## The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and NumbersIn elementary introductions to mathematical analysis, the treatment of the logical and algebraic foundations of the subject is necessarily rather skeletal. This book attempts to flesh out the bones of such treatment by providing an informal but systematic account of the foundations of mathematical analysis written at an elementary level. This book is entirely self-contained but, as indicated above, it will be of most use to university or college students who are taking, or who have taken, an introductory course in analysis. Such a course will not automatically cover all the material dealt with in this book and so particular care has been taken to present the material in a manner which makes it suitable for self-study. In a particular, there are a large number of examples and exercises and, where necessary, hints to the solutions are provided. This style of presentation, of course, will also make the book useful for those studying the subject independently of taught course. |

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### Contents

Introduction page | 1 |

Set operations | 21 |

Relations | 28 |

Functions | 33 |

Real numbers | 44 |

Principle of induction | 54 |

Real numbers II | 62 |

10+ Construction of the number systems | 78 |

3+ Natural numbers | 79 |

20+ Complex numbers | 92 |

20+ Radicals | 107 |

20+ Transcendental numbers | 121 |

### Common terms and phrases

addition argument arithmetic asserts assume assumption axiom bijection called cardinality chapter collection complex numbers consider consistent construct contains count countable course Dedekind deduce defined definition denote diagram difference discussion elements equal equation example exercise exists expressed fact factor false finite follows formal formula function f given Hence hold idea illustrate implies important induction infinite integers interval introduced irrational laws length loves mathematical maximum means measuring multiplication natural numbers non-empty set notation Note objects Observe obtained ordered field ordering pair particular polynomial positive possible precisely predicate prime principle problem Proof properties Prove quadratic question rational numbers real number system represent require result roots rules satisfies Show smallest solution square statement subset Suppose symbols theorem theory true truth uncountable unique universal set upper bound usual well-ordering write