Discrete Mathematics with Applications

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The framework seems totally consistent. I don't see any problems. From what I know of the subjects involved, the terminology seems appropriate and consistent Introduction. Possibility Trees and the Multiplication Rule. Counting Elements of Disjoint Sets: The Addition Rule. The Pigeonhole Principle. Counting Subsets of a Set: Combinations. r-Combinations with Repetition Allowed. Pascal’s Formula and the Binomial Theorem. Probability Axioms and Expected Value. Conditional Probability, Bayes’ Formula, and Independent Events. No problems at all. I find the author's mechanics very good and his style is a joyful and as stated above enthusiastic. Oscar Levin is an Associate 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.

Over 500 worked examples in problem-solution format guide students in building a conceptual understanding of how to solve problems. Proof solutions are intuitively developed in two steps, a discussion on how to approach the proof and a summary of the solution, allowing students the choice of faster or more deliberate instruction depending on how well they understand the problem. Functions Defined on General Sets. One-to-one, Onto, and Inverse Functions. Composition of Functions. Cardinality with Applications to Computability. It's in its third edition and the author mentions making corrections and thanking others for pointing out errors. I didn't find any errors so I would imagine the book is highly accurate.

We have a criticism of the index, which contains some inaccuracies and omissions. For example, the term “adjacency matrix” appears in the index as a main entry, with the sublistings “of directed graph” and “of undirected graph.” The main entries “matrix” and “undirected graph,” however, contain no sublistings for “adjacency matrix.” Exercise numbers with solutions or hints given in an appendix are printed in blue ink. The idea of breaking the pattern of giving answers to only odd-numbered problems is good but, in a nonscientific sample of eight professors, secretaries, students, and spouses, three could not distinguish the blue, three could distinguish it only with difficulty, and two had no difficulty. (The lighter the shade of blue, the less difficulty people had.) Overall, this excellent text is mathematically sound and well written. In fact, it is among the best textbooks for a lower-division discrete mathematics course that we have seen. The author begins the exposition of each topic in a manner that is highly accessible to the typical first-year college student (who has preparation for calculus, except possibly for trigonometry) and then develops higher levels of abstraction. Throughout, the author makes a painstaking, successful effort to give information, to motivate, and to explain. Even the table of contents gives a nice description of the contents of each section. Especially beneficial are the thorough, complete informal discussions of propositions followed by a formal statement and then a complete formal proof. The text is relevant in its content and examples. Mathematical concepts and techniques covered in the textbook will only become more relevant in applications. I found the text to be extremely well written. It has a joyful, upbeat, and enthusiastic voice. I found it very engaging and not at all boring. Professor Levin is a talented writer.

Direct Proof and Counterexample I: Introduction. Direct Proof and Counterexample II: Writing Advice. Direct Proof and Counterexample III: Rational Numbers. Direct Proof and Counterexample IV: Divisibility. Direct Proof and Counterexample V: Division into Cases and the Quotient-Remainder Theorem. Direct Proof and Counterexample VI: Floor and Ceiling. Indirect Argument: Contradiction and Contraposition. Indirect Argument: Two Famous Theorems. Application: Algorithms. There are just a few images in the text. Graphics used to illustrate set operations and graph theory concepts are well laid out. Some tree and Venn diagrams might be improved. 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”). I think the writing is superlative and very clear and totally logical. I don't see how it can be improved. The author states in the preface that “the feature that most distinguishes this book from other discrete mathematics texts is that it teaches—explicitly but in a way that can be understood by typical freshman and sophomore college students—the unspoken logic and reasoning that underlie mathematical thought.” She pays considerable attention to logic and proof, blending well-selected examples and exercises with appropriate levels of abstraction, all designed to teach students to understand mathematical arguments and create their own proofs.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. 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, 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. 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.