Discrete Math with Proof (Hardcover)
暫譯: 離散數學與證明 (精裝版)

Eric Gossett

Discrete Math with Proof (Hardcover)

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Summary

For one-semester undergraduate Discrete Mathematics courses for math and computer science majors.

This exceedingly well-written text equips students to apply discrete mathematics and provides opportunities for practice of the concepts presented. A semester of Calculus is useful for reasons of sophistication.

Table of Contents

1. Introduction.

What Is Discrete Mathematics? The Stable Marriage Problem. Other Examples. Exercises.

2. Sets, Logic, and Boolean Algebras.

Sets. Logic in Daily Life. Propositional Logic. A Detailed Look at Tautologies. Boolean Algebras. Predicate Logic. Analyzing Claims (Optional). Quick Check Solutions. Projects.

3. Proof.

Introduction to Mathematical Proof. Proof Strategies. Mathematical Induction. Creating Proofs: Hints and Suggestions. Quick Check Solutions.

4. Algorithms.

Expressing Algorithms. Measuring Algorithm Efficiency. Pattern Matching. The Halting Problem. Quick Check Solutions.

5. Counting.

Permutations and Combinations. Combinatorial Proofs. Pigeon-Hole: Inclusion-Exclusion. Quick Check Solutions.

6. Finite Probability Theory.

The Language of Probabilities. Conditional Probabilities and Independent Events. Counting and Probability. Expected Value. Bayes' Theorem. Quick Check Solutions.

7. Recursion.

Recursive Algorithms. Recurrence Relations. Generating Functions. The Josephus Problem. Quick Check Solutions. Projects.

8. Combinatorics.

Partitions, Occupancy Problems, and Stirling Numbers. Latin Squares; Finite Projective Planes. Balanced Incomplete Block Designs. The Knapsack Problem. Error-Correcting Codes. Distinct Representatives; Ramsey Numbers. Quick Check Solutions.

9. Formal Models in Computer Science.

Information. Finite-State Machines. Formal Languages. Regular Expressions. The Three Faces of Regular. A Glimpse at More Advanced Topics. Quick Check Solutions.

10. Graphs.

Terminology. Connectivity and Adjacency. Euler and Hamilton. Representation and Isomorphism. The Big Theorems: Planarity, Polyhedra, Chromatic Number. Directed Graphs and Weighted Graphs. Quick Check Solutions.

11. Trees.

Terminology, Counting. Traversal, Searching, and Sorting. More Applications of Trees. Spanning Trees. Quick Check Solutions.

12. Functions, Relations, Databases, and Circuits.

Functions and Relations. Equivalence Relations. n-ary Relations and Relational Databases. Binary Functions and Binary Expressions. Combinatorial Circuits. Quick Check Solutions.

Appendix A. Number Systems.

The Natural Numbers. The Integers. The Rational Numbers. The Real Numbers. The Complex Numbers. Other Number Systems. Representation of Numbers.

Appendix B. Summation Notation.
Appendix C. Logic Puzzles.

Logic Puzzles about AND, OR, NOT. Logic Puzzles about Implication, Biconditional, and Equivalence. Exercises. Quick Check Solutions.

Appendix D. The Golden Ratio.
Appendix E. Matrices.
References.
Index.

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摘要

這本極為優秀的教材為數學和計算機科學專業的本科生提供了一個學期的離散數學課程。該書幫助學生應用離散數學，並提供了練習所呈現概念的機會。一學期的微積分對於提升理解能力是有幫助的。

目錄

1. 介紹
離散數學是什麼？穩定婚姻問題。其他範例。練習題。

2. 集合、邏輯與布林代數
集合。日常生活中的邏輯。命題邏輯。對同義反覆的詳細探討。布林代數。謂詞邏輯。分析主張（可選）。快速檢查解答。專案。

3. 證明
數學證明入門。證明策略。數學歸納法。創建證明：提示與建議。快速檢查解答。

4. 演算法
表達演算法。測量演算法效率。模式匹配。停機問題。快速檢查解答。

5. 計數
排列與組合。組合證明。鴿巢原理：包含-排除。快速檢查解答。

6. 有限概率論
概率的語言。條件概率與獨立事件。計數與概率。期望值。貝葉斯定理。快速檢查解答。

7. 遞迴
遞迴演算法。遞迴關係。生成函數。約瑟夫問題。快速檢查解答。專案。

8. 組合數學
分割、佔用問題與斯特林數。拉丁方；有限射影平面。平衡不完全區塊設計。背包問題。錯誤更正碼。不同代表；拉姆齊數。快速檢查解答。

9. 計算機科學中的形式模型
資訊。有限狀態機。形式語言。正則表達式。正則的三個面向。更高級主題的簡介。快速檢查解答。

10. 圖
術語。連通性與鄰接性。歐拉與哈密頓。表示法與同構。重要定理：平面性、多面體、色數。有向圖與加權圖。快速檢查解答。

11. 樹
術語、計數。遍歷、搜尋與排序。樹的更多應用。生成樹。快速檢查解答。

12. 函數、關係、資料庫與電路
函數與關係。等價關係。n-元關係與關聯資料庫。二元函數與二元表達式。組合電路。快速檢查解答。

附錄A. 數字系統
自然數。整數。有理數。實數。複數。其他數字系統。數字的表示。

附錄B. 總和符號
附錄C. 邏輯謎題
關於 AND、OR、NOT 的邏輯謎題。關於蘊涵、雙條件與等價的邏輯謎題。練習題。快速檢查解答。

附錄D. 黃金比例
附錄E. 矩陣
參考文獻
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