Sets, Groups, and Mappings: An Introduction to Abstract Mathematics (Pure and Applied Undergraduate Texts)

"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."-- Back cover 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