Discrete Mathematics for Computer Science

This book discusses the role of proofs in mathematics and computer science. In mathematics, a proof involves validating a proposition through logical deductions from axioms. Computer scientists focus on demonstrating program accuracy, given the increasing error susceptibility of software. A community of specialists aims to enhance program precision, extending to verifying computer processor chips for leading manufacturers. Creating mathematical models to affirm program validity is an active study area. A proof, in this context, involves a sequence of logical deductions from axioms and established statements, leading to the desired proposition. While crafting proofs may seem daunting, standard templates offer a framework. Some templates can be interconnected, providing both high-level structure and detailed guidance. The Principle of Mathematical Induction is applied to validate algorithms without computer reliance. Sets underpin modern mathematics and software engineering, introduced with language and typical tasks. Primary set operations' understanding enables proof techniques for functions, relations, and graphs, validating algorithms for specific tasks. The book delves into language describing element collections and sets, providing proof templates for comprehension and construction. The book covers common set operations, introduces additional proof templates, and addresses numbering elements and the Principle of Mathematical Induction. This exploration deepens the understanding of mathematical proofs and their role in computer science applications. Cover HalfTitle Page Title Page Copyright About the Editor Table of Contents List of Figures List of Abbreviations Abstract Preface Chapter 1: Mathematical Logic and Proofs 1.1. Proving Conditional Statements 1.2. Proving Biconditional Statements 1.3. Proof By Contradiction 1.4. Proof By Contrapositive 1.5. Proof By Cases 1.6. Variables and Quantification Chapter 2: Basic Mathematics on the Real Numbers 2.1. Introduction 2.2. Real Number Notions And Operations Chapter 3: Fundamental Mathematical Objects 3.1. Number Theory 3.2. Combinatorics 3.3. Set Theory 3.4. Geometry 3.5. Linear Algebra 3.6. Abstract Algebra Chapter 4: Modular Arithmetic and Polynomials 4.1. Introduction 4.2. Modular Arithmetic 4.3. Polynomials 4.4. Modular Arithmetic With Polynomials 4.5. Polynomials and Finite Fields 4.6. Polynomials and Modular Equations 4.7. Summary of Key Points Chapter 5: Mathematical Functions 5.1. Introduction 5.2. Fundamentals of Functions 5.3. Function Properties and Graphs 5.4. Function Analysis and Applications 5.5. Special Functions and Advanced Topics 5.6. Complex Functions and Analyticity 5.7. Applications of Complex Functions 5.8. Numerical Methods for Functions 5.9. Advanced Topics in Functional Analysis Chapter 6: Linear Algebra in Mathematics 6.1. Linear Algebra Topics 6.2. Eigen Values and Eigen Vectors 6.3. Orthogonal Matrices 6.4. Projections 6.5. Matrix Operations 6.6. Linear Dependence 6.7. Module Theory 6.8. Multilinear Algebra and Tensors 6.9. Topological Vector Spaces 6.10. Applications of Linear Algebra in Mathematics Chapter 7: Mathematical Graphs 7.1. Introduction 7.2. Cartesian Graphs 7.3. Directed Graphs (digraphs) 7.4. Weighted Graphs 7.5. Complete Graphs 7.6. Bipartite Graphs 7.7. Planar Graphs 7.8. Hypergraph 7.9. Eulerian Graphs 7.10. Hamiltonian Graphs Chapter 8: Mathematical Counting and Combinatorics 8.1. Introduction 8.2. Concepts in Combinatorial Theory 8.3. Combinatorics Applications Chapter 9: Discrete Probability in Mathematics 9.1. Introduction 9.2. Basic Concepts in Discrete Probability 9.3. Counting Principles and Probability 9.4. Discrete Random Variables 9.5. Common Discrete Probability Distributions 9.6. Law Of Large Numbers and Central Limit Theorem 9.7. Applications of Discrete Probability Chapter 10: Recurrence Relations 10.1. Introduction 10.2. Definition and Terminology 10.3. Solving Recurrence Relations 10.4. Recurrence Relations Obtained from “solutions” 10.5. Homogeneous Recurrence Relation 10.6. Solution Of Nonhomogeneous Finite Order Linear Relations Bibliography Index Back Cover