Discrete and Combinatorial Mathematics: An Applied Introduction, 4/e (精裝)
暫譯: 離散與組合數學：應用導論（第4版）

Ralph Grimaldi

Discrete and Combinatorial Mathematics: An Applied Introduction, 4/e (精裝)
  • 出版商: Addison Wesley
  • 售價: $1,078
  • 語言: 英文
  • 頁數: 896
  • 裝訂: Hardcover
  • ISBN: 0201199122
  • ISBN-13: 9780201199123

    • 已絕版

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Description

This fourth edition continues to improve on the features that have made it the market leader. The text offers a flexible organization, enabling instructors to adapt the book to their particular courses: discrete mathematics, graph theory, modern algebra, and/or combinatorics. More elementary problems were added, creating a greater variety of level in problem sets, which allows students to perfect skills as they practice. This new edition continues to feature numerous computer science applications-making this the ideal text for preparing students for advanced study.

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Features

  • This text has an enhanced mathematical approach, with carefully thought out examples, including many examples with computer sciences applications.
  • Historical reviews and biographies bring a human element to their assignments.
  • Chapter summaries allow students to review what they have learned.

I. FUNDAMENTALS OF DISCRETE MATHEMATICS.

1. Fundamental Principles of Counting.
The Rules of Sum and Product.
Permutations.
Combinations: The Binomial Theorem.
Combinations with Repetition.
An Application in the Physical Sciences (Optional).
The Catalan Numbers (Optional).

2. Fundamentals of Logic.
Basic Connectives and Truth Tables.
Logical Equivalence: The Laws of Logic.
Logical Implication: Rules of Inference.
The Use of Quantifiers.
Quantifiers, Definitions, and the Proofs of Theorems.

3. Set Theory.
Sets and Subsets.
Set Operations and the Laws of Set Theory.
Counting and Venn Diagrams.
A Word on Probability.

4. Properties of the Integers: Mathematical Induction.
The Well-Ordering Principle: Mathematical Induction.
Recursive Definitions.
The Division Algorithm: Prime Numbers.
The Greatest Common Divisor: The Euclidean Algorithm.
The Fundamental Theorem of Arithmetic.

5. Relations and Functions.
Cartesian Products and Relations.
Functions: Plain and One-to-One.
Onto Functions: Stirling Numbers of the Second Kind.
Special Functions.
The Pigeonhole Principle.
Function Composition and Inverse Functions.
Computational Complexity.
Analysis of Algorithms.

6. Languages: Finite State Machines.
Language: The Set Theory of Strings.
Finite State Machines: A First Encounter.
Finite State Machines: A Second Encounter.

7. Relations: The Second Time Around.
Relations Revisited: Properties of Relations.
Computer Recognition: Zero-One Matrices and Directed Graphs.
Partial Orders: Hasse Diagrams.
Equivalence Relations and Partitions.
Finite State Machines: The Minimization Process.

II. FURTHER TOPICS IN ENUMERATION.

8. The Principle of Inclusion and Exclusion.
The Principle of Inclusion and Exclusion.
Generalizations of the Principle (Optional).
Derangements: Nothing Is in Its Right Place.
Rook Polynomials.
Arrangements with Forbidden Positions.

9. Generating Functions.
Introductory Examples.
Definition and Examples: Calculational Techniques.
Partitions of Integers.
The Exponential Generating Functions.
The Summation Operator.

10. Recurrence Relations.
The First-Order Linear Recurrence Relation.
The Second-Order Linear Recurrence Relation with Constant Coefficients.
The Nonhomogeneous Recurrence Relation.
The Method of Generating Functions.
A Special Kind of Nonlinear Recurrence Relation (Optional).
Divide and Conquer Algorithms (Optional).

III. GRAPH THEORY AND APPLICATIONS.

11. An Introduction to Graph Theory.
Definitions and Examples.
Subgraphs, Complements, and Graph Isomorphism.
Vertex Degree: Euler Trails and Circuits.
Planar Graphs.
Hamilton Paths and Cycles.
Graph Coloring and Chromatic Polynomials.

12. Trees.
Definitions, Properties, and Examples.
Rooted Trees.
Trees and Sorting.
Weighted Trees and Prefix Codes.
Biconnected Components and Articulation Points.

13. Optimization and Matching.
Dijkstra's Shortest Path Algorithm.
Minimal Spanning Trees: The Algorithms of Kruskal and Prim.
Transport Networks: The Max-Flow Min-Cut Theorem.
Matching Theory.

IV. MODERN APPLIED ALGEBRA.

14. Rings and Modular Arithmetic.
The Ring Structure: Definition and Examples.
Ring Properties and Substructures.
The Integers Modulo n.
Ring Homomorphisms and Isomorphisms.

15. Boolean Algebra and Switching Functions.
Switching Functions: Disjunctive and Conjunctive Normal Forms.
Gating Networks: Minimal Sums of Products: Karnaugh Maps.
Further Applications: Don't Care Conditions.
The Structure of a Boolean Algebra (Optional).

