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
Subject Term:
Added Author:
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
---|---|---|---|---|---|
Searching... | 30000010144417 | QA9 D85 2006 | Open Access Book | Book | Searching... |
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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 |