## Tools of the Trade: Introduction to Advanced MathematicsThis 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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#### Review: Tools of the Trade

User Review - Brian - GoodreadsI 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 |

189 | |

### Common terms and phrases

accumulation point addition and multiplication ak)keN Axiom of Choice basis bijection called cardinality Cauchy sequence Chapter closed sets coefficients collection commutative ring complete metric space complex numbers consider contains continuous function countable cyclic group decimal expansion define Definition denoted dense empty set equivalence classes equivalence relation example Exercise exists field F finite number finite set finite subcover function f given Hence homeomorphism identity iii)Show infinite set intersection isomorphic least upper bound Lemma Let F linear transformation linearly independent linearly independent vectors n)neN natural numbers non-empty subset nonzero number of elements open covering open interval open set ordered field orthogonal p-adic polynomial function positive integer Proof Prove rotations satisfies scalar Show that f subgroup of G subspace Suppose Theorem topological space topology unique unit ball upper bound property usual metric vector space write