Sets for Mathematics

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Cambridge University Press, 27.01.2003 - 261 Seiten
Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion.
 

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Inhalt

Abstract Sets and Mappings
1
12 Listings Properties and Elements
4
13 Surjective and Injective Mappings
8
14 Associativity and Categories
10
15 Separators and the Empty Set
11
16 Generalized Elements
15
17 Mappings as Properties
17
18 Additional Exercises
23
62 Truth Values for TwoStage Variable Sets
114
63 Additional Exercises
117
Consequences and Uses of Exponentials
120
72 The Distributive Law
126
73 Cantors Diagonal Argument
129
74 Additional Exercises
134
More on Power Sets
136
82 The Covariant Power Set Functor
141

Sums Monomorphisms and Parts
26
22 Monomorphisms and Parts
32
23 Inclusion and Membership
34
24 Characteristic Functions
38
25 Inverse Image of a Part
40
26 Additional Exercises
44
Finite Inverse Limits
48
32 Isomorphism and Dedekind Finiteness
54
33 Cartesian Products and Graphs
58
34 Equalizers
66
35 Fullbacks
69
36 Inverse Limits
71
37 Additional Exercises
75
Colimits Epimorphisms and the Axiom of Choice
78
42 Epimorphisms and Split Surjections
80
43 The Axiom of Choice
84
44 Partitions and Equivalence Relations
85
45 Split Images
89
46 The Axiom of Choice as the Distinguishing Property of ConstantRandom Sets
92
47 Additional Exercises
94
Mapping Sets and Exponentials
96
52 Exponentiation
98
53 Functoriality of Function Spaces
102
54 Additional Exercises
108
Summary of the Axioms and an Example of Variable Sets
111
83 The Natural Map PX2²ˣ
145
84 Measuring Averaging and Winning with VValued Quantities
148
85 Additional Exercises
152
Introduction to Variable Sets
154
92 Recursion
157
93 Arithmetic of N
160
94 Additional Exercises
165
Models of Additional Variation
167
102 Actions
171
103 Reversible Graphs
176
104 Chaotic Graphs
180
105 Feedback and Control
186
106 To and from Idempotents
189
107 Additional Exercises
191
Logic as the Algebra of Parts
193
A1 Basic Operators and Their Rules of Inference
195
A2 Fields Nilpotents Idempotents
212
The Axiom of Choice and Maximal Principles
220
Definitions Symbols and the Greek Alphabet
231
C2 Mathematical Notations and Logical Symbols
251
C3 The Greek Alphabet
252
Bibliography
253
Index
257
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Sets for Mathematics
F. William Lawvere,Robert Rosebrugh
Keine Leseprobe verfügbar - 2003

Häufige Begriffe und Wortgruppen

A₁ abstract sets Additional Exercises adjoint algebra arbitrary maps arrows axiom of choice Boolean called Cantor category of abstract category of sets characteristic function codomain cograph colimits commutative composition consider construction defined Definition denoted diagonal diagram domain dual elements endomap epimorphism equal equations equivalent example exists exponentiation fiber finite limits functor given graph hence Hint homomorphism idempotent implies injective inverse image isomorphism linear logic M-sets mapping ƒ mathematical Maximal Principle means monoid monomapping monomorphism morphisms natural transformation nilpotent notation one-element operation pair parameterization partially ordered sets poset Proof Proposition prove pullback real numbers relation retraction right A-set rule of inference satisfies set theory sets and mappings Show slice category specific statement structure surjective t₁ terminal object theorem topos true truth values unique map universal mapping property variable sets vector space X₁ Y₁

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Bibliografische Informationen

TitelSets for Mathematics
AutorenF. William Lawvere, Robert Rosebrugh
Ausgabeillustriert
VerlagCambridge University Press, 2003
ISBN0521010608, 9780521010603
Länge261 Seiten
  
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