# Tools of the Trade: Introduction to Advanced Mathematics

American Mathematical Soc., 2008 - Mathematics - 193 pages
This book provides a transition from the formula-full aspects of the beginning study of college level mathematics to the rich and creative world of more advanced topics. It is designed to assist the student in mastering the techniques of analysis and proof that are required to do mathematics. Along with the standard material such as linear algebra, construction of the real numbers via Cauchy sequences, metric spaces and complete metric spaces, there are three projects at the end of each chapter that form an integral part of the text. These projects include a detailed discussion of topics such as group theory, convergence of infinite series, decimal expansions of real numbers, point set topology and topological groups. They are carefully designed to guide the student through the subject matter. Together with numerous exercises included in the book, these projects may be used as part of the regular classroom presentation, as self-study projects for students, or for Inquiry Based Learning activities presented by the students.

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User Review  - Brian - Goodreads

I am almost certainly biased, being a big fan of Mr. Sally and his brilliant pedagogy. Nevertheless, this book is a very clear, flowing primer to a great number of the basic mathematical entities and ... Read full review

### Contents

 Sets Functions and Other Basic Ideas 1 1 Sets and Elements 2 3 The Algebra of Sets 4 4 Cartesian Products Counting and Power Sets 8 5 Some Sets of Numbers 10 6 Equivalence Relations and the Construction of Q 15 7 Functions 22 8 Countability and Other Basic Ideas 30
 4 Intervals 97 5 The Construction of the Real Numbers 98 6 Convergence in R 102 7 Automorphisms of Fields 107 8 Construction of the Complex Numbers 108 9 Convergence in C 110 10 Independent Projects 115 Metric and Euclidean Spaces 125

 9 Axiom of Choice 38 10 Independent Projects 41 Linear Algebra 47 1 Fundamentals of Linear Algebra 48 2 Linear Transformations 54 3 Linear Transformations and Matrices 56 4 Determinants 59 5 Geometric Linear Algebra 67 6 Independent Projects 76 The Construction of the Real and Complex Numbers 89 1 The Least Upper Bound Property and the Real Numbers 90 2 Consequences of the Least Upper Bound Property 92 3 Rational Approximation 94
 2 Definition and Basic Properties of Metric Spaces 126 3 Topology of Metric Spaces 129 4 Limits and Continuous Functions 137 5 Compactness Completeness and Connectedness 145 6 Independent Projects 155 Complete Metric Spaces and the padic Completion of Q 167 1 The Contraction Mapping Theorem and Its Applications 168 2 The Baire Category Theorem and Its Applications 170 3 The StoneWeierstrass Theorem 172 4 The p adic Completion of Q 176 5 Challenge Problems 184 Index 189 Copyright