Discrete Mathematics: An Open Introduction
Oscar Levin, University of Northern Colorado
Pub Date: 2016
ISBN 13: 978-1-5349707-4-8
Conditions of Use
This textbook, “Discrete Mathematics: An Open Introduction”, by Oscar Levin, provides a good overview of topics in Discrete Mathematics. The primary read more
This textbook, “Discrete Mathematics: An Open Introduction”, by Oscar Levin, provides a good overview of topics in Discrete Mathematics. The primary focus of this text is not to provide a rigorous mathematical foundation for Computer Science students; instead, it is targeted towards first and second year undergraduate math majors who will go on to teach middle school and high school mathematics. The text starts with a brief but useful introduction to mathematical concepts (mathematical statements, sets and functions), and then goes on to cover a range of topics in depth, broken up into four main sections: Combinatorics, Sequences, Symbolic Logic and Proofs, and Graph Theory, as well an Additional Topics section that touches on Generating Functions and provides an introduction to Number Theory. The material touches on a wide array of concepts such as the Pigeonhole principle, The text has several features that I found quite innovative and helpful. The presentation takes an inquiry-based approach, and most topics start with an “Investigate!” section that poses a number of questions or problems to help motivate students to understand the context for the topic they’re about to start – for example, the Combinatorial Proofs topic is preceded by an Investigate! Section that uses the Stanley cup tournament rules to make students think about how many ways a team can win, and how to generalize the problem space. The text also approaches mathematical proofs in a friendly, non-intimidating manner and provides different approaches to proving a given identity or theorem, helping students to broaden their mathematical toolkit. The text has a comprehensive index, and has both a PDF version and a well-designed interactive online format, with a contents tab and expandable solutions (allowing students to attempt a question before unveiling the solution).
The material in the book was well-edited and proof-read. I didn’t encounter obvious mistakes or omissions in my first reading of the text, and only a few typos (e.g. “bijectitve”).
The content of this text is relevant to current undergraduate courses in Discrete Mathematics, particularly for those students intending to pursue careers in middle and high school education. The topics are of fundamental, enduring importance, and not subject to obsolescence.
The author writes clearly and successfully manages to make the subject material approachable, interesting and comprehensible, while not shirking from exploring the more complex aspects of each topic. Mathematical proofs are exceptionally well explained, focusing on helping students understand why an identity is true rather than merely the mechanical aspects of stepping through a number of steps in a proof that may lead to losing sight of the forest for the trees. There are a few places where additional editing might improve clarity, but overall, the quality of the writing is commendable.
The text is well organized and structured, the terminology used is consistent and pedagogically sound, and the overall presentation is designed so that students find that each topic is presented in a logical, evolutionary manner.
Within the constraints of the subject matter, where topics frequently require understanding of preceding concepts, the text is organized in a reasonably modular fashion. The online interactive format is particularly engaging and likely, in my opinion, to be found useful by students.
The text is well organized and structured, allowing the material to flow and be built up in an accessible manner. The use of the introductory Investigate! sections through-out the text is an excellent tool to motivate students to think about topics before getting into the details.
The book’s design and interface is well-thought out, particularly the interactive online version, which is cleanly designed, non-distracting, functional and approachable, with simple and straight-forward navigational controls.
I found the writing to be high-quality, well-proofed, and free of grammatical issues.
Given the nature of the text’s material, cultural relevance is not a major concern. However, the examples used in the text appeared to be appropriate, without any cultural or gender stereo-typing.
I found this text to be well written and structured, and will be considering using it as the text for a Discrete Mathematics course that I teach.
There are many topics in discrete mathematics. This book does a fine job of covering numerous topics in this area, including among several other read more
There are many topics in discrete mathematics. This book does a fine job of covering numerous topics in this area, including among several other topics, symbolic logic, counting, sets, and a short section on number theory. There is very good index that links to pages in the text. I did not find a glossary, but because the index links to the text, that is not really necessary. There is clearly enough material here for a very meaty undergraduate course.
