Name: Period: Date: NON-CALCULATOR SECTION
Name: Period: Date: NON-CALCULATOR SECTION Vocabulary: Define each word and give an example. 1. discrete mathematics 2. dependent outcomes 3. series Short Answer: 4. Describe when to use a combination.
More informationConvolution Pyramids. Zeev Farbman, Raanan Fattal and Dani Lischinski SIGGRAPH Asia Conference (2011) Julian Steil. Prof. Dr.
Zeev Farbman, Raanan Fattal and Dani Lischinski SIGGRAPH Asia Conference (2011) presented by: Julian Steil supervisor: Prof. Dr. Joachim Weickert Fig. 1.1: Gradient integration example Seminar - Milestones
More informationUCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Lecture 3 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ Lecture 3 Notes Goal for today: CL Section 3 Subsets,
More informationPROOFS OF SOME BINOMIAL IDENTITIES USING THE METHOD OF LAST SQUARES
PROOFS OF SOME BINOMIAL IDENTITIES USING THE METHOD OF LAST SQUARES MARK SHATTUCK AND TAMÁS WALDHAUSER Abstract. We give combinatorial proofs for some identities involving binomial sums that have no closed
More information1 = 3 2 = 3 ( ) = = = 33( ) 98 = = =
Math 115 Discrete Math Final Exam December 13, 2000 Your name It is important that you show your work. 1. Use the Euclidean algorithm to solve the decanting problem for decanters of sizes 199 and 98. In
More informationMAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
More informationOlympiad Combinatorics. Pranav A. Sriram
Olympiad Combinatorics Pranav A. Sriram August 2014 Chapter 2: Algorithms - Part II 1 Copyright notices All USAMO and USA Team Selection Test problems in this chapter are copyrighted by the Mathematical
More informationWith Question/Answer Animations. Chapter 6
With Question/Answer Animations Chapter 6 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and
More informationDISCUSSION #8 FRIDAY MAY 25 TH Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics
DISCUSSION #8 FRIDAY MAY 25 TH 2007 Sophie Engle (Teacher Assistant) ECS20: Discrete Mathematics 2 Homework 8 Hints and Examples 3 Section 5.4 Binomial Coefficients Binomial Theorem 4 Example: j j n n
More informationSection 8.1. Sequences and Series
Section 8.1 Sequences and Series Sequences Definition A sequence is a list of numbers. Definition A sequence is a list of numbers. A sequence could be finite, such as: 1, 2, 3, 4 Definition A sequence
More informationPattern Avoidance in Unimodal and V-unimodal Permutations
Pattern Avoidance in Unimodal and V-unimodal Permutations Dido Salazar-Torres May 16, 2009 Abstract A characterization of unimodal, [321]-avoiding permutations and an enumeration shall be given.there is
More informationWeek 1. 1 What Is Combinatorics?
1 What Is Combinatorics? Week 1 The question that what is combinatorics is similar to the question that what is mathematics. If we say that mathematics is about the study of numbers and figures, then combinatorics
More informationProblem Set 8 Solutions R Y G R R G
6.04/18.06J Mathematics for Computer Science April 5, 005 Srini Devadas and Eric Lehman Problem Set 8 Solutions Due: Monday, April 11 at 9 PM in Room 3-044 Problem 1. An electronic toy displays a 4 4 grid
More informationJong C. Park Computer Science Division, KAIST
Jong C. Park Computer Science Division, KAIST Today s Topics Basic Principles Permutations and Combinations Algorithms for Generating Permutations Generalized Permutations and Combinations Binomial Coefficients
More informationThe Apprentices Tower of Hanoi
Journal of Mathematical Sciences (2016) 1-6 ISSN 272-5214 Betty Jones & Sisters Publishing http://www.bettyjonespub.com Cory B. H. Ball 1, Robert A. Beeler 2 1. Department of Mathematics, Florida Atlantic
More informationCONTENTS PREFACE. Part One THE DESIGN PROCESS: PROPERTIES, PARADIGMS AND THE EVOLUTIONARY STRUCTURE
Copyrighted Material Dan Braha and Oded Maimon, A Mathematical Theory of Design: Foundations, Algorithms, and Applications, Springer, 1998, 708 p., Hardcover, ISBN: 0-7923-5079-0. PREFACE Part One THE
More informationArithmetic Sequences Read 8.2 Examples 1-4
