There are many discrete mathematics textbooks available, so why did I decide to invest my time and energy to work on something that perhaps only I myself would appreciate?
Mathematical writings are full of jargon and conventions that, without proper guidance, are difficult for beginners to follow. In the past, students were expected to pick them up along the way on their own. Those who failed to do so would be left behind. Looking back, I consider myself lucky. It was by God’s grace that I survived all those years. Now, when I teach a mathematical concept, I discuss its motivation, explain why it is important, and provide a lot of examples. I dissect the proofs thoroughly to make sure everyone understands them. In brief, I want to show my students how to analyze mathematical problems.
Most textbooks typically hide all these details. They only show you the final polished products. By training, mathematicians love short and elegant proofs. This is reflected in their own writing. Yes, the results are beautiful, but it is a mystery how mathematicians come up with such ideas. I want a textbook that discusses mathematical concepts in greater detail. I want to teach my students how to read and write mathematical arguments. Since I could not find a textbook that suited my needs, I started writing lecture notes to supplement the main text. Marginal notes, hands-on exercises, summaries, and section exercises were subsequently added at different stages. The lecture notes have evolved into a full-length text.
Discrete mathematics is a rich subject, full of many interesting topics. Often, it is taught to both mathematics and computer science majors. Due to the limit in space, this text addresses mainly the needs of the mathematics majors. Consequently, we will concentrate on logic and proof techniques, and apply them to sets, basic number theory, and functions. In the last two chapters, we discuss relations and combinatorics, as many students will find them useful in other courses.
Since the intended audience of the text is mathematics majors, I use a number of examples from calculus. By design, I hope this can help the students review what they have learned, and see that discrete mathematics forms the foundation of many mathematical arguments.
Discrete mathematics is often a required course in computer science. I find it hard and unjust to serve two different groups of students in the same textbook. Although this text could be used in a typical first semester discrete mathematics class for the computer science majors, they need to consult another text for the second semester course. Here are two that serve this purpose well:
Alan Doerr and Kenneth Levasseur, Applied Discrete Structures.
Miguel A. Lerma, Notes on Discrete Mathematics.
Both are available on-line.
Why do I call this a workbook? There are many hands-on exercises designed to help students understand a new concept before they move on to the next. I believe the title Workbook reflects the nature of the book, because I expect the students to work on the hands-on exercises. But why spiral? Because the pedagogy is inspired by the spiral method. The idea is to revisit some themes and results several times throughout the course and each time further deepen your understanding. You will find some problems pop up more than once, and are solved in a different way each time. In other instances, a concept you learned earlier will be viewed from a new perspective, thus adding a new dimension to it.
I am indebted to the anonymous reviewers, whose numerous valuable comments helped to shape the workbook in its current form. I would also like to express my great appreciation to Scott Richmond of Reed Library at the State University of New York at Fredonia, who provided many helpful suggestions and editorial assistance.
The reason I developed this workbook is to help students learn discrete mathematics. If this workbook proves to be a failure, I am the one to blame. If you find this workbook serves its intended purposes, I give all the glory to God, in whom I believe and trust.
April 21, 2015