SET THEORY AND ITS APPLICATIONS IN PHYSICS AND COMPUTING

Why learn set theory? This book provides the answer — it is interesting, and also useful! Taking a new approach and looking from a fresh perspective, the discussion flows in a friendly and transparent way, supplemented with a lot of examples and figures. This makes the theory easily comprehensible: the proofs get vivid and visual, enveloped with interesting applications for students in (applied) math, physics, and engineering.

Given the theory and the applications, the book could serve as a textbook in four (undergraduate) math courses: Introduction to set theory and its application; Chaos theory and stability — a geometrical point of view; Functional analysis — Han-Banach theory; and Cryptography with quantum computing. It teaches set theory from the basics, including the axiom of choice, the well ordering theorem, and Zorn's lemma. Furthermore, it uses Cantor's set to introduce chaos theory from a geometrical point of view. Moreover, it introduces the binomial formula (and other related formulas), and uses them in quantum statistical mechanics. And finally, it uses Zorn's lemma in functional analysis, general relativity, and quantum mechanics. There are also practical applications in cryptography, error correction, quantum computing and programming.

Contents:
Introduction to Set Theory:Sets and Their CardinalityOrdinals and Zorn's LemmaApplications in Functional Analysis:Zorn's Lemma in Han-Banach TheoryCantor Set and Stability:The Pigeonhole Principle and Stability and Its Applications in Calculus and Classical MechanicsCantor Set and Its ApplicationsIs the Universe Infinite?Binary Trees and Chaos TheoryEntropy and InformationThe Binomial Formula and Quantum Statistical Mechanics:Newton's Binomial and Trinomial FormulasApplications in Quantum Statistical MechanicsTowards General Relativity and Quantum Mechanics:Spacetime and Local CoordinatesZorn's Lemma in Quantum MechanicsApplications in Cryptography and Error Correction:Coding–Decoding: the RSA Key ExchangeFast Fourier Transform: A Virtual Binary TreeError Correction: The Reed-Solomon CodeApplication in Computational BiologyTowards Quantum Computing:Quantum FFTShor's Factoring AlgorithmTowards Feynman DiagramsAppendix: Applications in C++:Dynamic Trees and Tensors in C++Nonlinear Maxwell Solver in C++ReferencesIndex
Readership: Advanced undergraduate and graduate students of set theory and its applications in chaos, functional analysis, cryptography, and related courses.

Key Features: Takes a new approach, looking at things from a fresh angle: not only theoretical, but also practical The discussion flows in a friendly and transparent way, supplemented with a lot of examples and figures Serve as a textbook in a few (undergraduate) math courses, thanks to the theory and the applications Teaches set theory from scratch, including the axiom of choice, the well ordering theorem, and Zorn's lemma Also included are practical applications in cryptography, error correction, quantum computing and programming Self-contained and requires no prerequisite at all