CC Algebra II HW #8 Name Period Row Date Arithmetic Sequences Read 8.2 Examples -4 Section 8.2 In Exercises 3 0, tell whether the sequence is arithmetic. Explain your reasoning. (See Example.) 4. 2, 6,
More informationTABLEAU DES MODIFICATIONS
TABLEAU DES MODIFICATIONS APPORTÉES AUX STATUTS REFONDUS, 1964 ET AUX LOIS PUBLIQUES POSTÉRIEURES DANS CE TABLEAU Ab. = Abrogé Ann. = Annexe c. = Chapitre cc. = Chapitres Form. = Formule R. = Statuts refondus,
More informationCardinality revisited
Cardinality revisited A set is finite (has finite cardinality) if its cardinality is some (finite) integer n. Two sets A,B have the same cardinality iff there is a one-to-one correspondence from A to B
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationGenerating indecomposable permutations
Discrete Mathematics 306 (2006) 508 518 www.elsevier.com/locate/disc Generating indecomposable permutations Andrew King Department of Computer Science, McGill University, Montreal, Que., Canada Received
More informationMathematics Competition Practice Session 6. Hagerstown Community College: STEM Club November 20, :00 pm - 1:00 pm STC-170
2015-2016 Mathematics Competition Practice Session 6 Hagerstown Community College: STEM Club November 20, 2015 12:00 pm - 1:00 pm STC-170 1 Warm-Up (2006 AMC 10B No. 17): Bob and Alice each have a bag
More informationGraph Application in The Strategy of Solving 2048 Tile Game
Graph Application in The Strategy of Solving 2048 Tile Game Harry Setiawan Hamjaya and 13516079 Program Studi Teknik Informatika Sekolah Teknik Elektro dan Informatika Institut Teknologi Bandung, Jl. Ganesha
More informationMidterm practice super-problems
Midterm practice super-problems These problems are definitely harder than the midterm (even the ones without ), so if you solve them you should have no problem at all with the exam. However be aware that
More informationCSE 1400 Applied Discrete Mathematics Permutations
CSE 1400 Applied Discrete Mathematics Department of Computer Sciences College of Engineering Florida Tech Fall 2011 1 Cyclic Notation 2 Re-Order a Sequence 2 Stirling Numbers of the First Kind 2 Problems
More informationPermutations of a Multiset Avoiding Permutations of Length 3
Europ. J. Combinatorics (2001 22, 1021 1031 doi:10.1006/eujc.2001.0538 Available online at http://www.idealibrary.com on Permutations of a Multiset Avoiding Permutations of Length 3 M. H. ALBERT, R. E.
More informationRESTRICTED PERMUTATIONS AND POLYGONS. Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, Haifa, Israel
RESTRICTED PERMUTATIONS AND POLYGONS Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, 905 Haifa, Israel {gferro,toufik}@mathhaifaacil abstract Several authors have examined
More informationCombinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3 Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers Hacène Belbachir and Amine
More informationSPICE for Power Electronics and Electric Power
SPICE for Power Electronics and Electric Power Third Edition Muhammad H. Rashid Life Fellow IEEE /^0\ \Cf*' CRC Press I Taylor & Francis eis Crou Group Boca Raton London New York CRC Press is an imprint
More informationWhat Is Leaps and Bounds? A Research Foundation How to Use Leaps and Bounds Frequently Asked Questions Components
Contents Program Overview What Is Leaps and Bounds? A Research Foundation How to Use Leaps and Bounds Frequently Asked Questions Components ix x xiv xvii xix Teaching Notes Strand: Number Number Strand
More informationGames on graphs. Keywords: positional game, Maker-Breaker, Avoider-Enforcer, probabilistic
Games on graphs Miloš Stojaković Department of Mathematics and Informatics, University of Novi Sad, Serbia milos.stojakovic@dmi.uns.ac.rs http://www.inf.ethz.ch/personal/smilos/ Abstract. Positional Games
More informationDiscrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions
CS 70 Discrete Mathematics and Probability Theory Spring 2018 Ayazifar and Rao Midterm 2 Solutions PRINT Your Name: Oski Bear SIGN Your Name: OS K I PRINT Your Student ID: CIRCLE your exam room: Pimentel
More informationName:... Date:... Use your mathematical skills to solve the following problems. Remember to show all of your workings and check your answers.