I found no errors in the text and found no bias of any kind in the text.
This subject is essentially timeless because the principles are mathematical and will always be true and valid. There is one problem involving Continental Airlines that no longer exists, but that is a minor quibble. This does not make the text obsolete.
This is the book's strongest suit. It is written in an upbeat enthusiastic style that comes through. The reader can tell that the author is an energetic teacher who genuinely enjoys the subject. The prose is clear and inviting to the reader. The "Investigate!" sections at the beginning of each lesson are designed to and do pique the student's curiosity.
There are no problems here at all. The book uses terms and concepts consistently throughout the book/
Actually I think the book could be improved with more headings and subheadings to help the reader understand where the next paragraph or section is going. Since the topics do not necessarily build on one another, I think it would be possible to reorganize the text to build a course which would deal only with selected topics. There is not excessive self-reference within the book. I think an instructor would be able to pick and choose among the topics without much trouble.
All topics are introduced by an "Investigate!" section which has the reader puzzle over a problem or set of problems. These "Investigate!" sections are tremendous and whet the reader's appetite for what follows. The problems are of varying degrees of difficultly and many are quite thought provoking. The book has a nice logical flow.
Some modern textbooks have many more pictures, sidebars, and bells and whistles. This book does not have a lot of that, but the limited numbers of illustrations are clear and do not confuse the reader. The links from the index are excellent. This reviewer tends to think that a lot of textbooks simply distract the reader with all of the pictures and sidebars. The book has a simple clear interface. It is not a fancy book and it does not need to be.
I found no grammar errors.
The book is not culturally insensitive or offensive in any way. I note that one of the problems refers to a Christmas party. Maybe there should be references to other religious parties or traditions. it is a math book about discrete mathematics so it is difficult to work in examples that include other races, ethnicities or backgrounds, but with a little creativity such examples could probably be included.
The best thing about this book is the clear tone of enthusiasm for the subject that comes through loud and clear. The tone is infectious and I found myself as I read the book feeling as if I were in a lecture hall attentively listening to the author, Oscar Levin. There is an informality to the book which does not sacrifice any rigor. This is a definite plus. I was very impressed with this book.
Table of Contents
0 Introduction and Preliminaries 1
- 0.1 What is Discrete Mathematics?
- 0.2 Mathematical Statements
- 0.3 Sets
- 1.1 Additive and Multiplicative Principles
- 1.2 Binomial Coefficients
- 1.3 Combinations and Permutations
- 1.4 Combinatorial Proofs
- 1.5 Stars and Bars
- 1.6 Advanced Counting Using PIE
- 1.7 Chapter Summary
- 2.1 Definitions
- 2.2 Arithmetic and Geometric Sequences
- 2.3 Polynomial Fitting
- 2.4 Solving Recurrence Relations
- 2.5 Induction
- 2.6 Chapter Summary
3 Symbolic Logic and Proofs
- 3.1 Propositional Logic
- 3.2 Proofs
- 3.3 Chapter Summary
4 Graph Theory
- 4.1 Definitions
- 4.2 Planar Graphs
- 4.3 Coloring
- 4.4 Euler Paths and Circuits
- 4.5 Matching in Bipartite Graphs
- 4.6 Chapter Summary
5 Additional Topics
- 5.1 Generating Functions
- 5.2 Introduction to Number Theory
About the Book
Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. The textbook has been developed while teaching the Discrete Mathematics course at the University of Northern Colorado. Primitive versions were used as the primary textbook for that course since Spring 2013, and have been used by other instructors as a free additional resource. Since then it has been used as the primary text for this course at UNC, as well as at other institutions.
About the Contributors
Oscar Levin is an Assistant Professor at the University of Northern Colorado in the School of Mathematical Sciences. He has taught mathematics at the college level for over 10 years and has received multiple teaching awards. He received his Ph.D. in mathematics from the University of Connecticut in 2009.