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Sets, Groups, and Mappings: An Introduction to Abstract Mathematics

Andrew D. Hwang College of the Holy Cross, Worcester, MA
Available Formats:
Hardcover ISBN: 978-1-4704-4932-2
Product Code: AMSTEXT/39
List Price: $82.00 MAA Member Price:$73.80
AMS Member Price: $65.60 Electronic ISBN: 978-1-4704-5352-7 Product Code: AMSTEXT/39.E List Price:$82.00
MAA Member Price: $73.80 AMS Member Price:$65.60
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $123.00 MAA Member Price:$110.70
AMS Member Price: $98.40 Click above image for expanded view Sets, Groups, and Mappings: An Introduction to Abstract Mathematics Andrew D. Hwang College of the Holy Cross, Worcester, MA Available Formats:  Hardcover ISBN: 978-1-4704-4932-2 Product Code: AMSTEXT/39  List Price:$82.00 MAA Member Price: $73.80 AMS Member Price:$65.60
 Electronic ISBN: 978-1-4704-5352-7 Product Code: AMSTEXT/39.E
 List Price: $82.00 MAA Member Price:$73.80 AMS Member Price: $65.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$123.00 MAA Member Price: $110.70 AMS Member Price:$98.40
• Book Details

Pure and Applied Undergraduate Texts
Volume: 392019; 304 pp
MSC: 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 easy-going 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 non-trivial computation.

The main part of the book can be used as the basis for a transition-to-proofs 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 finitely-generated 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

• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 392019; 304 pp
MSC: 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 easy-going 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 non-trivial computation.

The main part of the book can be used as the basis for a transition-to-proofs 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 finitely-generated 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.