Name:... Date:... Use your mathematical skills to solve the following problems. Remember to show all of your workings and check your answers. There has been a zombie virus outbreak in your school! The
More informationGreedy Flipping of Pancakes and Burnt Pancakes
Greedy Flipping of Pancakes and Burnt Pancakes Joe Sawada a, Aaron Williams b a School of Computer Science, University of Guelph, Canada. Research supported by NSERC. b Department of Mathematics and Statistics,
More informationThe Art of Counting. Bijections, Double Counting. Peng Shi. September 16, Department of Mathematics Duke University
The Art of Counting Bijections, Double Counting Peng Shi Department of Mathematics Duke University September 16, 2009 What we focus on in this talk? Enumerative combinatorics is a huge branch of mathematics,
More informationLecture 1. Permutations and combinations, Pascal s triangle, learning to count
18.440: Lecture 1 Permutations and combinations, Pascal s triangle, learning to count Scott Sheffield MIT 1 Outline Remark, just for fun Permutations Counting tricks Binomial coefficients Problems 2 Outline
More informationCMath 55 PROFESSOR KENNETH A. RIBET. Final Examination May 11, :30AM 2:30PM, 100 Lewis Hall
CMath 55 PROFESSOR KENNETH A. RIBET Final Examination May 11, 015 11:30AM :30PM, 100 Lewis Hall Please put away all books, calculators, cell phones and other devices. You may consult a single two-sided
More informationUse the given information to write the first 5 terms of the sequence and the 20 th term. 6. a1= 4, d= 8 7. a1= 10, d= -6 8.
Arithmetic Sequences Class Work Find the common difference in sequence, and then write the next 3 terms in the sequence. 1. 3, 7,11, 15, 2. 1, 8, 15, 22, 3. 5, 2, -1, -4, 4. 68, 56, 44, 32, 5. 1.3, 2.6,
More informationA slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal
The Slope of a Line (2.2) Find the slope of a line given two points on the line (Objective #1) A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal
More informationDVA325 Formal Languages, Automata and Models of Computation (FABER)
DVA325 Formal Languages, Automata and Models of Computation (FABER) Lecture 1 - Introduction School of Innovation, Design and Engineering Mälardalen University 11 November 2014 Abu Naser Masud FABER November
More informationHOMEWORK ASSIGNMENT 5
HOMEWORK ASSIGNMENT 5 MATH 251, WILLIAMS COLLEGE, FALL 2006 Abstract. These are the instructor s solutions. 1. Big Brother The social security number of a person is a sequence of nine digits that are not
More informationSimple Counting Problems
Appendix F Counting Principles F1 Appendix F Counting Principles What You Should Learn 1 Count the number of ways an event can occur. 2 Determine the number of ways two or three events can occur using
More informationAPPLICATION FOR APPROVAL OF A IENG EMPLOYER-MANAGED FURTHER LEARNING PROGRAMME
APPLICATION FOR APPROVAL OF A IENG EMPLOYER-MANAGED FURTHER LEARNING PROGRAMME When completing this application form, please refer to the relevant JBM guidance notably those setting out the requirements
More informationError-Correcting Codes for Rank Modulation
ISIT 008, Toronto, Canada, July 6-11, 008 Error-Correcting Codes for Rank Modulation Anxiao (Andrew) Jiang Computer Science Department Texas A&M University College Station, TX 77843, U.S.A. ajiang@cs.tamu.edu
More informationON THE PERMUTATIONAL POWER OF TOKEN PASSING NETWORKS.
