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A Review of Discrete Mathematical Structures with Applications to Computer Science by J.P. Tremblay and R. Manohar

This book is a comprehensive introduction to discrete mathematics, covering topics such as sets, relations, functions, logic, algebraic structures, graphs, languages, and automata. The book aims to provide a solid foundation for students of computer science and related fields, as well as to develop their skills in mathematical reasoning and problem-solving.

The book is divided into six parts, each consisting of several chapters. The first part introduces the basic concepts and tools of discrete mathematics, such as sets, relations, functions, induction, recursion, and proof techniques. The second part deals with logic and its applications, such as propositional and predicate calculus, resolution method, and logic programming. The third part explores algebraic structures and their properties, such as groups, rings, fields, lattices, Boolean algebras, and cryptography. The fourth part focuses on graph theory and its applications, such as trees, planarity, coloring, connectivity, network flows, and matching. The fifth part covers languages and automata theory, such as regular expressions, finite automata, grammars, pushdown automata, Turing machines, and decidability. The sixth part discusses some advanced topics in discrete mathematics, such as combinatorics, recurrence relations, generating functions, and complexity theory.

The book is well-written and organized, with clear definitions, examples, theorems, proofs, exercises, and references. The book also includes some historical notes and biographical sketches of prominent mathematicians who contributed to the development of discrete mathematics. The book is suitable for undergraduate or graduate courses in discrete mathematics or computer science. It can also be used as a reference or a self-study guide for anyone interested in learning more about discrete mathematics.In this section, I will provide a brief overview of each chapter of the book and highlight some of the key concepts and applications.

Part I: Basic Concepts

Chapter 1: Sets

This chapter introduces the notion of sets and their operations, such as union, intersection, difference, complement, and Cartesian product. It also defines some important types of sets, such as finite, infinite, countable, uncountable, and power sets. The chapter also discusses some applications of sets in computer science, such as data structures, databases, and hashing.

Chapter 2: Relations

This chapter defines the concept of relations and their properties, such as reflexivity, symmetry, transitivity, equivalence, and order. It also introduces some special types of relations, such as functions, partial orders, total orders, and lattices. The chapter also explores some applications of relations in computer science, such as sorting, searching, encryption, and inheritance.

Chapter 3: The Foundations: Logic and Proofs

This chapter covers the basics of logic and proofs, such as propositions, truth values, logical connectives, truth tables, tautologies, contradictions, logical equivalence, implication, negation, quantifiers, and predicates. It also explains some methods of proof, such as direct proof, proof by contradiction, proof by contrapositive, proof by cases, and mathematical induction.

Chapter 4: Basic Structures: Sets and Functions

This chapter revisits the topics of sets and functions from a more formal perspective. It defines some concepts such as cardinality, bijection, injection, surjection, inverse function, composition function, identity function, and permutation. It also discusses some applications of sets and functions in computer science, such as counting techniques and hashing functions.

Chapter 5: The Fundamentals: Algorithms

This chapter introduces the concept of algorithms and their characteristics. It defines some terms such as input/output specification,

precondition/postcondition,

correctness,

efficiency,

complexity,

and asymptotic notation.

It also explains some techniques for designing algorithms,

such as pseudocode,

flowcharts,

recursion,

and iteration.

It also presents some examples of algorithms for various problems,

such as searching,

sorting,

and matrix multiplication.

Chapter 6: Algorithms with Numbers

This chapter deals with algorithms that involve numerical computations. It covers some topics such as modular arithmetic,

divisibility,

prime numbers,

greatest common divisor,

Euclidean algorithm,

congruences,

Chinese remainder theorem,

and Fermat's little theorem.

It also discusses some applications of algorithms with numbers in computer science,

such as cryptography,

error detection/correction codes,

and hashing functions. 061ffe29dd