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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Basic Mathematics
|title=Basic Mathematics
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|image=[[File:Lawvere Sets for Mathematics Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|author=[https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere]
|language=English
|language=English
|series=
|series=
|genre=
|genre=
|publisher=Springer
|publisher=Cambridge University Press
|publicationdate=1 July 1988
|publicationdate=10 April 2003
|pages=496
|pages=276
|isbn10=0387967877
|isbn10=0521010608
|isbn13=978-0387967875
|isbn13=978-0521010603
}}
}}
The textbook '''''Basic Mathematics''''' by [https://en.wikipedia.org/wiki/Serge_Lang Serge Lang] provides an overview of mathematical topics usually encountered through the end of high school/secondary school, specifically arithmetic, algebra, trigonometry, logic, and geometry. It serves as a solid review no matter how far along one may be in their studies, be it just beginning or returning to strengthen one's foundations.
The textbook '''''Sets for Mathematics''''' by [https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere] uses categorical algebra to introduce set theory.


Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.
In parallel to Grothendieck, Lawvere developed the notion of a topos as a collection of objects/points behaving as sets and arrows as maps between sets. The utility of this is for a characterization of sets via mappings only - there is a unique (equivalence class of) set(s) with one element that can alternatively be described as only receiving one map from each other set. The number of maps in the other direction count the elements of sets conversely. Two element sets are an instance of "subobject classifiers" in the topos of sets such that the maps into them correspond to subsets of the source set of the map. The language of toposes is particularly accessible here, and plays a universal role in modern mathematics, e.g. for toposes emerging from sets parametrized by a topological space (presheaf toposes) even inspiring functional programming languages such as Haskell due to the logical properties of toposes.