ON THE PERMUTATIONAL POWER OF TOKEN PASSING NETWORKS. M. H. ALBERT, N. RUŠKUC, AND S. LINTON Abstract. A token passing network is a directed graph with one or more specified input vertices and one or more
More informationOnline Computation and Competitive Analysis
Online Computation and Competitive Analysis Allan Borodin University of Toronto Ran El-Yaniv Technion - Israel Institute of Technology I CAMBRIDGE UNIVERSITY PRESS Contents Preface page xiii 1 Introduction
More informationRestricted Permutations Related to Fibonacci Numbers and k-generalized Fibonacci Numbers
Restricted Permutations Related to Fibonacci Numbers and k-generalized Fibonacci Numbers arxiv:math/0109219v1 [math.co] 27 Sep 2001 Eric S. Egge Department of Mathematics Gettysburg College 300 North Washington
More informationMA 524 Midterm Solutions October 16, 2018
MA 524 Midterm Solutions October 16, 2018 1. (a) Let a n be the number of ordered tuples (a, b, c, d) of integers satisfying 0 a < b c < d n. Find a closed formula for a n, as well as its ordinary generating
More informationECS 20 (Spring 2013) Phillip Rogaway Lecture 1
ECS 20 (Spring 2013) Phillip Rogaway Lecture 1 Today: Introductory comments Some example problems Announcements course information sheet online (from my personal homepage: Rogaway ) first HW due Wednesday
More information17. Symmetries. Thus, the example above corresponds to the matrix: We shall now look at how permutations relate to trees.
7 Symmetries 7 Permutations A permutation of a set is a reordering of its elements Another way to look at it is as a function Φ that takes as its argument a set of natural numbers of the form {, 2,, n}
More informationUNC Charlotte 2012 Comprehensive
March 5, 2012 1. In the English alphabet of capital letters, there are 15 stick letters which contain no curved lines, and 11 round letters which contain at least some curved segment. How many different
More informationSequential games. We may play the dating game as a sequential game. In this case, one player, say Connie, makes a choice before the other.
Sequential games Sequential games A sequential game is a game where one player chooses his action before the others choose their. We say that a game has perfect information if all players know all moves
More informationMAT3707. Tutorial letter 202/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/202/1/2017
MAT3707/0//07 Tutorial letter 0//07 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Semester Department of Mathematical Sciences SOLUTIONS TO ASSIGNMENT 0 BARCODE Define tomorrow university of south africa
More informationFall. Spring. Possible Summer Topics
Fall Paper folding: equilateral triangle (parallel postulate and proofs of theorems that result, similar triangles), Trisect a square paper Divisibility by 2-11 and by combinations of relatively prime
More informationWhat is counting? (how many ways of doing things) how many possible ways to choose 4 people from 10?
Chapter 5. Counting 5.1 The Basic of Counting What is counting? (how many ways of doing things) combinations: how many possible ways to choose 4 people from 10? how many license plates that start with
More informationDesign. EMI Filter. Timothy THIRD EDITION. Richard Lee Ozenbaugh. M. Pullen. CRC Press. Taylor & Francis Croup. Taylor & Francis Croup,
EMI Filter Design THIRD EDITION Richard Lee Ozenbaugh Timothy M. Pullen CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
More informationThe Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n
Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product
More informationFrom permutations to graphs
From permutations to graphs well-quasi-ordering and infinite antichains Robert Brignall Joint work with Atminas, Korpelainen, Lozin and Vatter 28th November 2014 Orderings on Structures Pick your favourite
More informationGraphs of Tilings. Patrick Callahan, University of California Office of the President, Oakland, CA
Graphs of Tilings Patrick Callahan, University of California Office of the President, Oakland, CA Phyllis Chinn, Department of Mathematics Humboldt State University, Arcata, CA Silvia Heubach, Department
More information11.3B Warmup. 1. Expand: 2x. 2. Express the expansion of 2x. using combinations. 3. Simplify: a 2b a 2b
11.3 Warmup 1. Expand: 2x y 4 2. Express the expansion of 2x y 4 using combinations. 3 3 3. Simplify: a 2b a 2b 4. How many terms are there in the expansion of 2x y 15? 5. What would the 10 th term in
More information5. (1-25 M) How many ways can 4 women and 4 men be seated around a circular table so that no two women are seated next to each other.
A.Miller M475 Fall 2010 Homewor problems are due in class one wee from the day assigned (which is in parentheses. Please do not hand in the problems early. 1. (1-20 W A boo shelf holds 5 different English
More informationThree of these grids share a property that the other three do not. Can you find such a property? + mod
PPMTC 22 Session 6: Mad Vet Puzzles Session 6: Mad Veterinarian Puzzles There is a collection of problems that have come to be known as "Mad Veterinarian Puzzles", for reasons which will soon become obvious.
More informationTeacher s Notes. Problem of the Month: Courtney s Collection
Teacher s Notes Problem of the Month: Courtney s Collection Overview: In the Problem of the Month, Courtney s Collection, students use number theory, number operations, organized lists and counting methods
More informationMath 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
More information