Softcover ISBN:  9781470465148 
Product Code:  TEXT/68 
List Price:  $85.00 
MAA Member Price:  $63.75 
AMS Member Price:  $63.75 
eBook ISBN:  9781470467616 
Product Code:  TEXT/68.E 
List Price:  $85.00 
MAA Member Price:  $63.75 
AMS Member Price:  $63.75 
Softcover ISBN:  9781470465148 
eBook: ISBN:  9781470467616 
Product Code:  TEXT/68.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $127.50 $95.63 
AMS Member Price:  $127.50 $95.63 
Softcover ISBN:  9781470465148 
Product Code:  TEXT/68 
List Price:  $85.00 
MAA Member Price:  $63.75 
AMS Member Price:  $63.75 
eBook ISBN:  9781470467616 
Product Code:  TEXT/68.E 
List Price:  $85.00 
MAA Member Price:  $63.75 
AMS Member Price:  $63.75 
Softcover ISBN:  9781470465148 
eBook ISBN:  9781470467616 
Product Code:  TEXT/68.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $127.50 $95.63 
AMS Member Price:  $127.50 $95.63 

Book DetailsAMS/MAA TextbooksVolume: 68; 2021; 334 ppMSC: Primary 00;
Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics that is built around proofs and concepts.
The spirit of the book is that the basic tools of abstract mathematics are best developed in context and that creativity and imagination are at the core of mathematics. So, while the book has chapters on statements and sets and functions and induction, the bulk of the book focuses on core mathematical ideas and on developing intuition. Along with chapters on elementary combinatorics and beginning number theory, this book contains introductory chapters on real analysis, group theory, and graph theory that serve as gentle first exposures to their respective areas. The book contains hundreds of exercises, both routine and nonroutine.
This book has been used for a transition to advanced mathematics courses at California State University, Northridge, as well as for a general education course on mathematical reasoning at Krea University, India.
ReadershipUndergraduate students interested in an introduction to proofs.

Table of Contents

Title page

Copyright

Contents

Preface

Chapter 1. Introduction

1.1. Further Exercises

Chapter 2. The Pigeonhole Principle

2.1. Pigeonhole Principle (PHP)

2.2. PHP Generalized Form

2.3. Further Exercises

Chapter 3. Statements

3.1. Statements

3.2. Negation of a Statement

3.3. Compound Statements

3.4. Statements Related to the Conditional

3.5. Remarks on the Implies Statement: Alternative Phrasing, Negations

3.6. Further Exercises

Chapter 4. Counting, Combinations

4.1. Fundamental Counting Principle

4.2. Permutations and Combinations

4.3. Binomial Relations and Binomial Theorem

4.4. Further Exercises

Chapter 5. Sets and Functions

5.1. Sets

5.2. Equality of Sets, Subsets, Supersets

5.3. New Sets From Old

5.4. Functions Between Sets

5.5. Composition of Functions, Inverses

5.6. Examples of Some Sets Commonly Occurring in Mathematics

5.7. Further Exercises

Chapter 6. Interlude: So, How to Prove It? An Essay

Chapter 7. Induction

7.1. Principle of Induction

7.2. Another Form of the Induction Principle

7.3. Further Exercises

7.4. Notes

Chapter 8. Cardinality of Sets

8.1. Finite and Infinite Sets, Countability, Uncountability

8.2. Cardinalities of Q and R

8.3. The SchröderBernstein Theorem

8.4. Cantor Set

8.5. Further Exericses

Chapter 9. Equivalence Relations

9.1. Relations, Equivalence Relations, Equivalence Classes

9.2. Examples

9.3. Further Exercises

Chapter 10. Unique Prime Factorization in the Integers

10.1. Notion of Divisibility

10.2. Greatest Common Divisor, Relative Primeness

10.3. Proof of Unique Prime Factorization Theorem

10.4. Some Consequences of the Unique Prime Factorization Theorem

10.5. Further Exercises

Chapter 11. Sequences, Series, Continuity, Limits

11.1. Sequences

11.2. Convergence

11.3. Continuity of Functions

11.4. Limits of Functions

11.5. Relation between limits and continuity

11.6. Series

11.7. Further Exercises

Chapter 12. The Completeness of R

12.1. Least Upper Bound Property (LUB)

12.2. Greatest Lower Bound Property

12.3. Archimedean Property

12.4. Monotone Convergence Theorem

12.5. BolzanoWeierstrass Theorem

12.6. Nested Intervals Theorem

12.7. Cauchy sequences

12.8. Convergence of Series

12.9. 𝑛th roots of positive real numbers

12.10. Further Exercises

Notes

Chapter 13. Groups and Symmetry

13.1. Symmetries of an equilateral triangle

13.2. Symmetries of a square

13.3. Symmetries of an 𝑛element set

Groups

13.4. Subgroups

13.5. Cosets, Lagrange’s Theorem

13.6. Symmetry

13.7. Isomorphisms Between Groups

13.8. Further Exercises

Chapter 14. Graphs: An Introduction

14.1. Königsberg Bridge Problem and Graphs

14.2. Walks, Paths, Trails, Connectedness

14.3. Existence of Eulerian Trails and Circuits: Sufficiency

14.4. Further Exercises

Index


Additional Material

Reviews

It is well suited for selfstudy, but several possible selections of the chapters are suggested to set up courses of varying lengths. No special prerequisites are needed besides a basic mathematical formation and a certain desire to move to a higher mathematical level.
Adhemar François Bultheel, zbMATH


