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In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.
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.
This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight into Fuzzy Logic. Fuzzy Sets have been introduced by Lotfi Zadeh in 1965 and since then, they have been used in many applications. As a consequence, there is a vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. The main purpose of the present book is to reduce this gap by providing a theoretical introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory. Well-known applications, as for example fuzzy control, are also discussed in this book and placed on new ground, a theoretical foundation. Moreover, a few advanced chapters and several new results are included. These comprise, among others, a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types, that investigates their approximation capability by providing new error estimates.
This undergraduate text develops its subject through observations of the physical world, covering finite sets, cardinal numbers, infinite cardinals, and ordinals. Includes exercises with answers. 1958 edition.
"This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--
This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.
Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.

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