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Bridge to Abstract Mathematics

Table of Contents

• Some Notes on Notation
• To the Students
  • To Those Beginning the Journey into Proof Writing
  • How to Use This Text
  • Do the Exercises!
  • Acknowledgments
• For the Professors
  • To Those Leading the Development of Proof Writing for Students in a Broad Range of Disciplines
• I. THE AXIOMATIC METHOD
  • 1. Introduction
    • 1.1 The History of Numbers
    • 1.2 The Algebra of Numbers
    • 1.3 The Axiomatic Method
    • 1.4 Parallel Mathematical Universes
  • 2. Statements in Mathematics
    • 2.1 Mathematical Statements
    • 2.2 Mathematical Connectives
    • 2.3 Symbolic Logic
    • 2.4 Compound Statements in English
    • 2.5 Predicates and Quantifiers
    • 2.6 Supplemental Exercises
  • 3. Proofs in Mathematics
    • 3.1 What is Mathematics?
    • 3.2 Direct Proof
    • 3.3 Contraposition and Proof by Contradiction
    • 3.4 Proof by Induction
    • 3.5 Proof by Complete Induction
    • 3.6 Examples and Counterexamples
    • 3.7 Supplemental Exercises
    • How to THINK about mathematics: A Summary
    • How to COMMUNICATE mathematics: A Summary
    • How to DO mathematics: A Summary
• II. SET THEORY
  • 4. Basic Set Operations
    • 4.1 Introduction
    • 4.2 Subsets
    • 4.3 Intersections and Unions
    • 4.4 Intersections and Unions of Arbitrary Collections
    • 4.5 Differences and Complements
    • 4.6 Power Sets
    • 4.7 Russell's Paradox
    • 4.8 Supplemental Exercises
  • 5. Functions
    • 5.1 Functions as Rules
    • 5.2 Cartesian Products, Relations, and Functions
    • 5.3 Injective, Surjective, and Bijective Functions
    • 5.4 Compositions of Functions
    • 5.5 Inverse Functions and Inverse Images of Functions
    • 5.6 Another Approach to Compositions
    • 5.7 Supplemental Exercises
  • 6. Relations on a Set
    • 6.1 Properties ofRelations
    • 6.2 Order Relations
    • 6.3 Equivalence Relations
    • 6.4 Supplemental Exercises
  • 7. Cardinality
    • 7.1 Cardinality of Sets: Introduction
    • 7.2 Finite Sets
    • 7.3 Infinite Sets
    • 7.4 Countable Sets
    • 7.5 Uncountable Sets
    • 7.6 Supplemental Exercises
• III. NUMBER SYSTEMS
  • 8. Algebra of Number Systems
    • 8.1 Introduction: A Road Map
    • 8.2 Primary Properties of Number Systems
    • 8.3 Secondary Properties
    • 8.4 Isomorphisms and Embeddings
    • 8.5 Archimedean Ordered Fields
    • 8.6 Supplemental Exercises
  • 9. The Natural Numbers
    • 9.1 Introduction
    • 9.2 Zero, the Natural Numbers, and Addition
    • 9.3 Multiplication
    • 9.4 Supplemental Exercises
    • Summary of the Properties of the Nonnegative Integers
  • 10. The Integers
    • 10.1 Introduction: Integers as Equivalence Classes
    • 10.2 A Total Ordering of the Integers
    • 10.3 Addition of Integers
    • 10.4 Multiplication of Integers
    • 10.5 Embedding the Natural Numbers in the Integers
    • 10.6 Supplemental Exercises
    • Summary of the Properties of the Integers
  • 11. The Rational Numbers
    • 11.1 Introduction: Rationals as Equivalence Classes
    • 11.2 A Total Ordering of the Rationals
    • 11.3 Addition of Rationals
    • 11.4 Multiplication of Rationals
    • 11.5 An Ordered Field Containing the Integers
    • 11.6 Supplemental Exercises
    • Summary of the Properties of the Rationals
  • 12. The Real Numbers
    • 12.1 Dedekind Cuts
    • 12.2 Order and Addition of Real Numbers
    • 12.3 Multiplication of Real Numbers
    • 12.4 Embedding the Rationals in the Reals
    • 12.5 Uniqueness of the Set of Real Numbers
    • 12.6 Supplemental Exercises
  • 13. Cantor's Reals
    • 13.1 Convergence of Sequences of Rational Numbers
    • 13.2 Cauchy Sequences of Rational Numbers
    • 13.3 Cantor's Set of Real Numbers
    • 13.4 The Isomorphism from Cantor's to Dedekind's Reals
    • 13.5 Supplemental Exercises
  • 14. The Complex Numbers
    • 14.1 Introduction
    • 14.2 Algebra of Complex Numbers
    • 14.3 Order on the Complex Field
    • 14.4 Embedding the Reals in the Complex Numbers
    • 14.5 Supplemental Exercises
• IV. TIME SCALES
  • 15. Time Scales
    • 15.1 Introduction
    • 15.2 Preliminary Results
    • 15.3 The Time Scale and its Jump Operators
    • 15.4 Limits and Continuity
    • 15.5 Supplemental Exercises
  • 16. The Delta Derivative
    • 16.1 Delta Differentiation
    • 16.2 Higher Order Delta Differentiation
    • 16.3 Properties of the Delta Derivative
    • 16.4 Supplemental Exercises
• V. HINTS
  • 17. Hints for (and Comments on) the Exercises
    • Hints for Chapter 2
    • Hints for Chapter 3
    • Hints for Chapter 4
    • Hints for Chapter 5
    • Hints for Chapter 6
    • Hints for Chapter 7
    • Hints for Chapter 8
    • Hints for Chapter 9
    • Hints for Chapter 10
    • Hints for Chapter 11
    • Hints for Chapter 12
    • Hints for Chapter 13
    • Hints for Chapter 14
    • Hints for Chapter 15
    • Hints for Chapter 16
• Bibliography
• Index
• About the Authors

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