Editing Sets for Mathematics (Book)

{{InfoboxBook
|title=Basic Mathematics
|image=[[File:Lawvere Sets for Mathematics Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere]
|language=English
|series=
|genre=
|publisher=Cambridge University Press
|publicationdate=10 April 2003
|pages=276
|isbn10=0521010608
|isbn13=978-0521010603
}}
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 ==

{| class="wikitable"
|-
! Chapter/Section # !! Title !! Page #
|- 
! colspan="2" | Foreword || ix
|-
! colspan="2" | Contributors to Sets for Mathematics || xiii
|-
! 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
|-
| 1.5 || Separators and the Empty Set || 11
|-
| 1.6 || Generalized Elements || 15
|- 
| 1.7 || Mappings as Properties || 17
|- 
| 1.8 || Additional Exercises || 23
|- 
! colspan="3" | 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
|-
! colspan="3" | 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
|-
! 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
|-
| 4.5 || 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 <math>Placeholder</math> || 145
|-
| 8.4 || Measuring, Averaging, and Winning with <math>V</math>-Valued Quantities || 148
|-
| 8.5 || Additional Exercises || 152
|-
! colspan="3" | 9. Introduction to Variable Sets
|-
| 9.1 || The Axiom of Infinity: Number Theory || 154
|-
| 9.2 || Recursion || 157
|-
| 9.3 || Arithmetic of <math>N</math> || 160
|-
| 9.4 || Additional Exercises || 165
|-
! colspan="3" | 10. Models of Additional Variation
|-
| 10.1 || Monoids, Posets, 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" | 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 || 257
|-
|}

[[Category:Mathematics]]