Sets for mathematics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2003
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 261 S. graph. Darst. |
| ISBN: | 0521804442 0521010608 |
Staff View
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| 245 | 1 | 0 | |a Sets for mathematics |c F. William Lawvere ; Robert Rosebrugh |
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Record in the Search Index
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| adam_text | Contents
Foreword page ix
Contributors to Sets for Mathematics xiii
1 Abstract Sets and Mappings 1
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 26
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 48
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
v
vi Contents
4 Colimits, Epimorphisms, and the Axiom of Choice 78
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 96
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 111
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 120
7.1 Concrete Duality: The Behavior of Monies 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 136
8.1 Images 136
8.2 The Covariant Power Set Functor 141
8.3 The Natural Map VX —*22 145
8.4 Measuring, Averaging, and Winning with V Valued Quantities 148
8.5 Additional Exercises 152
9 Introduction to Variable Sets 154
9.1 The Axiom of Infinity: Number Theory 154
9.2 Recursion 157
9.3 Arithmetic of N 160
9.4 Additional Exercises 165
10 Models of Additional Variation 167
10.1 Monoids, Posets, and Groupoids 167
10.2 Actions 171
10.3 Reversible Graphs 176
10.4 Chaotic Graphs 180
Contents vii
10.5 Feedback and Control 186
10.6 To and from Idempotents 189
10.7 Additional Exercises 191
Appendixes 193
A Logic as the Algebra of Parts 193
A.O Why Study Logic? 193
A. 1 Basic Operators and Their Rules of Inference 195
A.2 Fields, Nilpotents, Idempotents 212
B The Axiom of Choice and Maximal Principles 220
C Definitions, Symbols, and the Greek Alphabet 231
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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| author | Lawvere, Francis W. 1937-2023 Rosebrugh, Robert |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
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| dewey-search | 511.3/22 511.322 |
| dewey-sort | 3511.3 222 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T19:03:10Z |
| institution | BVB |
| isbn | 0521804442 0521010608 |
| language | English |
| lccn | 2002071478 |
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| physical | XIII, 261 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Cambridge Univ. Press |
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| spelling | Lawvere, Francis W. 1937-2023 Verfasser (DE-588)108079104 aut Sets for mathematics F. William Lawvere ; Robert Rosebrugh 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2003 XIII, 261 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Ensembles, Théorie des Verzamelingen (wiskunde) gtt Set theory Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 s DE-604 Rosebrugh, Robert Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009884576&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lawvere, Francis W. 1937-2023 Rosebrugh, Robert Sets for mathematics Ensembles, Théorie des Verzamelingen (wiskunde) gtt Set theory Mengenlehre (DE-588)4074715-3 gnd |
| subject_GND | (DE-588)4074715-3 |
| title | Sets for mathematics |
| title_auth | Sets for mathematics |
| title_exact_search | Sets for mathematics |
| title_full | Sets for mathematics F. William Lawvere ; Robert Rosebrugh |
| title_fullStr | Sets for mathematics F. William Lawvere ; Robert Rosebrugh |
| title_full_unstemmed | Sets for mathematics F. William Lawvere ; Robert Rosebrugh |
| title_short | Sets for mathematics |
| title_sort | sets for mathematics |
| topic | Ensembles, Théorie des Verzamelingen (wiskunde) gtt Set theory Mengenlehre (DE-588)4074715-3 gnd |
| topic_facet | Ensembles, Théorie des Verzamelingen (wiskunde) Set theory Mengenlehre |
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