《离散数学及应用》深入剖析了数字化时代背景下, 离散数学作为计算机科学、信息工程等领域核心基础学科的理论价值与实践意义, 为科研创新、技术开发及产业应用提供关键逻辑支撑与方法指导。全书共分为六章, 第一章系统梳理离散数学的学科概貌, 阐释其核心分支、基本理论框架及与关联学科的交叉逻辑; 第二章聚焦集合论与关系代数, 深入讲解集合运算、关系性质及应用场景; 第三章围绕图论展开, 涵盖图的基本结构、经典算法及实际应用; 第四章阐述代数系统的核心内容, 包括群、环、域的定义、性质及典型案例; 第五章聚焦数理逻辑, 详解命题逻辑、谓词逻辑的推理规则及在程序验证中的应用; 第六章以“理论落地”为核心, 结合计算机科学、信息安全、人工智能等领域的实际案例, 展现离散数学的应用价值与创新方向。本书旨在帮助读者夯实离散数学理论基础, 掌握逻辑推理与抽象建模能力, 同时提升将理论转化为实际解决方案的实践素养。

Discrete Mathematics and Applications offers an in-depth exploration of the theoretical underpinnings and practical relevance of discrete mathematics as a fundamental discipline for computer science and information engineering in the digital era, providing critical logical scaffolding and methodological guidance for research, technological development, and industrial applications. The book comprises six chapters: Chapter 1 outlines the disciplinary landscape, core branches, and basic theoretical framework of discrete mathematics, along with its interdisciplinary connections. Chapter 2 focuses on set theory and relational algebra, delving into set operations, relational properties, and their applications. Chapter 3 is dedicated to graph theory, covering fundamental graph structures, classical algorithms, and real-world use cases. Chapter 4 elaborates on algebraic systems, including the definitions, properties, and typical examples of groups, rings, and fields. Chapter 5 centers on mathematical logic, detailing the inference rules of propositional and predicate logic and their application in program verification. Chapter 6, emphasizing the translation of “theory into practice, ” showcases the applied value and innovative directions of discrete mathematics through concrete case studies drawn from computer science, information security, and artificial intelligence. This work is designed to strengthen readers' theoretical foundation in discrete mathematics, cultivate their capacity for logical reasoning and abstract modeling, and enhance their proficiency in transforming theoretical principles into practical solutions.