The Discrete Math Workbook
This practically-oriented textbook presents an accessible introduction to discrete mathematics through a substantial collection of classroom-tested exercises. Each chapter opens with concise coverage of the theory underlying the topic, reviewing the basic concepts and establishing the terminology, as well as providing the key formulae and instructions on their use. This is then followed by a detailed account of the most common problems in the area, before the reader is invited to practice solving such problems for themselves through a varied series of questions and assignments.
The Discrete Math Workbook
The book does not cover graphs, discrete probability, random variables and expectations. It also does not cover some counting problems like combinations with repetition and permutations with indistinguishable objects. It does not cover all rules...read more
The book does not cover graphs, discrete probability, random variables and expectations. It also does not cover some counting problems like combinations with repetition and permutations with indistinguishable objects. It does not cover all rules of inference in propositional logic. I teach all these topics in CS 317 (Discrete Information Structures), a required course for computer science majors at my university. The indices are effective. There is no glossary, but this is not a problem because the topic indices are effective. Most other textbooks on discrete mathematics do not have a glossary either.
It is easy to update this book with changing times or syllabi. Some exercies have only symbols, numbers and mathematical operators and mathematical terms with no mention of real world. Exercises 6.3 and 6.4 about one-to-one and onto functions fall in this category. My examples of relating these topics to real world include mapping people to e-mail addresses or companies of employment or web pages or social media accounts. The quantity of problems in some exercises is higher that what is needed to understand the concepts perfectly. More diversity in the problems will be more useful than more problems in some exercises. It will be good to mention that a bijection is also called ``one-to-one correspondence''. Inclusion of topological sorting after Hasse diagram in the chapter on relations will be useful.
Some statements are vague. On page 43, the author says that some proofs only require direct computation. I think what the author means is that some proofs involve only basic mathematical facts, laws, and transformations leading to the goal to be proved and do not use one of the standard proof methods which include vacuous proof, trivial proof, proof by contradiction, direct proof, and proof by cases. I think the author's point can be better explained by first explaining the standard proof methods and then telling that there are proofs that do not fall into any of these categories. The definition of a function on page 163 is spread over two bullets. The definition should be covered by one bullet. If a student reads only one of the two bullets, he/she will leave with an incorrect understanding of what functions are. Bullets should be ideally used for certain kinds of information like facts, opinions, goals, guidelines, and observations. If bullets are used for describing a multi-step procedure, then one bullet should be used for describing all steps of the same procedure. The author describes different steps of mathematical induction using different bullets. If a student does not read all bullets, he/she may have an incorrect understanding of mathematical induction. Spreading N related steps of the same procedure over N bullets may lead a student to think that there are N different one-step procedures. Though most of the book is very clear, there are parts where clarity should be improved. There are many places where the author has given extra information not found in other textbooks to avoid common misunderstanding among students. I applaud these efforts of the author. The author has related multiple representations at many places, like relating a set containing sets to cubes within cubes and I am sure that students will appreciate this. This certainly helps different learning and thinking styles. I found the notation for logical XOR very confusing.For XOR of propositions p and q, a dash below OR is put between p and q. Some students may think that this is OR of NOT or NOT of OR. I suggest using + within a circle for XOR or some other notation that does not combine existing notation in an ambiguous manner.
(I think this criterion is less relevant to engineering, mathematics and computer science, and more relevant to other disciplines which include geography, economics, politics, religious studies, architecture, history, healthcare, law, and linguistics.) The book does not have content that will offend any population. The topic of the book neither requires nor prohibits inclusion of races, ethnicities, and backgrounds in examples and problems.
This book covers the main topics in a discrete mathematics text. It does not include an analysis of algorithms, graphs, trees, and other topics that would be of interest to computer science students. The presentation of logic and the techniques...read more
This book covers the main topics in a discrete mathematics text. It does not include an analysis of algorithms, graphs, trees, and other topics that would be of interest to computer science students. The presentation of logic and the techniques for writing proofs are thorough and nicely laid out. The problems presented to the students have sufficient variety. New definitions could be included in the summary of the section in which they are presented.
The mathematical content is accurate, though there are a few instances where definitions in this text deviate a bit from those presented in mathematics textbooks. For example, I have read mathematical books that define a function, f, with domain A and co-domain B as a subset of AxB satisfying two properties: for every element of A, a, there exists an element of B, b, such that (a, b) is an element of f, and if (a, b) and (a, c) are elements of f, then b = c. In this textbook, the graph of a function is defined this way (sort of), but functions are not presented as a collection of ordered pairs when initially being defined. These differences can be used to point out the importance of learning definitions and terminology, and using those definitions to create and explore conjectures.
The textbook covers traditional material included in a discrete mathematics class. It includes examples and problems that are typically used in other textbooks in this field. This textbook does not include examples that are particularly modern, or that reference "pop culture" which helps with longevity.
The text is very well written. The material on formulating proofs is extremely well written, and could be incorporated into any course that helps students make the transition from computational mathematics to abstract mathematics. The explanations are clear, the examples highlight what students should do and what they should avoid. Common mistakes are pointed out with clear explanations and examples. The format of the mathematical steps involved in solving a problem makes it easy to follow along and understand the calculations being presented. There are a few proofs that include the word "obviously" as a reason, however, and it would be helpful to have all definitions presented similarly to how theorems are stated (blocked to stand out and labeled as such).
Each subsection presents a topic and explores the ideas in more depth. The sequence of subsections in a chapter follows a natural progression that is typical for a discrete mathematics textbook. Each chapter is well-organized and the content being discussed builds on previously presented ideas.
Some of the "hands-on exercises" are split over two pages, making it difficult to understand the task needing to be practiced (for example, exercise 6.2.1). The experience of scrolling through the pdf file was as expected. Leaving space between the hands-on exercise problems reflect that it is a workbook that will be printed, but if a student only accesses the text as a pdf file, the space is not needed. There are pros and cons to each arrangement.
This is a standard mathematics textbook without references to "pop culture" or terminology that could be considered offensive. It does not contain images of events or people with the potential for being problematic in the future. On the other hand, having specific examples of how the concepts being introduced are relevant to computer science (for example, RSA encryption, hash functions etc...) could help motivate students to learn this material.
Overall, I thought that the foundations of mathematics are well-presented. The advice to students in the Introduction could be shared in every math class! The material is clearly presented, there are ample hands-on exercises, and students would benefit from reading this textbook. Including a few applications of the material being covered would strengthen the textbook, and including topics related to an analysis of algorithms (and big-Oh notation), graphs and trees would complete the topics presented in a typical discrete mathematics class.
This is a text that covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions, relations, and elementary combinatorics, with an emphasis on motivation. It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a different perspective or at a higher level of complexity. The goal is to slowly develop students' problem-solving and writing skills.Open SUNY Textbooks is an open access textbook publishing initiative established by State University of New York libraries and supported by SUNY Innovative Instruction Technology Grants. This initiative publishes high-quality, cost-effective course resources by engaging faculty as authors and peer-reviewers, and libraries as publishing service and infrastructure. The pilot launched in 2012, providing an editorial framework and service to authors, students and faculty, and establishing a community of practice among libraries. Participating libraries in the 2012- 2013 pilot include SUNY Geneseo, College at Brockport, College of Environmental Science and Forestry, SUNY Fredonia, Upstate Medical University, and University at Buffalo, with support from other SUNY libraries and SUNY Press. More information can be found at 041b061a72