Cardinalities of Fuzzy Sets

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Counting is one of the basic elementary mathematical activities. It comes with two complementary aspects: to determine the number of elements of a set - and to create an ordering between the objects of counting just by counting them over. For finite sets of objects these two aspects are realized by the same type of num bers: the natural numbers. That these complementary aspects of the counting pro cess may need different kinds of numbers becomes apparent if one extends the process of counting to infinite sets. As general tools to determine numbers of elements the cardinals have been created in set theory, and set theorists have in parallel created the ordinals to count over any set of objects. For both types of numbers it is not only counting they are used for, it is also the strongly related process of calculation - especially addition and, derived from it, multiplication and even exponentiation - which is based upon these numbers. For fuzzy sets the idea of counting, in both aspects, looses its naive foundation: because it is to a large extent founded upon of the idea that there is a clear distinc tion between those objects which have to be counted - and those ones which have to be neglected for the particular counting process.
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Dec 6, 2012
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Computers / Data Processing
Language Arts & Disciplines / Library & Information Science / General
Mathematics / Algebra / Abstract
Mathematics / Discrete Mathematics
Mathematics / General
Mathematics / Group Theory
Science / System Theory
Technology & Engineering / General
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The last two decades have witnessed an enormous growth with regard to ap plications of information theoretic framework in areas of physical, biological, engineering and even social sciences. In particular, growth has been spectac ular in the field of information technology,soft computing,nonlinear systems and molecular biology. Claude Shannon in 1948 laid the foundation of the field of information theory in the context of communication theory. It is in deed remarkable that his framework is as relevant today as was when he 1 proposed it. Shannon died on Feb 24, 2001. Arun Netravali observes "As if assuming that inexpensive, high-speed processing would come to pass, Shan non figured out the upper limits on communication rates. First in telephone channels, then in optical communications, and now in wireless, Shannon has had the utmost value in defining the engineering limits we face". Shannon introduced the concept of entropy. The notable feature of the entropy frame work is that it enables quantification of uncertainty present in a system. In many realistic situations one is confronted only with partial or incomplete information in the form of moment, or bounds on these values etc. ; and it is then required to construct a probabilistic model from this partial information. In such situations, the principle of maximum entropy provides a rational ba sis for constructing a probabilistic model. It is thus necessary and important to keep track of advances in the applications of maximum entropy principle to ever expanding areas of knowledge.
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Thinking in Systems, is a concise and crucial book offering insight for problem solving on scales ranging from the personal to the global. Edited by the Sustainability Institute’s Diana Wright, this essential primer brings systems thinking out of the realm of computers and equations and into the tangible world, showing readers how to develop the systems-thinking skills that thought leaders across the globe consider critical for 21st-century life.

Some of the biggest problems facing the world—war, hunger, poverty, and environmental degradation—are essentially system failures. They cannot be solved by fixing one piece in isolation from the others, because even seemingly minor details have enormous power to undermine the best efforts of too-narrow thinking.

While readers will learn the conceptual tools and methods of systems thinking, the heart of the book is grander than methodology. Donella Meadows was known as much for nurturing positive outcomes as she was for delving into the science behind global dilemmas. She reminds readers to pay attention to what is important, not just what is quantifiable, to stay humble, and to stay a learner.

In a world growing ever more complicated, crowded, and interdependent, Thinking in Systems helps readers avoid confusion and helplessness, the first step toward finding proactive and effective solutions.

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