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Softcover ISBN:  9781470471880 
Product Code:  AMSTEXT/62 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470474164 
Product Code:  AMSTEXT/62.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470471880 
eBook ISBN:  9781470474164 
Product Code:  AMSTEXT/62.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 62; 2023; 415 ppMSC: Primary 00; 03
This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and culture.
Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces.
Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecturebased or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge.
ReadershipAppropriate for an Introduction to Proofs course and for undergraduate students interested in mathematical thinking and language.

Table of Contents

Cover

Title page

Copyright

Contents

Preface

Who is this book for?

Acknowledgments

To the Student

To the Teacher

Prerequisites

Advice for teaching from this book

Chapter 1. Sets

1.1. Sets, informally

1.2. Proving set membership

1.3. Subsets

1.4. Sets whose elements are sets

1.5. Proving set equality

1.6. Uniqueness of certain elements

1.7. Additional exercises

Chapter 2. Sets with Structure

2.1. Groups

2.2. Metric spaces

2.3. Graphs

2.4. The natural numbers

2.5. Application: Symmetry groups

2.6. Appendix: Euclidean metric

Chapter 3. Logic, Briefly

3.1. Statements, predicates, and quantifiers

3.2. Conjunctions and disjunctions

3.3. Negations

3.4. Implications

3.5. A remark on uniqueness

3.6. Basic exercises in logic

3.7. Russell’s paradox

3.8. Application: The halting problem

Chapter 4. Basic Proof Techniques, Briefly

4.1. Direct proof

4.2. Proof by contraposition

4.3. Proof by contradiction

4.4. Existence

4.5. Uniqueness

4.6. Application: 𝑝values and scientific reasoning

4.7. Writing well

4.8. Additional proofs

Chapter 5. Building Sets

5.1. Subsets

5.2. Complements

5.3. Intersections

5.4. Unions

5.5. Power sets

5.6. Cartesian products

5.7. The persistence of structure

5.8. Application: Configuration spaces

5.9. Application: The geometric structure of data

5.10. Additional problems

Chapter 6. Optional: Set Theory Axiomatics

6.1. The ZFC axioms

6.2. The controversies

6.3. The existence of a natural number system

6.4. The existence of the Cartesian product

6.5. Functions, formally

Chapter 7. Equivalence Relations

7.1. Partitions

7.2. Equivalence relations

7.3. Equivalence classes

7.4. Quotient sets

7.5. Equivalence relations vs. partitions

7.6. Angle addition

7.7. Constructing the integers and rationals

7.8. Modular arithmetic

7.9. Application: Configuration spaces of unlabeled points

7.10. Additional problems

Chapter 8. Functions

8.1. The definition of a function

8.2. Visualizing functions

8.3. Important functions

8.4. Extended examples

8.5. Combining and adapting functions

8.6. Being well defined

8.7. Properties of functions

8.8. Application: Affine encryption

8.9. Application: Campanology

8.10. Application: Probability functions

8.11. Application: Electrical circuits

8.12. Additional problems

Chapter 9. Advanced Proof Techniques

9.1. Regular old induction

9.2. Complete induction

9.3. Wellordering principle

9.4. Constructing sequences recursively

9.5. Other induction methods

9.6. Application: Probability

9.7. Application: Iterated function systems

9.8. Application: Paths in graphs

9.9. Additional exercises

9.10. Appendix: The wellordering theorem

Chapter 10. The Sizes of Sets

10.1. Finite sets

10.2. Infinite sets

10.3. Countable sets

10.4. Uncountable sets

10.5. Producing larger cardinalities

10.6. The Cantor–Bernstein theorem

10.7. Application: Transcendental numbers

10.8. Application: Countable sets and probability

10.9. The cardinal numbers

10.10. Application: Cardinality and symmetry

10.11. Application: Dimension and spacefilling curves

10.12. Application: Infinity in the humanities

Chapter 11. Sequences: From Numbers to Spaces

11.1. Subsequences

11.2. Convergent sequences

11.3. Completeness

11.4. Sequences and subsequences in R

11.5. Application: Circular billiards

11.6. Additional problems

Chapter 12. New Numbers from Completed Spaces

12.1. Metric completions

12.2. The 10adic numbers

12.3. Constructing R

Appendix A. Axioms

Appendix B. A Summary of Proof Techniques

Appendix C. Typography

Bibliography

Index

Back Cover


Additional Material

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This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and culture.
Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces.
Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecturebased or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge.
Appropriate for an Introduction to Proofs course and for undergraduate students interested in mathematical thinking and language.

