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Full Description
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, problemsand insight panels.
Contents
Introduction.- Part I: Basics.- Membership and Edge Relations.- Sets, Graphs, and Set Universes.- Part II: Graphs as Sets.- The Undirected Structure Underlying Sets.- Graphs as Transitive Sets.- Part III: Sets as Graphs.- Counting and Encoding Sets.- Random Generation of Sets.- Infinite Sets and Finite Combinatorics.- Appendix: Excerpts from a Referee-Checked Proof-Script.
