Using material from a successful course on fuzzy logic, this book is an introduction to the theory of fuzzy sets: mathematical objects modeling the vagueness of our natural language when we describe phenomena that do not have sharply defined boundaries. The book provides background information necessary to apply fuzzy set theory in various areas, including engineering fuzzy logic and decision-making. The exercises at the end of each chapter serve to deepen the reader's understanding of the concepts, and to test his or her ability to make the necessary calculations.
Using material from a successful course on fuzzy logic, this book is an introduction to the theory of fuzzy sets: mathematical objects modeling the vagueness of our natural language when we describe phenomena that do not have sharply defined boundaries. The book provides background information necessary to apply fuzzy set theory in various areas, including engineering fuzzy logic and decision-making. The exercises at the end of each chapter serve to deepen the reader's understanding of the concepts, and to test his or her ability to make the necessary calculations.
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Inhaltsangabe
The Concept of Fuzziness
Examples. Mathematical modeling. Some operations on fuzzy sets. Fuzziness as uncertainty.
Some Algebra of Fuzzy Sets
Boolean algebras and lattices. Equivalence relations and partitions. Composing mappings. Isomorphisms and homomorphisms. Alpha-cuts. Images of alpha-level sets.
Fuzzy Quantities
Fuzzy quantities. Fuzzy numbers. Fuzzy intervals.
Logical Aspects of Fuzzy Sets
Classical two-valued logic. A three-valued logic. Fuzzy logic. Fuzzy and Lukasiewicz logics. Interval-valued fuzzy logic.
Basic Connectives
t-norms. Generators of t-norms. Isomorphisms of t-norms. Negations. Nilpotent t-norms and negations. T-conforms. De Morgan systems. Groups and t-norms. Interval-valued fuzzy sets. Type-2 fuzzy sets.
Additional Topics on Connectives
Fuzzy implications. Averaging operators. Powers of t-norms. Sensitivity of connectives. Copulas and t-norms.
Fuzzy Relations
Definitions and examples. Binary fuzzy relations. Operations on fuzzy relations. Fuzzy partitions. Fuzzy relations as Chu spaces. Approximate reasoning. Approximate reasoning in expert systems. A simple form of generalized modus ponens. The compositional rule of inference.
Universal Approximation
Fuzzy rule bases. Design methodologies. Some mathematical background. Approximation capability.
Possibility Theory
Probability and uncertainty. Random sets. Possibility measures.
Partial Knowledge
Motivations. Belief functions and incidence algebras. Monotonicity. Beliefs, densities, and allocations. Belief functions on infinite sets. Mobius transforms of set-functions. Reasoning with belief functions. Decision making using belief functions. Rough sets. Conditional events.
Fuzzy Measures
Motivation and definitions. Fuzzy measures and lower probabilities. Fuzzy measures in other areas. Conditional fuzzy measures.
The Choquet Integral
The Lebesgue integral. The Sugeno integral. The Choquet integral.
Fuzzy Modeling and Control
Motivation for fuzzy control. The methodology of fuzzy control. Optimal fuzzy control. An analysis of fuzzy control techniques.
Examples. Mathematical modeling. Some operations on fuzzy sets. Fuzziness as uncertainty.
Some Algebra of Fuzzy Sets
Boolean algebras and lattices. Equivalence relations and partitions. Composing mappings. Isomorphisms and homomorphisms. Alpha-cuts. Images of alpha-level sets.
Fuzzy Quantities
Fuzzy quantities. Fuzzy numbers. Fuzzy intervals.
Logical Aspects of Fuzzy Sets
Classical two-valued logic. A three-valued logic. Fuzzy logic. Fuzzy and Lukasiewicz logics. Interval-valued fuzzy logic.
Basic Connectives
t-norms. Generators of t-norms. Isomorphisms of t-norms. Negations. Nilpotent t-norms and negations. T-conforms. De Morgan systems. Groups and t-norms. Interval-valued fuzzy sets. Type-2 fuzzy sets.
Additional Topics on Connectives
Fuzzy implications. Averaging operators. Powers of t-norms. Sensitivity of connectives. Copulas and t-norms.
Fuzzy Relations
Definitions and examples. Binary fuzzy relations. Operations on fuzzy relations. Fuzzy partitions. Fuzzy relations as Chu spaces. Approximate reasoning. Approximate reasoning in expert systems. A simple form of generalized modus ponens. The compositional rule of inference.
Universal Approximation
Fuzzy rule bases. Design methodologies. Some mathematical background. Approximation capability.
Possibility Theory
Probability and uncertainty. Random sets. Possibility measures.
Partial Knowledge
Motivations. Belief functions and incidence algebras. Monotonicity. Beliefs, densities, and allocations. Belief functions on infinite sets. Mobius transforms of set-functions. Reasoning with belief functions. Decision making using belief functions. Rough sets. Conditional events.
Fuzzy Measures
Motivation and definitions. Fuzzy measures and lower probabilities. Fuzzy measures in other areas. Conditional fuzzy measures.
The Choquet Integral
The Lebesgue integral. The Sugeno integral. The Choquet integral.
Fuzzy Modeling and Control
Motivation for fuzzy control. The methodology of fuzzy control. Optimal fuzzy control. An analysis of fuzzy control techniques.
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