16. Groups, Coding Theory, and Polya's Method of Enumeration.
Definition, Examples, and Elementary Properties.
Homomorphisms, Isomorphisms, and Cyclic Groups.
Cosets and Lagrange's Theorem.
Elements of Coding Theory.
The Hamming Metric.
The Parity-Check and Generator trices.
Group Codes: Decoding with Coset Leaders.
Hamming Matrices.
Counting and Equivalence: Burnside's Theorem.
The Cycle Index.
The Pattern Inventory: Polya's Method of Enumeration.

17. Finite Fields and Combinatorial Designs.
Polynomial Rings.
Irreducible Polynomials: Finite Fields.
Latin Squares.
Finite Geometries and Affine Planes.
Block Designs and Projective Planes.

Appendices.
Exponential and Logarithmic Functions.
Matrices, Matrix Operations, and Determinants.
Countable and Uncountable Sets.

Solutions.
Index.

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商品描述(中文翻譯)

描述
本書的第四版持續改進其使其成為市場領導者的特點。文本提供靈活的組織結構，使教師能夠根據其特定課程調整書籍內容：離散數學、圖論、現代代數和/或組合數學。新增了更多基礎問題，創造了更大的問題集層次多樣性，讓學生在練習中完善技能。本新版本繼續包含眾多計算機科學應用，使其成為為學生準備進階學習的理想教材。

特點
- 本書具有增強的數學方法，包含經過深思熟慮的範例，包括許多與計算機科學應用相關的範例。
- 歷史回顧和傳記為作業增添了人性化的元素。
- 章節摘要讓學生能夠回顧所學內容。

I. 離散數學基礎
1. 計數的基本原則
- 加法和乘法法則
- 排列
- 組合：二項式定理
- 帶重複的組合
- 物理科學中的應用（可選）
- 卡塔蘭數（可選）

2. 邏輯基礎
- 基本連接詞和真值表
- 邏輯等價：邏輯法則
- 邏輯蘊涵：推理規則
- 量詞的使用
- 量詞、定義和定理的證明

3. 集合論
- 集合和子集
- 集合運算和集合論法則
- 計數和維恩圖
- 概率簡介

4. 整數的性質：數學歸納法
- 良序原則：數學歸納法
- 遞歸定義
- 除法算法：質數
- 最大公因數：歐幾里得算法
- 算術基本定理

5. 關係和函數
- 笛卡爾積和關係
- 函數：平凡和一對一
- 滿射函數：第二類斯特林數
- 特殊函數
- 鳩巢原理
- 函數組合和反函數
- 計算複雜度
- 算法分析

6. 語言：有限狀態機
- 語言：字符串的集合論
- 有限狀態機：第一次接觸
- 有限狀態機：第二次接觸

7. 關係：第二次回顧
- 關係重訪：關係的性質
- 計算機識別：零一矩陣和有向圖
- 偏序：哈斯圖
- 等價關係和劃分
- 有限狀態機：最小化過程

II. 列舉中的進一步主題
8. 包含與排除原則
- 包含與排除原則
- 原則的概括（可選）
- 置換：沒有任何東西在其正確位置
- 騎士多項式
- 禁止位置的排列

9. 生成函數
- 入門範例
- 定義和範例：計算技術
- 整數的劃分
- 指數生成函數
- 求和運算子

10. 遞迴關係
- 一階線性遞迴關係
- 二階線性遞迴關係（常數係數）
- 非齊次遞迴關係
- 生成函數法
- 一種特殊的非線性遞迴關係（可選）
- 分治算法（可選）

III. 圖論及其應用
11. 圖論簡介
- 定義和範例
- 子圖、補圖和圖同構
- 頂點度數：歐拉路徑和回路
- 平面圖
- 哈密頓路徑和回路
- 圖著色和色彩多項式

12. 樹
- 定義、性質和範例
- 根樹
- 樹和排序
- 加權樹和前綴碼
- 二連通分量和關鍵點

13. 優化和匹配
- Dijkstra的最短路徑算法
- 最小生成樹：Kruskal和Prim的算法
- 運輸網絡：最大流最小割定理
- 匹配理論

IV. 現代應用代數
14. 環和模運算
- 環結構：定義和範例
- 環的性質和子結構
- 模n的整數
- 環同態和同構

15. 布爾代數和開關函數
- 開關函數：析取和合取標準形式
- 閘網絡：最小乘積和Karnaugh圖
- 進一步應用：不在意條件
- 布爾代數的結構（可選）

16. 群、編碼理論和Polya的列舉方法
- 定義、範例和基本性質
- 同態、同構和循環群
- 余類和拉格朗日定理
- 編碼理論的要素
- Hamming度量
- 奇偶檢查和生成矩陣
- 群碼：使用余類領導者解碼
- Hamming矩陣
- 計數和等價：Burnside定理
- 週期指數
- 模式清單：Polya的列舉方法

17. 有限域和組合設計
- 多項式環
- 不可約多項式：有限域
- 拉丁方
- 有限幾何和仿射平面
- 區塊設計和射影平面

附錄
- 指數和對數函數
- 矩陣、矩陣運算和行列式
- 可數集和不可數集

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