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E-raamat: On Sets and Graphs: Perspectives on Logic and Combinatorics

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-May-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319549811
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On Sets and Graphs: Perspectives on Logic and CombinatoricsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 11-May-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319549811
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This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels.
1 Introduction
1(30)
1.1 Tiny Motivating Examples
4(2)
1.2 Why Sets as Graphs?
6(12)
1.3 Why Graphs as Sets?
18(13)
Part I Basics
2 Membership and Edge Relations
31(28)
2.1 Edge/Membership Languages and Their Structures
32(4)
2.1.1 Terms, Formulae, and Sentences
32(4)
2.1.2 Interpretations vs. Models
36(1)
2.2 Set Theories
36(12)
2.2.1 Axioms
36(5)
2.2.2 Extended Syntax
41(1)
2.2.3 Setformers: A Perspicuous Set-Abstraction Construct
42(4)
2.2.4 Transitive Sets
46(1)
2.2.5 Anti-Foundation
47(1)
2.3 Graph Theories
48(11)
2.3.1 Directed Graphs
51(3)
2.3.2 Undirected Graphs
54(2)
Exercises
56(3)
3 Sets, Graphs, and Set Universes
59(42)
3.1 Hereditarily Finite Sets and Hypersets
64(2)
3.2 The Full-Fledged Cumulative Hierarchy
66(4)
3.3 The Ackermann Encoding of Hereditarily Finite Sets
70(2)
3.4 Sets and Hypersets as Membership Graphs
72(11)
3.5 Graphs for Set-Formulae
83(18)
3.5.1 Ground-Level Satisfaction
83(4)
3.5.2 Quantified-Level Satisfaction
87(8)
Exercises
95(6)
Part II Graphs as Sets
4 The Undirected Structure Underlying Sets
101(28)
4.1 Set Graphs
103(2)
4.2 Graph Classes Included in the Class of Set Graphs
105(4)
4.3 Characterizations of Set Graphs in Some Graph Classes
109(5)
4.3.1 Unicyclic Graphs
109(2)
4.3.2 Complete Multipartite Graphs
111(1)
4.3.3 Block Graphs
112(1)
4.3.4 A Generalization of Claw-Free Graphs and Block Graphs
113(1)
4.4 Set Graph Transformations
114(3)
4.5 Computational Complexity
117(12)
4.5.1 Recognizing Set Graphs
117(4)
4.5.2 Hyperextensional Orientations
121(3)
Exercises
124(5)
5 Graphs as Transitive Sets
129(46)
5.1 Proofs + Sets = Checkable Proofs
131(3)
5.2 Formalizing Our Mathematical Roadmap
134(14)
5.2.1 Acyclic Orientations and Extensionalization
137(2)
5.2.2 Injective Digraph Decorations
139(2)
5.2.3 Claw-Free Graphs Mirrored into Transitive Finite Sets
141(4)
5.2.4 Perfect Matchings, Hamiltonian Cycles, and Claw-Freeness
145(3)
5.3 Set-Based Proof Verification
148(20)
5.3.1 The Referee System
149(8)
5.3.2 Proof-Checking on Graphs as Transitive Sets
157(11)
5.4 Conclusions About Our Proof-Checking Experiment
168(7)
Exercises
170(5)
Part III Sets as Graphs
6 Counting and Encoding Sets
175(26)
6.1 Counting Sets as Graphs
176(5)
6.2 A New Look at the Ackermann Order
181(7)
6.2.1 Successive Partition Refinements
184(1)
6.2.2 Computational Complexity
185(3)
6.3 Encoding Sets and Hypersets
188(13)
6.3.1 Peddicord's Encoding of Transitive Sets
188(2)
6.3.2 An Ackermann-Like Ordering (and Encoding) of Hypersets
190(7)
Exercises
197(4)
7 Random Generation of Sets
201(16)
7.1 Approaches Based on Combinatorial Decompositions
202(8)
7.1.1 The Recursive Method
207(1)
7.1.2 Ranking and Unranking
208(2)
7.2 A Markov Chain Monte Carlo Approach
210(7)
Exercises
215(2)
8 Infinite Sets and Finite Combinatorics
217(34)
8.1 The Bernays-Schonfinkel-Ramsey Class of Formulae
218(3)
8.2 Taming Infinite Set Models
221(23)
8.2.1 Stating Infinity by Means of a BSR Sentence
221(4)
8.2.2 Stating Infinity, Under Different Set-Theoretic Axioms
225(2)
8.2.3 Basic Pattern for an Infinite Well-Founded Spiral
227(3)
8.2.4 Well-Founded Infinities, Under (Anti-)Foundation or Without It
230(3)
8.2.5 Basic Pattern for an Infinite Non-well-founded Spiral
233(4)
8.2.6 Non-well-founded, or Just Ill-Founded, Infinities
237(7)
8.3 The Spectrum of a BSR Formula
244(7)
Exercises
250(1)
Appendix A Excerpts from a Referee-Checked Proof-Script
251(12)
A.1 A Tricky Proof that Connected Graphs Have Non-cut Vertices
252(2)
A.2 Ref's Proof that Connected Graphs Have Non-cut Vertices
254(9)
Exercises
261(2)
List of Symbols 263(2)
References 265(6)
Index 271
Dr. Eugenio G. Omodeo is a professor in the Department of Mathematics and Geosciences at the University of Trieste, Italy. His other publications include the Springer title Computational Logic and Set Theory.

Dr. Alberto Policriti is a Professor of Computer Science in the Department of Mathematics, Computer Science, and Physics at the University of Udine, Italy. Together with Dr. Eugenio G. Omodeo, he is co-author of the Springer title Set Theory for Computing.

Dr. Alexandru I. Tomescu is a postdoctoral researcher in the Department of Computer Science at the University of Helsinki, Finland.
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