Discrete Mathematics by Arthur T. Benjamin
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Discrete Mathematics written by
Arthur T. Benjamin, Ph.D. Professor of Mathematics, Harvey Mudd College.
Discrete mathematics can be described as an advanced look at the mathematics that we learned as children. In elementary school, we learned to count, did basic arithmetic, and amused ourselves with solving puzzles, ranging from connecting the dots, to coloring, to more sophisticated creative pursuits.
So what exactly is discrete mathematics? Perhaps it is easier to first say what it is not. Most of the mathematics that we are taught in high school from geometry through calculus—is continuous mathematics. Think of the second hand of a wristwatch or the path traveled by a ball as it is thrown in the air. These objects are typically described by real numbers and continuous functions. By contrast, discrete mathematics is concerned with processes that occur in separate chunks, such as how the seconds or minutes change on a digital watch, or the way the path of the ball would look if we took a few snapshots of its journey. The numbers used in discrete mathematics are whole numbers. Discrete mathematics is the foundation of computer science, where statements are true or false, numbers are represented with finite precision, and every piece of data is stored in a specific place.
In this course, we concentrate on 3 major fields of discrete mathematics: combinatorics, number theory, and graph theory. Combinatorics is the mathematics of counting. How many ways can we rearrange the letters of “Mississippi”? How many different lottery tickets can be printed? How many ways can we be dealt a full house in poker? Central to the answers to these questions is Pascal’s triangle, whose numbers contain some amazingly beautiful patterns, which we shall explore.
Discrete Mathematics written by
Arthur T. Benjamin
cover the following topics.
1. What Is Discrete Mathematics?
2. Basic Concepts of Combinatorics
3. The 12-Fold Way of Combinatorics
4. Pascal’s Triangle and the Binomial Theorem
5. Advanced Combinatorics—Multichoosing
6. The Principle of Inclusion-Exclusion
7. Proofs—Inductive, Geometric, Combinatorial
8. Linear Recurrences and Fibonacci Numbers
9. Gateway to Number Theory—Divisibility
10. The Structure of Numbers
11. Two Principles—Pigeonholes and Parity
12. Modular Arithmetic The Math of Remainders
13. Enormous Exponents and Card Shuffling
14. Fermat’s “Little” Theorem and Prime Testing
15. Open Secrets—Public Key Cryptography
16. The Birth of Graph Theory
17. Ways to Walk Matrices and Markov Chains
18. Social Networks and Stable Marriages
19. Tournaments and King Chickens
20. Weighted Graphs and Minimum Spanning Trees
21. Planarity—When Can a Graph Be Untangled?
22. Coloring Graphs and Maps
23. Shortest Paths and Algorithm Complexity
24. The Magic of Discrete Mathematics
Answers to Questions to Consider
Timeline
Glossary
Biographical Notes
Bibliography
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