Understand Mathematics, Understand Computing: Discrete Mathematics That All Computing Students Should Know

In this book the authors aim to endow the reader with an operational, conceptual, and methodological understanding of the discrete mathematics that can be used to study, understand, and perform computing. They want the reader to understand the elements of computing, rather than just know them. The basic topics are presented in a way that encourages readers to develop their personal way of thinking about mathematics. Many topics are developed at several levels, in a single voice, with sample applications from within the world of computing. Extensive historical and cultural asides emphasize the human side of mathematics and mathematicians. By means of lessons and exercises on “doing” mathematics, the book prepares interested readers to develop new concepts and invent new techniques and technologies that will enhance all aspects of computing. The book will be of value to students, scientists, and engineers engaged in the design and use of computing systems, and to scholars and practitioners beyond these technical fields who want to learn and apply novel computational ideas.

Table of Contents:
Introduction.- “Doing” Mathematics: A Toolkit for Mathematical Reasoning.- Sets and Their Algebras: The Stem Cells of Mathematics.- Numbers I: The Basics of Our Number System.- Arithmetic: Putting Numbers to Work.- Summations: Complex Operations from Simple Components.- The Vertigo of Infinity: Handling the Very Large and the Infinite.- Numbers II: Building the Integers and Building with the Integers.- Recurrences: Rendering Complex Structure Manageable.- Numbers III: Operational Representations and Their Consequences.- The Art of Counting: Combinatorics, Probability, and Statistics.- Graphs I: Representing Relationships Mathematically.- Graphs II: Graphs Within Computation and Communication.- Solutions to Exercises.- App. A, Pairing Functions.- App. B, A Deeper Look at the Fibonacci Numbers.- App. C, Two Recurrence-Defined Number Families.- App. D, Signed-Digit Numerals: Carry-Free Addition.- App. E, The Diverse Delights of de Bruijn Networks.- List of Symbols.- References.- Index.

About the Author :
Prof. Arnold Rosenberg is a distinguished university professor emeritus at the University of Massachusetts, Amherst. He also held research positions at Northeastern University and Colorado State University, a professorship at Duke University, and a staff research position at IBM Watson Research Center. He was elected a fellow of the ACM in 1996 for his work on graph-theoretic models of compuation, emphasizing theoretical studies of parallel algorithms and architectures, VLSI design and layout, and data structures. In 1997, he was elected as a fellow of the IEEE for fundamental contributions to theoretical aspects of computer science and engineering.  Prof. Denis Trystram is a distinguished professor at the Grenoble Institute of Engineering, an honorary member of the Institut Universitaire de France (IUF), and he works at the Laboratoire d'Informatique de Grenoble (LIG) in the team-project DataMove-INRIA. His research interestst include the design and analysis of efficient algorithms for optimizing resource use in parallel and distributed systems, approximation algorithms for scheduling and packing problems, and algorithms for data analytics. Both authors have considerable teaching and practical experience in the application of discrete mathematics approaches to computing tasks.

Review :
“There are a lot of things I like about this book … . The basic concept of the book, the overall structure of the book, and the material presented are all excellent: it deserves a better implementation.” (David J. Littleboy, SIGACT News, Vol. 54 (3), 2023) “The text is written in an easy to read format which generously incorporates narratives from the history of mathematics as well as rigorous proofs of the concepts presented. The appendices and references to other texts provide the reader with numerous sources of supplementary information for those wishing to delve into a subject at a deeper level … . chapters are organized and clearly labeled to express which sections are appropriate for a beginning learner, an intermediate learner, or the specialist.” (Tom French, MAA Reviews, October 3, 2021) “Each chapter comes with several exercises from easy to difficult, the latter with complete solutions in the appendix. To accommodate the book to readers with different backgrounds and goals, the authors provide a guide which gives paths through the book for several courses. The exposition is always clear and motivating, no prerequisites are presumed, all terms and concepts are defined precisely, and there are many look-and-see proofs.” (Dieter Riebesehl, zbMATH 1465.68004, 2021)