Difference between revisions of "Sets for Mathematics (Book)"
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{{InfoboxBook | {{InfoboxBook | ||
|title=Basic Mathematics | |title=Basic Mathematics | ||
|image=[[File: | |image=[[File:Lawvere Sets for Mathematics Cover.jpg]] | ||
|author=[https://en.wikipedia.org/wiki/ | |author=[https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere] | ||
|language=English | |language=English | ||
|series= | |series= | ||
|genre= | |genre= | ||
|publisher= | |publisher=Cambridge University Press | ||
|publicationdate= | |publicationdate=10 April 2003 | ||
|pages= | |pages=276 | ||
|isbn10= | |isbn10=0521010608 | ||
|isbn13=978- | |isbn13=978-0521010603 | ||
}} | }} | ||
The textbook ''''' | The textbook '''''Sets for Mathematics''''' by [https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere] uses categorical algebra to introduce set theory. | ||
== Table of Contents == | == Table of Contents == | ||
Line 23: | Line 21: | ||
! Chapter/Section # !! Title !! Page # | ! Chapter/Section # !! Title !! Page # | ||
|- | |- | ||
! colspan=" | ! colspan="2" | Foreword || ix | ||
|- | |- | ||
! colspan=" | ! colspan="2" | Contributors to Sets for Mathematics || xiii | ||
|- | |- | ||
| 1 | ! colspan="3" | 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 | ||
|- | |- | ||
| 6 || | | 1.5 || Separators and the Empty Set || 11 | ||
|- | |||
| 1.6 || Generalized Elements || 15 | |||
|- | |- | ||
| 1.7 || Mappings as Properties || 17 | |||
|- | |||
| 1 || | | 1.8 || Additional Exercises || 23 | ||
|- | |||
! colspan="3" | 2. Sums, Monomorphisms, and Parts | |||
|- | |||
| 1 || | |||
| | |||
|- | |||
! colspan="3" | |||
| 2 | |||
|- | |- | ||
| 2.1 || Sum as a Universal Property || 26 | |||
|- | |- | ||
| | | 2.2 || Monomorphisms and Parts || 32 | ||
|- | |- | ||
| 2 || | | 2.3 || Inclusion and Membership || 34 | ||
|- | |- | ||
| | | 2.4 || Characteristic Functions || 38 | ||
|- | |- | ||
| 2.5 || Inverse Image of a Part || 40 | |||
|- | |- | ||
| | | 2.6 || Additional Exercises || 44 | ||
|- | |- | ||
| | ! colspan="3" | 3. Finite Inverse Limits | ||
|- | |- | ||
| 3 || | | 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 | ||
|- | |- | ||
! colspan="3" | | ! colspan="3" | 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 || | | 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 | ||
|- | |- | ||
| | ! colspan="3" | 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 | ||
|- | |- | ||
| | ! colspan="3" | 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 | ||
|- | |- | ||
! colspan="3" | 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 | |||
|- | |- | ||
| | ! colspan="3" | 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 | |||
|- | |- | ||
! colspan="3" | | ! colspan="3" | 9. Introduction to Variable Sets | ||
|- | |- | ||
| 1 || | | 9.1 || The Axiom of Infinity: Number Theory || 154 | ||
|- | |- | ||
| 2 || | | 9.2 || Recursion || 157 | ||
|- | |- | ||
| 3 || | | 9.3 || Arithmetic of \(N\) || 160 | ||
|- | |- | ||
| 4 || | | 9.4 || Additional Exercises || 165 | ||
|- | |- | ||
| | ! colspan="3" | 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 | ||
|- | |- | ||
! colspan="3" | | ! colspan="3" | Appendixes | ||
|- | |- | ||
| | ! colspan="3" | 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 | |||
|- | |- | ||
| | ! colspan="2" | B. Logic as the Algebra of Parts || 220 | ||
|- | |- | ||
| | ! colspan="3" | 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 | ||
|- | |- | ||
| | ! colspan="2" | Bibliography || 253 | ||
|- | |- | ||
! colspan="2" | Index || | ! colspan="2" | Index || 257 | ||
|- | |- | ||
|} | |} | ||
[[Category:Mathematics]] | [[Category:Mathematics]] |
Revision as of 18:29, 20 September 2021
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 |