دانلود کتاب The Tools of Mathematical Reasoning (Pure and Applied Undergraduate Texts)

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Contents Preface Chapter 1. Language, Logic, and Proof 1.1. Language and logic Exercises 1.1 1.2. Proof Exercises 1.2 Chapter 2. Techniques of Proof 2.1. More direct proofs Exercises 2.1 2.2. Indirect proofs: Proofs by contradiction and contrapositive Exercises 2.2 2.3. Two important theorems Exercises 2.3 2.4. Proofs of statements involving mixed quantifiers Exercises 2.4 Chapter 3. Induction 3.1. Principle of Mathematical Induction Exercises 3.1 3.2. Strong induction Exercises 3.2 Chapter 4. Sets 4.1. The language of sets Exercises 4.1 4.2. Operations on sets Exercises 4.2 4.3. Arbitrary unions and intersections Exercises 4.3 4.4. Axiomatic set theory Chapter 5. Functions 5.1. Definitions Exercises 5.1 5.2. Function composition Exercises 5.2 5.3. One-to-one and onto functions Exercises 5.3 5.4. Invertible functions Exercises 5.4 5.5. Functions and sets Exercises 5.5 Chapter 6. An Introduction to Number Theory 6.1. The Division Algorithm and the Well-Ordering Principle Exercises 6.1 6.2. Greatest common divisors and the Euclidean Algorithm Exercises 6.2 6.3. Relatively prime integers and the Fundamental Theorem of Arithmetic Exercises 6.3 6.4. Congruences Exercises 6.4 6.5. Congruence classes Exercises 6.5 Chapter 7. Equivalence Relations and Partitions 7.1. Introduction Exercises 7.1 7.2. Equivalence relations Exercises 7.2 7.3. Partitions Exercises 7.3 Chapter 8. Finite and Infinite Sets 8.1. Introduction Exercises 8.1 8.2. Finite sets Exercises 8.2 8.3. Infinite sets Exercises 8.3 8.4. What next? Chapter 9. Foundations of Analysis 9.1. Introduction 9.2. The Completeness Axiom Exercises 9.2 9.3. The Archimedean Property and its consequences Exercises 9.3 9.4. What next? Appendix. Writing Mathematics Bibliography Index