# Sets for Mathematics (Book)

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Basic Mathematics | |

Information | |
---|---|

Author | F. William Lawvere |

Language | English |

Publisher | Cambridge University Press |

Publication Date | 10 April 2003 |

Pages | 276 |

ISBN-10 | 0521010608 |

ISBN-13 | 978-0521010603 |

The textbook * Sets for Mathematics* by F. William Lawvere uses categorical algebra to introduce set theory.

## Table of Contents[edit | edit source]

Chapter/Section # | Title | Page # |
---|---|---|

Foreword | ix | |

Contributors to Sets for Mathematics | xiii | |

1. Abstract Sets and Mappings | ||

1.1 | Sets, Mappings, and Composition | 1 |

1.2 | Listings, Properties, and Elements | 4 |

1.3 | Surjective and Injective Mappings | 8 |

1.4 | Associativity and Categories | 10 |

1.5 | Separators and the Empty Set | 11 |

1.6 | Generalized Elements | 15 |

1.7 | Mappings as Properties | 17 |

1.8 | Additional Exercises | 23 |

2. Sums, Monomorphisms, and Parts | ||

2.1 | Sum as a Universal Property | 26 |

2.2 | Monomorphisms and Parts | 32 |

2.3 | Inclusion and Membership | 34 |

2.4 | Characteristic Functions | 38 |

2.5 | Inverse Image of a Part | 40 |

2.6 | Additional Exercises | 44 |

3. Finite Inverse Limits | ||

3.1 | Retractions | 48 |

3.2 | Isomorphism and Dedekind Finiteness | 54 |

3.3 | Cartesian Products and Graphs | 58 |

3.4 | Equalizers | 66 |

3.5 | Pullbacks | 69 |

3.6 | Inverse Limits | 71 |

3.7 | Additional Exercises | 75 |

Colimits, Epimorphisms, and the Axiom of Choice | ||

4.1 | Colimits are Dual to Limits | 78 |

4.2 | Epimorphisms and Split Surjections | 80 |

4.3 | The Axiom of Choice | 84 |

4.4 | Partitions and Equivalence Relations | 85 |

4.5 | Split Images | 89 |

4.6 | The Axiom of Choice as the Distinguishing Property of Constant/Random Sets | 92 |

4.7 | Additional Exercises | 94 |

5. Mapping Sets and Exponentials | ||

5.1 | Natural Bijection and Functoriality | 96 |

5.2 | Exponentiation | 98 |

5.3 | Functoriality of Function Spaces | 102 |

5.4 | Additional Exercises | 108 |

6. Summary of the Axioms and an Example of Variable Sets | ||

6.1 | Axioms for Abstract Sets and Mappings | 111 |

6.2 | Truth Values for Two-Stage Variable Sets | 114 |

6.3 | Additional Exercises | 117 |

7. Consequences and Uses of Exponentials | ||

7.1 | Concrete Duality: The Behavior of Monics and Epics under the Contravariant Functoriality of Exponentiation | 120 |

7.2 | The Distributive Law | 126 |

7.3 | Cantor's Diagonal Argument | 129 |

7.4 | Additional Exercises | 134 |

8. More on Power Sets | ||

8.1 | Images | 136 |

8.2 | The Covariant Power Set Functor | 141 |

8.3 | The Natural Map \(Placeholder\) | 145 |

8.4 | Measuring, Averaging, and Winning with \(V\)-Valued Quantities | 148 |

8.5 | Additional Exercises | 152 |

9. Introduction to Variable Sets | ||

9.1 | The Axiom of Infinity: Number Theory | 154 |

9.2 | Recursion | 157 |

9.3 | Arithmetic of \(N\) | 160 |

9.4 | Additional Exercises | 165 |

10. Models of Additional Variation | ||

10.1 | Monoids, Podsets, and Groupoids | 167 |

10.2 | Actions | 171 |

10.3 | Reversible Graphs | 176 |

10.4 | Chaotic Graphs | 180 |

10.5 | Feedback and Control | 186 |

10.6 | To and from Idempotents | 189 |

10.7 | Additional Exercises | 191 |

Appendixes | ||

A. Logic as the Algebra of Parts | ||

A.0 | Why Study Logic? | 193 |

A.1 | Basic Operators and Their Rules of Inference | 195 |

A.2 | Fields, Nilpotents, Idempotents | 212 |

B. Logic as the Algebra of Parts | 220 | |

C. Definitions, Symbols, and the Greek Alphabet | ||

C.1 | Definitions of Some Mathematical and Logical Concepts | 231 |

C.2 | Mathematical Notations and Logical Symbols | 251 |

C.3 | The Greek Alphabet | 252 |

Bibliography | 253 | |

Index | 257 |