== Table of Contents ==
== Table of Contents ==
Line 23: Line 22:
! Chapter/Section # !! Title !! Page #
! Chapter/Section # !! Title !! Page #
|-  
|-  
! colspan="3" | PART I: ALGEBRA
! colspan="2" | Foreword || ix
|-
|-
! colspan="3" | Chapter 1: Numbers
! colspan="2" | Contributors to Sets for Mathematics || xiii
|-
|-
| 1 || The integers || 5
! colspan="3" | 1. Abstract Sets and Mappings
|-
|-
| 2 || Rules for addition || 8
| 1.1 || Sets, Mappings, and Composition || 1
|-
|-
| 3 || Rules for multiplication || 14
| 1.2 || Listings, Properties, and Elements || 4
|-
|-
| 4 || Even and odd integers; divisibility || 22
| 1.3 || Surjective and Injective Mappings || 8
|-
|-
| 5 || Rational numbers || 26
| 1.4 || Associativity and Categories || 10
|-
|-
| 6 || Multiplicative inverses || 42
| 1.5 || Separators and the Empty Set || 11
|-
| 1.6 || Generalized Elements || 15
|-  
|-  
! colspan="3" | Chapter 2: Linear Equations
| 1.7 || Mappings as Properties || 17
|-
|-  
| 1 || Equations in two unknowns || 53
| 1.8 || Additional Exercises || 23
|-
|-  
| 2 || Equations in three unknowns || 57
! colspan="3" | 2. Sums, Monomorphisms, and Parts
|-
! colspan="3" | Chapter 3: Real Numbers
|-
| 1 || Addition and multiplication || 61
|-
| 2 || Real numbers: positivity || 64
|-
| 3 || Powers and roots || 70
|-
| 4 || Inequalities || 75
|-
! colspan="3" | Chapter 4: Quadratic Equations
|-
! colspan="3" | Interlude: On Logic and Mathematical Expressions
|-
| 1 || On reading books || 93
|-
| 2 || Logic || 94
|-
| 3 || Sets and elements || 99
|-
| 4 || Notation || 100
|-
! colspan="3" | PART II: INTUITIVE GEOMETRY
|-
|-
! colspan="3" | Chapter 5: Distance and Angles
| 2.1 || Sum as a Universal Property || 26
|-
|-
| 1 || Distance || 107
| 2.2 || Monomorphisms and Parts || 32
|-
|-
| 2 || Angles || 110
| 2.3 || Inclusion and Membership || 34
|-
|-
| 3 || The Pythagoras theorem || 120
| 2.4 || Characteristic Functions || 38
|-
|-
! colspan="3" | Chapter 6: Isometries
| 2.5 || Inverse Image of a Part || 40
|-
|-
| 1 || Some standard mappings of the plane || 133
| 2.6 || Additional Exercises || 44
|-
|-
| 2 || Isometries || 143
! colspan="3" | 3. Finite Inverse Limits
|-
|-
| 3 || Composition of isometries || 150
| 3.1 || Retractions || 48
|-
|-
| 4 || Inverse of isometries || 155
| 3.2 || Isomorphism and Dedekind Finiteness || 54
|-
|-
| 5 || Characterization of isometries || 163
| 3.3 || Cartesian Products and Graphs || 58
|-
|-
| 6 || Congruences || 166
| 3.4 || Equalizers || 66
|-
|-
! colspan="3" | Chapter 7: Area and Applications
| 3.5 || Pullbacks || 69
|-
|-
| 1 || Area of a disc of radius ''r'' || 173
| 3.6 || Inverse Limits || 71
|-
|-
| 2 || Circumference of a circle of radius ''r'' || 180
| 3.7 || Additional Exercises || 75
|-
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
! colspan="3" | Colimits, Epimorphisms, and the Axiom of Choice
|-
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
| 4.1 || Colimits are Dual to Limits || 78
|-
|-
| 1 || Coordinate systems || 191
| 4.2 || Epimorphisms and Split Surjections || 80
|-
|-
| 2 || Distance between points || 197
| 4.3 || The Axiom of Choice || 84
|-
|-
| 3 || Equation of a circle || 203
| 4.4 || Partitions and Equivalence Relations || 85
|-
|-
| 4 || Rational points on a circle || 206
| 4.5 || Split Images || 89
|-
|-
! colspan="3" | Chapter 9: Operations on Points
| 4.6 || The Axiom of Choice as the Distinguishing Property of Constant/Random Sets || 92
|-
|-
| 1 || Dilations and reflections || 213
| 4.7 || Additional Exercises || 94
|-
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
! colspan="3" | 5. Mapping Sets and Exponentials
|-
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
| 5.1 || Natural Bijection and Functoriality || 96
|-
|-
| 1 || Segments || 229
| 5.2 || Exponentiation || 98
|-
|-
| 2 || Rays || 231
| 5.3 || Functoriality of Function Spaces || 102
|-
|-
| 3 || Lines || 236
| 5.4 || Additional Exercises || 108
|-
|-
| 4 || Ordinary equation for a line || 246
! colspan="3" | 6. Summary of the Axioms and an Example of Variable Sets
|-
|-
! colspan="3" | Chapter 11: Trigonometry
| 6.1 || Axioms for Abstract Sets and Mappings || 111
|-
|-
| 1 || Radian measure || 249
| 6.2 || Truth Values for Two-Stage Variable Sets || 114
|-
|-
| 2 || Sine and cosine || 252
| 6.3 || Additional Exercises || 117
|-
|-
| 3 || The graphs || 264
! colspan="3" | 7. Consequences and Uses of Exponentials
|-
|-
| 4 || The tangent || 266
| 7.1 || Concrete Duality: The Behavior of Monics and Epics under the Contravariant Functoriality of Exponentiation || 120
