D'Angelo J.P., West D.B. Mathematical Thinking: Problem-Solving and Proofs

2nd ed. — Upper Saddle River: Prentice Hall, 2000. — 432 p.This book arose from discussions about the undergraduate mathematics curriculum. We asked several questions. Why do students find it difficult to write proofs? What is the role of discrete mathematics? How can the curriculum better integrate diverse topics? Perhaps most important, why don't students enjoy and appreciate mathematics as much as we might hope? The excitement of mathematics springs from engaging problems. Students have natural mathematical curiosity about problems such as those listed in the Preface for the Student. They then care about the techniques used to solve them; hence we use these problems as a focus of development. We hope that students and instructors will enjoy this approach as much as we have. A course introducing techniques of proof should not specialize in one area of mathematics; later courses offer ample opportunities for specialization. This book considers diverse problems and demonstrates relationships among several areas of mathematics. One of the authors studies complex analysis in several variables, the other studies discrete mathematics. We explored the interactions between discrete and continuous mathematics to create a course on problem-solving and proofs.Preface for the Instructor.
Preface for the Student.
Elementary concepts.
Numbers, Sets and Functions.
Language and Proofs.
Induction.
Bijections and Cardinality.
Properties of numbers.
Combinatorial Reasoning.
Divisibility.
Modular Arithmetic.
The Rational Numbers.
Discrete mathematics.
Probability.
Two Principles of Counting.
Graph Theory.
Recurrence Relations.
Continuous mathematics.
The Real Numbers.
Sequences and Series.
Continuous Functions.
Differentiation.
Integration.
The Complex Numbers.
From N to R.
Hints for Selected Exercises.
Suggestions for Further Reading.
List of Notation.
Index.