Discrete Structures in Mathematics: a Problem-Solving Approach (Free PDF Textbook)

(with lots of practical applications, help, and hints to solve the hard problems)

by Prof. Gregory V. Bard

  • Chapter 0: Advice to the Student
    • Module 0.1: The Preface (How to Use This Book) (click here)
    • Module 0.2: The Seven Pitfalls of Students in Discrete Mathematics (click here)

  • Chapter 1: Set Theory
  • Chapter 2: The Past and Future of Mathematics
    • Module 2.1: Fermat's Last Theorem and Famous Unsolved Problems [Under Major Repairs]

  • Chapter 3: The Basics of Probability
    • Module 3.1: A Formal Introduction to Probability Theory (click here)
    • Module 3.2: Exploring Probability Through Problem Solving (click here)
    • Module 3.3: Expected Value and Insurance [Planned]
    • Module 3.4: You Can't Just Add Probabilities (click here)
    • Module 3.5: The Square Root of NPQ Rule (click here)

  • Chapter 4: The Basics of Combinatorics
    • Module 4.1: The Multiplication and Exponent Principles (click here)
    • Module 4.2: The Permutations and Factorial Principles (click here)
    • Module 4.3: The Combinations and Handshake Principles (click here)
    • Module 4.4: Which Combinatorial Formula Should I Use? (click here)

  • Chapter 5: Advanced Topics in Probability and Combinatorics
    • Module 5.1: Independence and Repetition (click here)
    • Module 5.2: The Binomial Distribution Formula (click here)
    • The Binomial Distribution Formula helper (powered by SageMathCell) (click here)
    • Module 5.3: Probability Tree Diagrams (click here)
    • Module 5.4: Conditional Probability Notation and Bayes' Rule [Planned]
    • Module 5.5: Probability and Dice Games [Underway]
    • Module 5.6: A Combinatorial View of Poker (5-Card Stud) [Planned]
    • Module 5.7: Pascal's Triangle and the Binomial Theorem (click here)
    • Module 5.8: Poisson's Rare Events Theorem [Underway]
    • Module 5.9: Some Advanced Combinatorial Principles [Under Revision]

  • Chapter 6: Logic
    • Module 6.1: Basic Truth Tables [Planned]
    • Module 6.2: The Logic Game: Ten Levels of Problems Toward Mathematical Logic, and Set Theory (without answers) (with answers)
    • Module 6.3: Contrapositives, Converses, and Counter-Examples (click here)
    • Module 6.4: More about Implications [Planned]
    • Module 6.5: Working with Quantifiers [Planned]

  • Chapter 7: Proof-Writing Techniques [Planned]

  • Chapter 8: Mathematical Induction and Recursive Sequences [Underway]

  • Chapter 9: The Theory of Digraphs and Graphs [Planned]

  • Chapter 10: Number Theory and Cryptography
    • Module 10.1: Exploring Steganography with the Baconian Cipher (click here)
    • Several Cayley Tables, useful for what follows (click here)
    • Modular Arithmetic Exploration Homework 1 (click here)
    • Modular Arithmetic Exploration Homework 2 (click here)
    • Modular Arithmetic Exploration Homework 3 (click here)
    • Modular Arithmetic Exploration Homework 4: The RSA Cipher (click here)

  • Appendix A: Good Old-Fashioned Mathematics
    • Module A.1: Different Number Bases [Planned]
    • Module A.2: Completing the Square [Underway]
    • Module A.3: Cardano's Cubic Formula [Underway]
    • Module A.4: Working in Binary [Planned]
    • Module A.5: The Pigeon-Hole Principle [Planned]
    • Module A.6: Injective, Surjective, and Bijective Functions (click here)
    • Module A.7: Equivalence Relations (Reflexive, Symmetric, Transitive) (click here)

  • Appendix B: A Lab about Dijkstra's Algorithm (without answers) (with answers)

Special thanks to my proofreaders:

I am very indebted to my chief proofreader, Joseph Bertino, who has now moved onto much bigger and better things, but who served me well for nine years; my senior proofreader, Russel Chamberlain; and my junior proofreaders, Trevor Kretschmann, Isaac Quella, and Tanner Verber. I must also mention my gratitude for students from MATH-270: Discrete Mathematics, many of whom have reported a typo or two. Thanks very much!

Other Discrete Mathematics Resources:

To see more great free stuff about discrete mathematics (by other authors), please visit the Discrete Math Resources and Help page. There are lots of useful resources there for both students and course instructors.

To Contact me, or to Notify me of Typos or Other Errors:

If you notice any typos, grammar errors, or mathematical issues, then please write to me (Prof. Gregory V. Bard) at the following email address. I will be very happy to hear from you. I would also happily accept any recommendations about online resources, tutorials, free textbooks, practice exercises, course syllabi, or anything that will help discrete math students solve problems. (The email address below is an image, to protect me from spam bots.) Please place "Discrete Math Hub" in the subject line. Feel free to also visit my professional webpage or my personal webpage.

An image to display my email address, which is the reverse of ude.tuotswu@gdrab (Read it backwards.)

Last Updated on March 1st, 2019.
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