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 Book Details
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Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics that is built around proofs and concepts.
The spirit of the book is that the basic tools of abstract mathematics are best developed in context and that creativity and imagination are at the core of mathematics. So, while the book has chapters on statements and sets and functions and induction, the bulk of the book focuses on core mathematical ideas and on developing intuition. Along with chapters on elementary combinatorics and beginning number theory, this book contains introductory chapters on real analysis, group theory, and graph theory that serve as gentle first exposures to their respective areas. The book contains hundreds of exercises, both routine and nonroutine.
This book has been used for a transition to advanced mathematics courses at California State University, Northridge, as well as for a general education course on mathematical reasoning at Krea University, India.
Undergraduate students interested in an introduction to proofs.

Title page

Copyright

Contents

Preface

Chapter 1. Introduction

1.1. Further Exercises

Chapter 2. The Pigeonhole Principle

2.1. Pigeonhole Principle (PHP)

2.2. PHP Generalized Form

2.3. Further Exercises

Chapter 3. Statements

3.1. Statements

3.2. Negation of a Statement

3.3. Compound Statements

3.4. Statements Related to the Conditional

3.5. Remarks on the Implies Statement: Alternative Phrasing, Negations

3.6. Further Exercises

Chapter 4. Counting, Combinations

4.1. Fundamental Counting Principle

4.2. Permutations and Combinations

4.3. Binomial Relations and Binomial Theorem

4.4. Further Exercises

Chapter 5. Sets and Functions

5.1. Sets

5.2. Equality of Sets, Subsets, Supersets

5.3. New Sets From Old

5.4. Functions Between Sets

5.5. Composition of Functions, Inverses

5.6. Examples of Some Sets Commonly Occurring in Mathematics

5.7. Further Exercises

Chapter 6. Interlude: So, How to Prove It? An Essay

Chapter 7. Induction

7.1. Principle of Induction

7.2. Another Form of the Induction Principle

7.3. Further Exercises

7.4. Notes

Chapter 8. Cardinality of Sets

8.1. Finite and Infinite Sets, Countability, Uncountability

8.2. Cardinalities of Q and R

8.3. The SchröderBernstein Theorem

8.4. Cantor Set

8.5. Further Exericses

Chapter 9. Equivalence Relations

9.1. Relations, Equivalence Relations, Equivalence Classes

9.2. Examples

9.3. Further Exercises

Chapter 10. Unique Prime Factorization in the Integers

10.1. Notion of Divisibility

10.2. Greatest Common Divisor, Relative Primeness

10.3. Proof of Unique Prime Factorization Theorem

10.4. Some Consequences of the Unique Prime Factorization Theorem

10.5. Further Exercises

Chapter 11. Sequences, Series, Continuity, Limits

11.1. Sequences

11.2. Convergence

11.3. Continuity of Functions

11.4. Limits of Functions

11.5. Relation between limits and continuity

11.6. Series

11.7. Further Exercises

Chapter 12. The Completeness of R

12.1. Least Upper Bound Property (LUB)

12.2. Greatest Lower Bound Property

12.3. Archimedean Property

12.4. Monotone Convergence Theorem

12.5. BolzanoWeierstrass Theorem

12.6. Nested Intervals Theorem

12.7. Cauchy sequences

12.8. Convergence of Series

12.9. 𝑛th roots of positive real numbers

12.10. Further Exercises

Notes

Chapter 13. Groups and Symmetry

13.1. Symmetries of an equilateral triangle

13.2. Symmetries of a square

13.3. Symmetries of an 𝑛element set

Groups

13.4. Subgroups

13.5. Cosets, Lagrange’s Theorem

13.6. Symmetry

13.7. Isomorphisms Between Groups

13.8. Further Exercises

Chapter 14. Graphs: An Introduction

14.1. Königsberg Bridge Problem and Graphs

14.2. Walks, Paths, Trails, Connectedness

14.3. Existence of Eulerian Trails and Circuits: Sufficiency

14.4. Further Exercises

Index

It is well suited for selfstudy, but several possible selections of the chapters are suggested to set up courses of varying lengths. No special prerequisites are needed besides a basic mathematical formation and a certain desire to move to a higher mathematical level.
Adhemar François Bultheel, zbMATH