|-
|-
| 5 || Addition formulas || 272
| 7.2 || The Distributive Law || 126
|-
|-
| 6 || Rotations || 277
| 7.3 || Cantor's Diagonal Argument || 129
|-
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
| 7.4 || Additional Exercises || 134
|-
|-
| 1 || The straight line again || 281
! colspan="3" | 8. More on Power Sets
|-
|-
| 2 || The parabola || 291
| 8.1 || Images || 136
|-
|-
| 3 || The ellipse || 297
| 8.2 || The Covariant Power Set Functor || 141
|-
|-
| 4 || The hyperbola || 300
| 8.3 || The Natural Map <math>Placeholder</math> || 145
|-
|-
| 5 || Rotation of hyperbolas || 305
| 8.4 || Measuring, Averaging, and Winning with <math>V</math>-Valued Quantities || 148
|-
|-
! colspan="3" | PART IV: MISCELLANEOUS
| 8.5 || Additional Exercises || 152
|-
|-
! colspan="3" | Chapter 13: Functions
! colspan="3" | 9. Introduction to Variable Sets
|-
|-
| 1 || Definition of a function || 313
| 9.1 || The Axiom of Infinity: Number Theory || 154
|-
|-
| 2 || Polynomial functions || 318
| 9.2 || Recursion || 157
|-
|-
| 3 || Graphs of functions || 330
| 9.3 || Arithmetic of <math>N</math> || 160
|-
|-
| 4 || Exponential function || 333
| 9.4 || Additional Exercises || 165
|-
|-
| 5 || Logarithms || 338
! colspan="3" | 10. Models of Additional Variation
|-
|-
! colspan="3" | Chapter 14: Mappings
| 10.1 || Monoids, Posets, and Groupoids || 167
|-
|-
| 1 || Definition || 345
| 10.2 || Actions || 171
|-
|-
| 2 || Formalism of mappings || 351
| 10.3 || Reversible Graphs || 176
|-
|-
| 3 || Permutations || 359
| 10.4 || Chaotic Graphs || 180
|-
|-
! colspan="3" | Chapter 15: Complex Numbers
| 10.5 || Feedback and Control || 186
|-
|-
| 1 || The complex plane || 375
| 10.6 || To and from Idempotents || 189
|-
|-
| 2 || Polar form || 380
| 10.7 || Additional Exercises || 191
|-
|-
! colspan="3" | Chapter 16: Induction and Summations
! colspan="3" | Appendixes
|-
|-
| 1 || Induction || 383
! colspan="3" | A. Logic as the Algebra of Parts
|-
|-
| 2 || Summations || 388
| A.0 || Why Study Logic? || 193
|-
|-
| 3 || Geometric series || 396
| A.1 || Basic Operators and Their Rules of Inference || 195
|-
|-
! colspan="3" | Chapter 17: Determinants
| A.2 || Fields, Nilpotents, Idempotents || 212
|-
|-
| 1 || Matrices || 401
! colspan="2" | B. Logic as the Algebra of Parts || 220
|-
|-
| 2 || Determinants of order 2 || 406
! colspan="3" | C. Definitions, Symbols, and the Greek Alphabet
|-
|-
| 3 || Properties of 2 x 2 determinants || 409
| C.1 || Definitions of Some Mathematical and Logical Concepts || 231
|-
|-
| 4 || Determinants of order 3 || 414
| C.2 || Mathematical Notations and Logical Symbols || 251
|-
|-
| 5 || Properties of 3 x 3 determinants || 418
| C.3 || The Greek Alphabet || 252
|-
|-
| 6 || Cramer's Rule || 424
! colspan="2" | Bibliography || 253
|-
|-
! colspan="2" | Index || 429
! colspan="2" | Index || 257
|-
|-
|}
|}


[[Category:Mathematics]]
[[Category:Mathematics]]

Latest revision as of 17:09, 19 February 2023

Basic Mathematics
Lawvere Sets for Mathematics Cover.jpg
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.

In parallel to Grothendieck, Lawvere developed the notion of a topos as a collection of objects/points behaving as sets and arrows as maps between sets. The utility of this is for a characterization of sets via mappings only - there is a unique (equivalence class of) set(s) with one element that can alternatively be described as only receiving one map from each other set. The number of maps in the other direction count the elements of sets conversely. Two element sets are an instance of "subobject classifiers" in the topos of sets such that the maps into them correspond to subsets of the source set of the map. The language of toposes is particularly accessible here, and plays a universal role in modern mathematics, e.g. for toposes emerging from sets parametrized by a topological space (presheaf toposes) even inspiring functional programming languages such as Haskell due to the logical properties of toposes.

Table of Contents[edit]

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 [math]\displaystyle{ Placeholder }[/math] 145
8.4 Measuring, Averaging, and Winning with [math]\displaystyle{ V }[/math]-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 [math]\displaystyle{ N }[/math] 160
9.4 Additional Exercises 165
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
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
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