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{{Stub}}
{{InfoboxBook
{{InfoboxBook
|title=Basic Mathematics
|title=Basic Mathematics
|image=[[File:Lawvere Sets for Mathematics Cover.jpg]]
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|language=English
|language=English
|series=
|series=
|genre=
|genre=
|publisher=Cambridge University Press
|publisher=Springer
|publicationdate=10 April 2003
|publicationdate=1 July 1988
|pages=276
|pages=496
|isbn10=0521010608
|isbn10=0387967877
|isbn13=978-0521010603
|isbn13=978-0387967875
}}
}}
The textbook '''''Sets for Mathematics''''' by [https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere] uses categorical algebra to introduce set theory.
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.


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.
Reading the Foreword and the Interlude is recommended for those unfamiliar with reading math texts.


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


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