Cover image for Transition to higher mathematics : structure and proof
Title:
Transition to higher mathematics : structure and proof
Personal Author:
Series:
The Walter Rudin student series in advanced mathematics
Publication Information:
New York, NY : McGraw Hill Higher Education, 2007
ISBN:
9780073533537

Available:*

Library
Item Barcode
Call Number
Material Type
Item Category 1
Status
Searching...
30000010144417 QA9 D85 2006 Open Access Book Book
Searching...

On Order

Summary

Summary

This text is intended for the Foundations of Higher Math bridge coursetaken by prospective math majors following completion of themainstream Calculus sequence, and is designed to help studentsdevelop the abstract mathematical thinking skills necessary forsuccess in later upper-level majors math courses. As lower-levelcourses such as Calculus rely more exclusively on computationalproblems to service students in the sciences and engineering, mathmajors increasingly need clearer guidance and more rigorous practicein proof technique to adequately prepare themselves for the advancedmath curriculum.


Table of Contents

Chapter 0 Introduction
0.1 Why this book is
0.2 What this book is
0.3 What this book is not
0.4 Advice to the Student
0.5 Advice to the Instructor
0.6 Acknowledgements
Chapter 1 Preliminaries
1.1 "And" "Or"
1.2 Sets
1.3 Functions
1.4 Injections, Surjections, Bijections
1.5 Images and Inverses
1.6 Sequences
1.7 Russell's Paradox
1.8 Exercises
1.9 Hints to Get Started on Some Exercises
Chapter 2 Relations
2.1 Definitions
2.2 Orderings
2.3 Equivalence Relations
2.4 Constructing Bijections
2.5 Modular Arithmetic
2.6 Exercises
Chapter 3 Proofs
3.1 Mathematics and Proofs
3.2 Propositional Logic
3.3 Formulas
3.4 Quantifiers
3.5 Proof Strategies
3.6 Exercises
Chapter 4 Principle of Induction
4.1 Well-Orderings
4.2 Principle of Induction
4.3 Polynomials
4.4 Arithmetic-Geometric Inequality
4.5 Exercises
Chapter 5 Limits
5.1 Limits
5.2 Continuity
5.3 Sequences of Functions
5.4 Exercises
Chapter 6 Cardinality
6.1 Cardinality
6.2 Infinite Sets
6.3 Uncountable Sets
6.4 Countable Sets
6.5 Functions and Computability
6.6 Exercises
Chapter 7 Divisibility
7.1 Fundamental Theorem of Arithmetic
7.2 The Division Algorithm
7.3 Euclidean Algorithm
7.4 Fermat's Little Theorem
7.5 Divisibility and Polynomials
7.6 Exercises
Chapter 8 The Real Numbers
8.1 The Natural Numbers
8.2 The Integers
8.3 The Rational Numbers
8.4 The Real Numbers
8.5 The Least Upper Bound Principle
8.6 Real Sequences
8.7 Ratio Test
8.8 Real Functions
8.9 Cardinality of the Real Numbers 8.10. Order-Completeness 8.11. Exercises
Chapter 9 Complex Numbers
9.1 Cubics
9.2 Complex Numbers
9.3 Tartaglia-Cardano Revisited
9.4 Fundamental Theorem of Algebra
9.5 Application to Real Polynomials
9.6 Further Remarks
9.7 Exercises
Appendix A The Greek Alphabet
Appendix B Axioms of Zermelo-Fraenkel with the Axiom of Choice
Bibliography
Index