Cover

Title page

Copyright

Contents

Preface

Who is this book for?

Acknowledgments

To the Student

To the Teacher

Prerequisites

Advice for teaching from this book

Chapter 1. Sets

1.1. Sets, informally

1.2. Proving set membership

1.3. Subsets

1.4. Sets whose elements are sets

1.5. Proving set equality

1.6. Uniqueness of certain elements

1.7. Additional exercises

Chapter 2. Sets with Structure

2.1. Groups

2.2. Metric spaces

2.3. Graphs

2.4. The natural numbers

2.5. Application: Symmetry groups

2.6. Appendix: Euclidean metric

Chapter 3. Logic, Briefly

3.1. Statements, predicates, and quantifiers

3.2. Conjunctions and disjunctions

3.3. Negations

3.4. Implications

3.5. A remark on uniqueness

3.6. Basic exercises in logic

3.7. Russell’s paradox

3.8. Application: The halting problem

Chapter 4. Basic Proof Techniques, Briefly

4.1. Direct proof

4.2. Proof by contraposition

4.3. Proof by contradiction

4.4. Existence

4.5. Uniqueness

4.6. Application: 𝑝values and scientific reasoning

4.7. Writing well

4.8. Additional proofs

Chapter 5. Building Sets

5.1. Subsets

5.2. Complements

5.3. Intersections

5.4. Unions

5.5. Power sets

5.6. Cartesian products

5.7. The persistence of structure

5.8. Application: Configuration spaces

5.9. Application: The geometric structure of data

5.10. Additional problems

Chapter 6. Optional: Set Theory Axiomatics

6.1. The ZFC axioms

6.2. The controversies

6.3. The existence of a natural number system

6.4. The existence of the Cartesian product

6.5. Functions, formally

Chapter 7. Equivalence Relations

7.1. Partitions

7.2. Equivalence relations

7.3. Equivalence classes

7.4. Quotient sets

7.5. Equivalence relations vs. partitions

7.6. Angle addition

7.7. Constructing the integers and rationals

7.8. Modular arithmetic

7.9. Application: Configuration spaces of unlabeled points

7.10. Additional problems

Chapter 8. Functions

8.1. The definition of a function

8.2. Visualizing functions

8.3. Important functions

8.4. Extended examples

8.5. Combining and adapting functions

8.6. Being well defined

8.7. Properties of functions

8.8. Application: Affine encryption

8.9. Application: Campanology

8.10. Application: Probability functions

8.11. Application: Electrical circuits

8.12. Additional problems

Chapter 9. Advanced Proof Techniques

9.1. Regular old induction

9.2. Complete induction

9.3. Wellordering principle

9.4. Constructing sequences recursively

9.5. Other induction methods

9.6. Application: Probability

9.7. Application: Iterated function systems

9.8. Application: Paths in graphs

9.9. Additional exercises

9.10. Appendix: The wellordering theorem

Chapter 10. The Sizes of Sets

10.1. Finite sets

10.2. Infinite sets

10.3. Countable sets

10.4. Uncountable sets

10.5. Producing larger cardinalities

10.6. The Cantor–Bernstein theorem

10.7. Application: Transcendental numbers

10.8. Application: Countable sets and probability

10.9. The cardinal numbers

10.10. Application: Cardinality and symmetry

10.11. Application: Dimension and spacefilling curves

10.12. Application: Infinity in the humanities

Chapter 11. Sequences: From Numbers to Spaces

11.1. Subsequences

11.2. Convergent sequences

11.3. Completeness

11.4. Sequences and subsequences in R

11.5. Application: Circular billiards

11.6. Additional problems

Chapter 12. New Numbers from Completed Spaces

12.1. Metric completions

12.2. The 10adic numbers

12.3. Constructing R

Appendix A. Axioms

Appendix B. A Summary of Proof Techniques

Appendix C. Typography

Bibliography

Index

Back Cover