Sets for Mathematics (Book) - Revision history

Aardvark at 17:09, 19 February 2023

2023-02-19T17:09:52Z

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	! colspan="3" \| 10. Models of Additional Variation		! colspan="3" \| 10. Models of Additional Variation
	\|-		\|-
	\| 10.1 \|\| Monoids, ~~Podsets~~, and Groupoids \|\| 167		\| 10.1 \|\| Monoids, Posets, and Groupoids \|\| 167
	\|-		\|-
	\| 10.2 \|\| Actions \|\| 171		\| 10.2 \|\| Actions \|\| 171

Aardvark at 17:06, 19 February 2023

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	\| 8.2 \|\| The Covariant Power Set Functor \|\| 141		\| 8.2 \|\| The Covariant Power Set Functor \|\| 141
	\|-		\|-
	\| 8.3 \|\| The Natural Map \(Placeholder\) \|\| 145		\| 8.3 \|\| The Natural Map <math>Placeholder</math> \|\| 145
	\|-		\|-
	\| 8.4 \|\| Measuring, Averaging, and Winning with \(V\)-Valued Quantities \|\| 148		\| 8.4 \|\| Measuring, Averaging, and Winning with <math>V</math>-Valued Quantities \|\| 148
	\|-		\|-
	\| 8.5 \|\| Additional Exercises \|\| 152		\| 8.5 \|\| Additional Exercises \|\| 152
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	\| 9.2 \|\| Recursion \|\| 157		\| 9.2 \|\| Recursion \|\| 157
	\|-		\|-
	\| 9.3 \|\| Arithmetic of \(N\) \|\| 160		\| 9.3 \|\| Arithmetic of <math>N</math> \|\| 160
	\|-		\|-
	\| 9.4 \|\| Additional Exercises \|\| 165		\| 9.4 \|\| Additional Exercises \|\| 165

Aardvark at 15:44, 15 February 2023

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	~~{{Stub}}~~
	{{InfoboxBook		{{InfoboxBook
	\|title=Basic Mathematics		\|title=Basic Mathematics

Anisomorphism: added a description of the book and a brief sense of topos theory

2023-02-15T03:26:27Z

added a description of the book and a brief sense of topos theory

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	}}		}}
	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 '''''Sets for Mathematics''''' by [https://en.wikipedia.org/wiki/William_Lawvere 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 ==		== Table of Contents ==

Aardvark at 18:29, 20 September 2021

2021-09-20T18:29:29Z

Show changes

Aardvark: Created page with "{{subst::Basic Mathematics (Book)}}"

2021-09-20T18:02:57Z

Created page with "{{subst::Basic Mathematics (Book)}}"

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{{Stub}}
{{InfoboxBook
|title=Basic Mathematics
|image=[[File:Lang Basic Mathematics Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Serge_Lang Serge Lang]
|language=English
|series=
|genre=
|publisher=Springer
|publicationdate=1 July 1988
|pages=496
|isbn10=0387967877
|isbn13=978-0387967875
}}
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.

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

== Table of Contents ==

{| class="wikitable"
|-
! Chapter/Section # !! Title !! Page #
|-
! colspan="3" | PART I: ALGEBRA
|-
! colspan="3" | Chapter 1: Numbers
|-
| 1 || The integers || 5
|-
| 2 || Rules for addition || 8
|-
| 3 || Rules for multiplication || 14
|-
| 4 || Even and odd integers; divisibility || 22
|-
| 5 || Rational numbers || 26
|-
| 6 || Multiplicative inverses || 42
|-
! colspan="3" | Chapter 2: Linear Equations
|-
| 1 || Equations in two unknowns || 53
|-
| 2 || Equations in three unknowns || 57
|-
! 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
|-
| 1 || Distance || 107
|-
| 2 || Angles || 110
|-
| 3 || The Pythagoras theorem || 120
|-
! colspan="3" | Chapter 6: Isometries
|-
| 1 || Some standard mappings of the plane || 133
|-
| 2 || Isometries || 143
|-
| 3 || Composition of isometries || 150
|-
| 4 || Inverse of isometries || 155
|-
| 5 || Characterization of isometries || 163
|-
| 6 || Congruences || 166
|-
! colspan="3" | Chapter 7: Area and Applications
|-
| 1 || Area of a disc of radius ''r'' || 173
|-
| 2 || Circumference of a circle of radius ''r'' || 180
|-
! colspan="3" | PART III: COORDINATE GEOMETRY
|-
! colspan="3" | Chapter 8: Coordinates and Geometry
|-
| 1 || Coordinate systems || 191
|-
| 2 || Distance between points || 197
|-
| 3 || Equation of a circle || 203
|-
| 4 || Rational points on a circle || 206
|-
! colspan="3" | Chapter 9: Operations on Points
|-
| 1 || Dilations and reflections || 213
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
|-
| 1 || Segments || 229
|-
| 2 || Rays || 231
|-
| 3 || Lines || 236
|-
| 4 || Ordinary equation for a line || 246
|-
! colspan="3" | Chapter 11: Trigonometry
|-
| 1 || Radian measure || 249
|-
| 2 || Sine and cosine || 252
|-
| 3 || The graphs || 264
|-
| 4 || The tangent || 266
|-
| 5 || Addition formulas || 272
|-
| 6 || Rotations || 277
|-
! colspan="3" | Chapter 12: Some Analytic Geometry
|-
| 1 || The straight line again || 281
|-
| 2 || The parabola || 291
|-
| 3 || The ellipse || 297
|-
| 4 || The hyperbola || 300
|-
| 5 || Rotation of hyperbolas || 305
|-
! colspan="3" | PART IV: MISCELLANEOUS
|-
! colspan="3" | Chapter 13: Functions
|-
| 1 || Definition of a function || 313
|-
| 2 || Polynomial functions || 318
|-
| 3 || Graphs of functions || 330
|-
| 4 || Exponential function || 333
|-
| 5 || Logarithms || 338
|-
! colspan="3" | Chapter 14: Mappings
|-
| 1 || Definition || 345
|-
| 2 || Formalism of mappings || 351
|-
| 3 || Permutations || 359
|-
! colspan="3" | Chapter 15: Complex Numbers
|-
| 1 || The complex plane || 375
|-
| 2 || Polar form || 380
|-
! colspan="3" | Chapter 16: Induction and Summations
|-
| 1 || Induction || 383
|-
| 2 || Summations || 388
|-
| 3 || Geometric series || 396
|-
! colspan="3" | Chapter 17: Determinants
|-
| 1 || Matrices || 401
|-
| 2 || Determinants of order 2 || 406
|-
| 3 || Properties of 2 x 2 determinants || 409
|-
| 4 || Determinants of order 3 || 414
|-
| 5 || Properties of 3 x 3 determinants || 418
|-
| 6 || Cramer's Rule || 424
|-
! colspan="2" | Index || 429
|-
|}

[[Category:Mathematics]]