Hardcover ISBN:  9781470449322 
Product Code:  AMSTEXT/39 
304 pp 
List Price:  $82.00 
MAA Member Price:  $73.80 
AMS Member Price:  $65.60 
Electronic ISBN:  9781470453527 
Product Code:  AMSTEXT/39.E 
304 pp 
List Price:  $82.00 
MAA Member Price:  $73.80 
AMS Member Price:  $65.60 

Book DetailsPure and Applied Undergraduate TextsVolume: 39; 2019MSC: Primary 20; Secondary 11; 00;
This book introduces students to the world of advanced mathematics using algebraic structures as a unifying theme. Having no prerequisites beyond precalculus and an interest in abstract reasoning, the book is suitable for students of math education, computer science or physics who are looking for an easygoing entry into discrete mathematics, induction and recursion, groups and symmetry, and plane geometry. In its presentation, the book takes special care to forge linguistic and conceptual links between formal precision and underlying intuition, tending toward the concrete, but continually aiming to extend students' comfort with abstraction, experimentation, and nontrivial computation.
The main part of the book can be used as the basis for a transitiontoproofs course that balances theory with examples, logical care with intuitive plausibility, and has sufficient informality to be accessible to students with disparate backgrounds. For students and instructors who wish to go further, the book also explores the Sylow theorems, classification of finitelygenerated Abelian groups, and discrete groups of Euclidean plane transformations.ReadershipUndergraduate students interested in the introduction to proofs and elementary concepts in abstract algebra.

