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
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 |