Difference between revisions of "Sets for Mathematics (Book)"

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{{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.


== Table of Contents ==
== Table of Contents ==
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! 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 \(Placeholder\) || 145
|-
|-
| 5 || Rotation of hyperbolas || 305
| 8.4 || Measuring, Averaging, and Winning with \(V\)-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 \(N\) || 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, Podsets, 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]]