Ebook: Vaguely Defined Objects: Representations, Fuzzy Sets and Nonclassical Cardinality Theory

In recent years, an impetuous development of new, unconventional theories, methods, techniques and technologies in computer and information sciences, systems analysis, decision-making and control, expert systems, data modelling, engineering, etc. , resulted in a considerable increase of interest in adequate mathematical description and analysis of objects, phenomena, and processes which are vague or imprecise by their very nature. Classical two-valued logic and the related notion of a set, together with its mathematical consequences, are then often inadequate or insufficient formal tools, and can even become useless for applications because of their (too) categorical character: 'true - false', 'belongs - does not belong', 'is - is not', 'black - white', '0 - 1', etc. This is why one replaces classical logic by various types of many-valued logics and, on the other hand, more general notions are introduced instead of or beside that of a set. Let us mention, for instance, fuzzy sets and derivative concepts, flou sets and twofold fuzzy sets, which have been created for different purposes as well as using distinct formal and informal motivations. A kind of numerical information concerning of 'how many' elements those objects are composed seems to be one of the simplest and more important types of information about them. To get it, one needs a suitable notion of cardinality and, moreover, a possibility to calculate with such cardinalities. Unfortunately, neither fuzzy sets nor the other nonclassical concepts have been equipped with a satisfactory (nonclassical) cardinality theory.

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This unique monograph explores the cardinal, or quantitative, aspects of objects in the presence of vagueness, called vaguely defined objects.
In the first part of the book such topics as fuzzy sets and derivative ideas, twofold fuzzy sets, and flow sets are concisely reviewed as typical mathematical representations of vaguely defined objects. Also, a unifying, approximative representation is presented.
The second part uses this representation, together with Lukasiewicz logic as a basis for constructing a complete, general and easily applicable nonclassical cardinality theory for vaguely defined objects. Applications to computer and information science are discussed.
Audience: This volume will be of interest to mathematicians, computer and information scientists, whose work involves mathematical aspects of vagueness, fuzzy sets and their methods, applied many-valued logics, expert systems and data bases.

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This unique monograph explores the cardinal, or quantitative, aspects of objects in the presence of vagueness, called vaguely defined objects.
In the first part of the book such topics as fuzzy sets and derivative ideas, twofold fuzzy sets, and flow sets are concisely reviewed as typical mathematical representations of vaguely defined objects. Also, a unifying, approximative representation is presented.
The second part uses this representation, together with Lukasiewicz logic as a basis for constructing a complete, general and easily applicable nonclassical cardinality theory for vaguely defined objects. Applications to computer and information science are discussed.
Audience: This volume will be of interest to mathematicians, computer and information scientists, whose work involves mathematical aspects of vagueness, fuzzy sets and their methods, applied many-valued logics, expert systems and data bases.
Content:
Front Matter....Pages i-xv
Front Matter....Pages 2-2
Basic Notions and Problems....Pages 3-10
Mathematical Approaches to Vaguely Defined Objects....Pages 11-27
Mathematical Approaches to Subdefinite Sets....Pages 28-31
A Unifying Approximative Approach to Vaguely Defined Objects....Pages 32-47
Front Matter....Pages 50-50
Equipotencies....Pages 51-70
Generalized Cardinal Numbers....Pages 71-106
Selected Applications....Pages 107-114
Inequalities....Pages 115-138
Many-Valued Generalizations....Pages 139-146
Towards Arithmetical Operations....Pages 147-153
Addition....Pages 154-164
Multiplication....Pages 165-174
Other Basic Operations....Pages 175-181
Generalized Arithmetical Operations....Pages 182-196
Cardinalities with Free Representing Pairs....Pages 197-210
Further Modifications and Final Remarks....Pages 211-215
Back Matter....Pages 217-265

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This unique monograph explores the cardinal, or quantitative, aspects of objects in the presence of vagueness, called vaguely defined objects.
In the first part of the book such topics as fuzzy sets and derivative ideas, twofold fuzzy sets, and flow sets are concisely reviewed as typical mathematical representations of vaguely defined objects. Also, a unifying, approximative representation is presented.
The second part uses this representation, together with Lukasiewicz logic as a basis for constructing a complete, general and easily applicable nonclassical cardinality theory for vaguely defined objects. Applications to computer and information science are discussed.
Audience: This volume will be of interest to mathematicians, computer and information scientists, whose work involves mathematical aspects of vagueness, fuzzy sets and their methods, applied many-valued logics, expert systems and data bases.
Content:
Front Matter....Pages i-xv
Front Matter....Pages 2-2
Basic Notions and Problems....Pages 3-10
Mathematical Approaches to Vaguely Defined Objects....Pages 11-27
Mathematical Approaches to Subdefinite Sets....Pages 28-31
A Unifying Approximative Approach to Vaguely Defined Objects....Pages 32-47
Front Matter....Pages 50-50
Equipotencies....Pages 51-70
Generalized Cardinal Numbers....Pages 71-106
Selected Applications....Pages 107-114
Inequalities....Pages 115-138
Many-Valued Generalizations....Pages 139-146
Towards Arithmetical Operations....Pages 147-153
Addition....Pages 154-164
Multiplication....Pages 165-174
Other Basic Operations....Pages 175-181
Generalized Arithmetical Operations....Pages 182-196
Cardinalities with Free Representing Pairs....Pages 197-210
Further Modifications and Final Remarks....Pages 211-215
Back Matter....Pages 217-265
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