Discrete Mathematics with Combinatorics, 2/e (Hardcover)

James A. Anderson

  • 出版商: Prentice Hall
  • 出版日期: 2003-08-29
  • 定價: $500
  • 售價: 9.8$490
  • 語言: 英文
  • 頁數: 928
  • 裝訂: Hardcover
  • ISBN: 0130457914
  • ISBN-13: 9780130457912
  • 相關分類: 離散數學 Discrete-mathematics
  • 立即出貨 (庫存=1)

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Summary

For one-/two- semester, freshman courses in Discrete Mathematics.

This carefully organized, very readable text covers every essential topic in discrete mathematics in a logical fashion. Placing each topic in context, it covers concepts associated with discrete mathematical systems that have applications in computer science, engineering, and mathematics. The author introduces more basic concepts at the freshman level than are found in other texts, in a simple, accessible form. Introductory material is balanced with extensive coverage of graphs, trees, recursion, algebra, theory of computing, and combinatorics. Extensive examples throughout the text reinforce concepts.

Table of Contents

1. Truth Tables, Logic, and Proofs.

Statements and Connectives. Conditional Statements. Equivalent Statements. Axiomatic Systems: Arguments and Proofs. Completeness in Propositional Logic. Karnaugh Maps. Circuit Diagrams.

2. Set Theory.

Introduction to Sets. Set Operations. Venn Diagrams. Boolean Algebras. Relations. Partially Ordered Sets. Equivalence Relations. Functions.

3. Logic, Integers, and Proofs.

Predicate Calculus. Basic Concepts of Proofs and the Structure of Integers. Mathematical Induction. Divisibility. Prime Integers. Congruence Relations.

4. Functions and Matrices.

Functions. Special Functions. Matrices. Cardinality. Cardinals Revisited.

5. Algorithms and Recursion.

The “for” Procedure and Algorithms for Matrices. Recursive Functions and Algorithms. Complexity of Algorithms. Sorting Algorithms. Prefix and Suffix Notation. Binary and Hexadecimal Numbers. Signed Numbers. Matrices Continued.

6. Graphs, Directed Graphs and Trees.

Graphs. Directed Graphs. Trees. Instant Insanity. Euler Paths and Cycles. Incidence and Adjacency Matrices. Hypercubes and Gray Code.

7. Number Theory.

Sieve of Eratosthenes. Fermat's Factorization Method. The Division and Euclidean Algorithms. Continued Fractions. Convergents.

8. Counting and Probability.

Basic Counting Principles. Inclusion-Exclusion Introduced. Permutations and Combinations. Generating Permutations and Combinations. Probability Introduced. Generalized Permutations and Combinations. Permutations and Combinations with Repetition. Pigeonhole Principle. Probability Revisited. Bayes' Theorem. Markov Chains.

9. Algebraic Structures.

Partially Ordered Sets Revisited. Semigroups and Semilattices. Lattices. Groups. Groups and Homomorphisms. Linear Algebra.

10. Number Theory Revisited.

Integral Solutions of Linear Equations. Solutions of Congruence Equations. Chinese Remainder Theorem. Order of an Integer.

11. Recursion Revisited.

Homogeneous Linear Recurrence Relations. Nonhomogeneous Linear Recurrence Relations. Finite Differences. Factorial Polynomials. Sums of Differences.

12. Counting Continued.

Occupancy Problems. Catalan Numbers. General Inclusion-Exclusion and Derangements. Rook Polynomials and Forbidden Positions.

13. Generating Functions.

Defining the Generating Function (optional). Generating Functions and Recurrence Relations. Generating Functions and Counting. Partitions. Exponential Generating Functions.

14. Graphs Revisited.

Algebraic Properties of Graphs. Planar Graphs. Coloring Graphs. Hamiltonian Paths and Cycles. Weighted Graphs and Shortest Path Algorithms.

15. Trees.

Properties of Trees. Binary Search Trees. Weighted Trees. Traversing Binary Trees. Spanning Trees. Minimal Spanning Trees.

16. Networks.

Networks and Flows. Matching. Petri Nets.

17. Theory of Computation.

Regular Languages. Automata. Finite State Machines with Output. Grammars. Turing Machines.

18. Theory of Codes.

Introduction. Generator Matrices. Hamming Codes.

19. Enumeration of Colors.

Burnside's Theorem. Polya's Theorem.

20. Rings, Integral Domains, and Fields.

Rings and Integral Domains. Integral Domains. Polynomials. Algebra and Polynomials.

21. Group and Semigroup Characters.

Complex Numbers. Group Characters. Semigroup Characters.

22. Applications of Number Theory.

Application: Pattern Matching. Application: Hashing Functions. Application: Cryptography.

Bibliography.
Hints and Solutions to Selected Exercises.
Index.

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摘要

這本精心組織且易讀的教材涵蓋了離散數學中的每個重要主題,並以邏輯的方式呈現。它將每個主題放入上下文中,涵蓋了與離散數學系統相關的概念,這些概念在計算機科學、工程和數學中都有應用。作者以一種簡單易懂的形式在大一水平上介紹了比其他教材更基礎的概念。入門材料與對圖形、樹、遞迴、代數、計算理論和組合學的廣泛涵蓋相平衡。教材中的大量例子強化了概念。

目錄

1. 真值表、邏輯和證明。
- 陳述和連接詞。
- 條件陳述。
- 等價陳述。
- 公理系統:論證和證明。
- 命題邏輯的完備性。
- 卡諾圖。
- 電路圖。

2. 集合論。
- 集合的介紹。
- 集合運算。
- 文氏圖。
- 布林代數。
- 關係。
- 部分有序集。
- 等價關係。
- 函數。

3. 邏輯、整數和證明。
- 謂詞演算。
- 證明的基本概念和整數的結構。
- 數學歸納法。
- 整除性。
- 質數。
- 同餘關係。

4. 函數和矩陣。
- 函數。
- 特殊函數。
- 矩陣。
- 基數。
- 基數的再討論。

5. 演算法和遞迴。
- "for"程序和矩陣的演算法。
- 遞迴函數和演算法。
- 演算法的複雜度。
- 排序演算法。
- 前綴和後綴表示法。
- 二進制和十六進制數字。
- 有符號數字。
- 矩陣的延續。

6. 圖形、有向圖和樹。
- 圖形。
- 有向圖。
- 樹。
- 瞬間瘋狂。
- 歐拉路徑和迴圈。
- 關聯和鄰接矩陣。
- 超立方體和格雷碼。

7. 數論。
- 埃拉托斯特尼篩法。
- 費馬因式分解法。
- 除法和歐幾里得算法。
- 連分數。
- 收斂。

8. 計數和概率。
- 基本計數原則。
- 引入包含-排除。
- 排列和組合。
- 生成排列和組合。
- 引入概率。
- 广义排列和组合。
- 重複排列和組合。
- 鴿巢原理。
- 概率的再討論。
- 貝葉斯定理。
- 馬可夫鏈。

9. 代數結構。
- 部分有序集的再討論。
- 半群和半格。
- 格。
- 群。
- 群和同態。
- 線性代數。

10. 數論的再討論。
- 線性方程的整數解。
- 同餘方程的解。
- 中國剩餘定理。
- 整數的次序。

11. 遞迴的再討論。
- 齊次線性遞迴關係。
- 非齊次線性遞迴關係。
- 有限差分。
- 階乘多項式。
- 差的總和。

12. 計數的延續。
- 佔用問題。
- 卡塔蘭數。
- 一般包含-排除和錯位。
- 車位多項式和禁止位置。

13. 生成函數。
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