Table of Contents

Cover

Title page

Contents

To the Instructor

To the Student

Chapter 1. Logic and Proofs

1.1. Statements, Negation, and Connectives

1.2. Quantification

1.3. Truth Tables and Applications

Exercises

Chapter 2. An Introduction to Sets

2.1. Specifying Sets

2.2. Complex Numbers

2.3. Sets and Logic, Partitions

Exercises

Chapter 3. The Integers

3.1. Counting and Arithmetic Operations

3.2. Consequences of the Axioms

3.3. The Division Algorithm

Exercises

Chapter 4. Mappings and Relations

4.1. Mappings, Images, and Preimages

4.2. Surjectivity and Injectivity

4.3. Composition and Inversion

4.4. Equivalence Relations

Exercises

Chapter 5. Induction and Recursion

5.1. Mathematical Induction

5.2. Applications

5.3. The Binomial Theorem

Exercises

Chapter 6. Binary Operations

6.1. Definitions and Equivalence

6.2. Algebraic Properties of Binary Operations

Exercises

Chapter 7. Groups

7.1. Definition and Basic Properties

7.2. The Law of Exponents

7.3. Subgroups

7.4. Generated Subgroups

7.5. Groups of Complex Numbers

Exercises

Chapter 8. Divisibility and Congruences

8.1. Residue Classes of Integers

8.2. Greatest Common Divisors

Exercises

Chapter 9. Primes

9.1. Definitions

9.2. Prime Factorizations

Exercises

Chapter 10. Multiplicative Inverses of Residue Classes

10.1. Invertibility

10.2. Linear Congruences

Exercises

Chapter 11. Linear Transformations

11.1. The Cartesian Vector Space

11.2. Plane Transformations

11.3. Cartesian Transformations

Exercises

Chapter 12. Isomorphism

12.1. Properties and Examples

12.2. Classification of Cyclic Groups

Exercises

Chapter 13. The Symmetric Group

13.1. Disjoint Cycle Structure of a Permutation

13.2. Cycle Multiplication

13.3. Parity and the Alternating Group

Exercises

Chapter 14. Examples of Finite Groups

14.1. Cayley’s Theorem

14.2. The Dihedral and Dicyclic Groups

14.3. Miscellaneous Examples

Exercises

Chapter 15. Cosets

15.1. Definitions and Examples

15.2. Normal Subgroups

Exercises

Chapter 16. Homomorphisms

16.1. Definition and Properties

16.2. Homomorphisms and Cyclic Groups

16.3. Quotient Groups

16.4. The Isomorphism Theorems

Exercises

Chapter 17. Group Actions

17.1. Actions and Automorphisms

17.2. Orbits and Stabilizers

17.3. The Sylow Theorems

17.4. Classification of Finite Abelian Groups

17.5. Notes on the Classification of Finite Groups

17.6. Finitely Generated Abelian Groups

Exercises

Chapter 18. Euclidean Geometry

18.1. The Cartesian Plane and Isometries

18.2. Structure of Cartesian Isometries

18.3. The Euclidean Isometry Group

18.4. Reflections

18.5. Discrete Groups of Plane Motions

Exercises

Appendix A. Euler’s Formula

Index

Back Cover


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This book introduces students to the world of advanced mathematics using algebraic structures as a unifying theme. Having no prerequisites beyond precalculus and an interest in abstract reasoning, the book is suitable for students of math education, computer science or physics who are looking for an easygoing entry into discrete mathematics, induction and recursion, groups and symmetry, and plane geometry. In its presentation, the book takes special care to forge linguistic and conceptual links between formal precision and underlying intuition, tending toward the concrete, but continually aiming to extend students' comfort with abstraction, experimentation, and nontrivial computation.
The main part of the book can be used as the basis for a transitiontoproofs course that balances theory with examples, logical care with intuitive plausibility, and has sufficient informality to be accessible to students with disparate backgrounds. For students and instructors who wish to go further, the book also explores the Sylow theorems, classification of finitelygenerated Abelian groups, and discrete groups of Euclidean plane transformations.
Undergraduate students interested in the introduction to proofs and elementary concepts in abstract algebra.

Cover

Title page

Contents

To the Instructor

To the Student

Chapter 1. Logic and Proofs

1.1. Statements, Negation, and Connectives

1.2. Quantification

1.3. Truth Tables and Applications

Exercises

Chapter 2. An Introduction to Sets

2.1. Specifying Sets

2.2. Complex Numbers

2.3. Sets and Logic, Partitions

Exercises

Chapter 3. The Integers

3.1. Counting and Arithmetic Operations

3.2. Consequences of the Axioms

3.3. The Division Algorithm

Exercises

Chapter 4. Mappings and Relations

4.1. Mappings, Images, and Preimages

4.2. Surjectivity and Injectivity

4.3. Composition and Inversion

4.4. Equivalence Relations

Exercises

Chapter 5. Induction and Recursion

5.1. Mathematical Induction

5.2. Applications

5.3. The Binomial Theorem

Exercises

Chapter 6. Binary Operations

6.1. Definitions and Equivalence

6.2. Algebraic Properties of Binary Operations

Exercises

Chapter 7. Groups

7.1. Definition and Basic Properties

7.2. The Law of Exponents

7.3. Subgroups

7.4. Generated Subgroups

7.5. Groups of Complex Numbers

Exercises

Chapter 8. Divisibility and Congruences

8.1. Residue Classes of Integers

8.2. Greatest Common Divisors

Exercises

Chapter 9. Primes

9.1. Definitions

9.2. Prime Factorizations

Exercises

Chapter 10. Multiplicative Inverses of Residue Classes

10.1. Invertibility

10.2. Linear Congruences

Exercises

Chapter 11. Linear Transformations

11.1. The Cartesian Vector Space

11.2. Plane Transformations

11.3. Cartesian Transformations

Exercises

Chapter 12. Isomorphism

12.1. Properties and Examples

12.2. Classification of Cyclic Groups

Exercises

Chapter 13. The Symmetric Group

13.1. Disjoint Cycle Structure of a Permutation

13.2. Cycle Multiplication

13.3. Parity and the Alternating Group

Exercises

Chapter 14. Examples of Finite Groups

14.1. Cayley’s Theorem

14.2. The Dihedral and Dicyclic Groups

14.3. Miscellaneous Examples

Exercises

Chapter 15. Cosets

15.1. Definitions and Examples

15.2. Normal Subgroups

Exercises

Chapter 16. Homomorphisms

16.1. Definition and Properties

16.2. Homomorphisms and Cyclic Groups

16.3. Quotient Groups

16.4. The Isomorphism Theorems

Exercises

Chapter 17. Group Actions

17.1. Actions and Automorphisms

17.2. Orbits and Stabilizers

17.3. The Sylow Theorems

17.4. Classification of Finite Abelian Groups

17.5. Notes on the Classification of Finite Groups

17.6. Finitely Generated Abelian Groups

Exercises

Chapter 18. Euclidean Geometry

18.1. The Cartesian Plane and Isometries

18.2. Structure of Cartesian Isometries

18.3. The Euclidean Isometry Group

18.4. Reflections

18.5. Discrete Groups of Plane Motions

Exercises

Appendix A. Euler’s Formula

Index

Back Cover