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# Bridge to Abstract Mathematics

Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence

• 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
• 4.8 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
• 10.4 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
• 11.4 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