A Transition to Abstract Mathematics - Learning Mathematical Thinking and Writing

Front Cover

A Transition to Abstract Mathematics

Copyright Page

Table of Contents

Why Read This Book?

Preface

Preface to the First Edition

Acknowledgments

Chapter 0. Notation and Assumptions

0.1 Set Terminology and Notation

0.2 Assumptions about the Real Numbers

0.2.1 Basic Algebraic Properties

0.2.2 Ordering Properties

0.2.3 Other Assumptions

Part 1: Foundations of Logic and ProofWriting

Chapter 1. Language and Mathematics

1.1 Introduction to Logic

1.1.1 Statements

1.1.2 Negation of a Statement

1.1.3 Combining Statements with AND

1.1.4 Combining Statements with OR

1.1.5 Logical Equivalence

1.1.6 Tautologies and Contradictions

1.2 If-Then Statements

1.2.1 If-Then Statements Defined

1.2.2 Variations on p . q

1.2.3 Logical Equivalence and Tautologies

1.3 Universal and Existential Quantifiers

1.3.1 The Universal Quantifier

1.3.2 The Existential Quantifier

1.3.3 Unique Existence

1.4 Negations of Statements

1.4.1 Negations of AND and OR Statements

1.4.2 Negations of If-Then Statements

1.4.3 Negations of Statements with the Universal Quantifier

1.4.4 Negations of Statements with the Existential Quantifier

1.5 How We Write Proofs

1.5.1 Direct Proof

1.5.2 Proof by Contrapositive

1.5.3 Proving a Logically Equivalent Statement

1.5.4 Proof by Contradiction

1.5.5 Disproving a Statement

Chapter 2. Properties of Real Numbers

2.1 Basic Algebraic Properties of Real Numbers

2.1.1 Properties of Addition

2.1.2 Properties of Multiplication

2.2 Ordering Properties of the Real Numbers

2.3 Absolute Value

2.4 The Division Algorithm

2.5 Divisibility and Prime Numbers

Chapter 3. Sets and Their Properties

3.1 Set Terminology

3.2 Proving Basic Set Properties

3.3 Families of Sets

3.4 The Principle of Mathematical Induction

3.5 Variations of the PMI

3.6 Equivalence Relations

3.7 Equivalence Classes and Partitions

3.8 Building the Rational Numbers

3.8.1 Defining Rational Equality

3.8.2 Rational Addition and Multiplication

3.9 Roots of Real Numbers

3.10 Irrational Numbers

3.11 Relations in General

Chapter 4. Functions

4.1 Definition and Examples

4.2 One-to-one and Onto Functions

4.3 Image and Pre-Image Sets

4.4 Composition and Inverse Functions

4.4.1 Composition of Functions

4.4.2 Inverse Functions

4.5 Three Helpful Theorems

4.6 Finite Sets

4.7 Infinite Sets

4.8 Cartesian Products and Cardinality

4.8.1 Cartesian Products

4.8.2 Functions Between Finite Sets

4.8.3 Applications

4.9 Combinations and Partitions

4.9.1 Combinations

4.9.2 Partitioning a Set

4.9.3 Applications

4.10 The Binomial Theorem

Part II: Basic Principles of Analysis

Chapter 5: The Real Numbers

5.1 The Least Upper Bound Axiom

5.1.1 Least Upper Bounds

5.1.2 Greatest Lower Bounds

5.2 The Archimedean Property

5.2.1 Maximum and Minimum of Finite Sets

5.3 Open and Closed Sets

5.4 Interior, Exterior, Boundary, and Cluster Points

5.4.1 Interior, Exterior, and Boundary

5.4.2 Cluster Points

5.5 Closure of Sets

5.6 Compactness

Chapter 6. Sequences of Real Numbers

6.1 Sequences Defined

6.1.1 Monotone Sequences

6.1.2 Bounded Sequences

6.2 Convergence of Sequences

6.2.1 Convergence to a Real Number

6.2.2 Convergence to Infinity

6.3 The Nested Interval Property

6.3.1 From LUB Axiom to NIP

6.3.2 The NIP Applied to Subsequences

6.3.3 From NIP to LUB Axiom

6.4 Cauchy Sequences

6.4.1 Convergence of Cauchy Sequences

6.4.2 From Completeness to the NIP

Chapter 7. Functions of a Real Variable

7.1 Bounded and Monotone Functions

7.1.1 Bounded Functions

7.1.2 Monotone Functions

7.2 Limits and Their Basic Properties

7.2.1 Definition of Limit

7.2.2 Basic Theorems of Limits

7.3 More on Limits

7.3.1 One-Sided Limits

7.3.2 Sequential Limits

7.4 Limits Involving Infinity

7.4.1 Limits at Infinity

7.4.2 Limits of Infinity

7.5 Continuity

7.5.1 Continuity at a Point

7.5.2 Continuity on a Set

7.5.3 One-Sided Continuity

7.6 Implications of Continuity

7.6.1 The Intermediate Value Theorem

7.6.2 Continuity and Open Sets

7.7 Uniform Continuity

7.7.1 Definition and Examples

7.7.2 Uniform Continuity and Compact Sets

Part III: Basic Principles of Algebra

Chapter 8. Groups

8.1 Introduction to Groups

8.1.1 Basic Characteristics of Algebraic Structures

8.1.2 Groups Defined

8.1.1 Basic Characteristics of Algebraic Structures

8.1.2 Groups Defined

8.2 Subgroups

8.2.1 Subgroups Defined

8.2.2 Generated Subgroups

8.2.3 Cyclic Subgroups

8.3 Quotient Groups

8.3.1 Integers Modulo n

8.3.2 Quotient Groups

8.3.3 Cosets and Lagrange’s Theorem

8.4 Permutation Groups

8.4.1 Permutation Groups Defined

8.4.2 The Symmetric Group

8.4.3 The Alternating Group

8.4.4 The Dihedral Group

8.5 Normal Subgroups

8.6 Group Morphisms

Chapter 9. Rings

9.1 Rings and Fields

9.1.1 Rings Defined

9.1.2 Fields Defined

9.2 Subrings

9.3 Ring Properties

9.4 Ring Extensions

9.4.1 Adjoining Roots of Ring Elements

9.4.2 Polynomial Rings

9.4.3 Degree of a Polynomial

9.5 Ideals

9.6 Generated Ideals

9.7 Prime and Maximal Ideals

9.8 Integral Domains

9.9 Unique Factorization Domains

9.10 Principal Ideal Domains

9.11 Euclidean Domains

9.12 Polynomials over a Field

9.13 Polynomials over the Integers

9.14 Ring Morphisms

9.14.1 Properties of Ring Morphisms

9.15 Quotient Rings

Index