Discrete Math with Proof (Hardcover)

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.

摘要

這本寫得非常好的教材適用於一學期的離散數學課程，適合數學和計算機科學專業的學生。修讀一學期的微積分有助於提高學生的學術水平。

目錄

1. 引言。
- 什麼是離散數學？穩定婚姻問題。其他例子。練習題。

2. 集合、邏輯和布林代數。
- 集合。日常生活中的邏輯。命題邏輯。深入研究重言式。布林代數。謂詞邏輯。分析主張（可選）。快速檢查解答。項目。

3. 證明。
- 數學證明入門。證明策略。數學歸納法。創建證明：提示和建議。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。
- 排列組合。組合證明。鴿籠原理：包含-排除。快速檢查解答。

6. 有限概率理論。
- 概率的語言。條件概率和獨立事件。計數和概率。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。項目。

8. 組合學。
- 分割、佔用問題和斯特林數。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. 電腦科學中的形式模型。
- 資訊。有限狀態機。形式語言。正則表達式。正則的三個面向。更高級主題的一瞥。快速檢查解答。

10. 圖。
- 術語。連通性和鄰接性。歐拉和漢密爾頓。表示和同構。重要定理：平面性、多面體、色數。有向圖和加權圖。快速檢查解答。

11. 樹。
- 術語、計數。遍歷、搜索和排序。樹的更多應用。生成樹。快速檢查解答。

12. 函數、關係、數據庫和電路。
- 函數和關係。等價關係。n元關係和關聯數據庫。二元函數和二元表達式。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。關於蘊含、雙向蘊含和等價的邏輯謎題。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。
索引。