## A Set Theory WorkbookThis book is a companion to A general topology workbook published by Birkhiiuser last year. In an ideal world the order of publication would have been reversed, for the notation and some of the results of the present book are used in the topology book and on the other hand (the reader may be assured) no topology is used here. Both books share the word Workbook in their titles. They are based on the principle that for at least some branches of mathematics a good way for a student to learn is to be presented with a clear statement of the definitions of the terms with which the subject is concerned and then to be faced with a collection of problems involving the terms just defined. In adopting this approach with my Dundee students of set theory and general topology I found it best not to differentiate too precisely between simple illustrative examples, easy exercises and results which in conventional textbooks would be labelled as Theorems. |

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### Contents

FIRST AXIOMS OF THE THEORY NBG | 7 |

RELATIONS | 13 |

FUNCTIONAL RELATIONS AND MAPPINGS | 19 |

FAMILIES OF SETS | 29 |

EQUIVALENCE RELATIONS | 35 |

ORDER RELATIONS | 41 |

WELLORDERING | 47 |

ORDINALS | 51 |

ANSWERS TO CHAPTER 2 | 81 |

ANSWERS TO CHAPTER 3 | 85 |

ANSWERS TO CHAPTER 4 | 95 |

ANSWERS TO CHAPTER 5 | 101 |

ANSWERS TO CHAPTER 6 | 107 |

ANSWERS TO CHAPTER 7 | 111 |

ANSWERS TO CHAPTER 8 | 115 |

ANSWERS TO CHAPTER 9 | 121 |

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### Common terms and phrases

A C B A U B ANSWERS TO CHAPTER Axiom of Choice Axiom of Extensionality Axiom of Foundation bijection C)-maximal canonical surjection Card P(a cardinals Choice holds codomain contradiction Conversely deduce define a mapping defined by setting denote disjoint sets domain equinumerous equivalence relation Exercise 30 f and g f is injective f is surjective finite character ft-least functional relation G ft g o f i?-least element i?-maximal element i?-upper bound inductively ordered initial segment Let E,R Let f mapping f mapping g mapping h natural number non-empty sets non-empty subset ordered pairs ordered set ordinal number Power Set Axiom proper subset property of finite reflexive relation of well-ordering respectively result of Exercise set theory sets equipotent symmetric relation including total order transitive unique isomorphism well-ordered sets Xi x X2 Zorn's Lemma