TALE OF DISCRETE MATHEMATICS, A

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. Contents: Propositional Logic: The Foundation of Mathematical Reasoning Set Theory and Introduction to Boolean Algebra: A Naïve Approach Prove It: Mathematical Proof Techniques Introduction to Predicate Logic: One Step Further Functions: Back to the Basics Elementary Number Theory Binary Relations Basic Combinatorics: The Art of Counting Without Counting Basics of Graph Theory More on Graph Theory Trees Readership: Undergraduate first-year and second-year students in mathematics, computer science, general science, physics and engineering. Anyone interested in learning the foundation of Mathematics. Key Features: Early introduction to the language of mathematics. Perhaps, one of the important features that sets this textbook apart from other textbooks the introduction of logic and mathematical reasoning at an early stage. A clear and thorough understanding of mathematical logic is key to fully grasp any mathematical concept. The author's experience in teaching some "transitional" courses like real analysis or abstract algebra shows that students' difficulties are not so much in grasping the theoretical concepts, but rather in properly expressing their reasoning and understanding of statements. A typical example is the confusion between the hypothesis and the conclusion in an implication Introduction to algorithms. The book contains several important algorithms in various topics. It is not very common for a typical Discrete Mathematics book to mix abstraction with practical algorithms. This combination enriches the book and widen the range of students who could benefit from it. Although the book does not include a chapter on writing and verifying algorithms, it lays down the mathematical foundation that fuels these algorithms. We explore, among others, Warshall algorithm to compute the transitive closure of a binary relation, the Euclidean algorithm to compute the GCD of two integers and express it as a linear combination, Havel–Hakimi algorithm to test graphical sequences, Ford–Fulkerson algorithm to find the maximum flow in a network A delicate balance between the theory and...