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Of the Properties of the Nonnegative IntegersThe Integers -- Introduction: Integers as Equivalence Classes -- A Total Ordering of the Integers -- Addition of Integers -- Multiplication of Integers -- Embedding the Natural Numbers in the Integers -- Supplemental Exercises -- Summary of the Properties...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
[Washington, DC] :
Mathematical Association of America,
©2012.
|
| Colección: | MAA textbooks.
|
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Front cover
- copyright page
- title page
- Contents
- Some Notes on Notation
- To the Students
- To Those Beginning the Journey into Proof Writing
- How to Use This Text
- Do the Exercises!
- Acknowledgments
- For the Professors
- To Those Leading the Development of Proof Writing for Students in a Broad Range of Disciplines
- I THE AXIOMATIC METHOD
- Introduction
- The History of Numbers
- The Algebra of Numbers
- The Axiomatic Method
- Parallel Mathematical Universes
- Statements in Mathematics
- Mathematical Statements
- Mathematical ConnectivesSymbolic Logic
- Compound Statements in English
- Predicates and Quantifiers
- Supplemental Exercises
- Proofs in Mathematics
- What is Mathematics?
- Direct Proof
- Contraposition and Proof by Contradiction
- Proof by Induction
- Proof by Complete Induction
- Examples and Counterexamples
- Supplemental Exercises
- How to THINK about mathematics: A Summary
- How to COMMUNICATE mathematics: A Summary
- How to DO mathematics: A Summary
- II SET THEORY
- Basic Set Operations
- Introduction
- Subsets
- Intersections and UnionsIntersections and Unions of Arbitrary Collections
- Differences and Complements
- Power Sets
- Russell's Paradox
- Supplemental Exercises
- Functions
- Functions as Rules
- Cartesian Products, Relations, and Functions
- Injective, Surjective, and Bijective Functions
- Compositions of Functions
- Inverse Functions and Inverse Images of Functions
- Another Approach to Compositions
- Supplemental Exercises
- Relations on a Set
- Properties of Relations
- Order Relations
- Equivalence Relations
- Supplemental ExercisesCardinality
- Cardinality of Sets: Introduction
- Finite Sets
- Infinite Sets
- Countable Sets
- Uncountable Sets
- Supplemental Exercises
- III NUMBER SYSTEMS
- Algebra of Number Systems
- Introduction: A Road Map
- Primary Properties of Number Systems
- Secondary Properties
- Isomorphisms and Embeddings
- Archimedean Ordered Fields
- Supplemental Exercises
- The Natural Numbers
- Introduction
- Zero, the Natural Numbers, and Addition
- Multiplication
- Supplemental Exercises
- Summary of the Properties of the Nonnegative IntegersThe Integers
- Introduction: Integers as Equivalence Classes
- A Total Ordering of the Integers
- Addition of Integers
- Multiplication of Integers
- Embedding the Natural Numbers in the Integers
- Supplemental Exercises
- Summary of the Properties of the Integers
- The Rational Numbers
- Introduction: Rationals as Equivalence Classes
- A Total Ordering of the Rationals
- Addition of Rationals
- Multiplication of Rationals
- An Ordered Field Containing the Integers
- Supplemental Exercises