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

✍ Scribed by Andrew D. Hwang

• Publisher: American Mathematical Society
• Year: 2019
• Tongue: English
• Leaves: 322
• Category: Library

✦ Synopsis

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.

✦ 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