Fuzzy Sets and Fuzzy Logic: Theory and ApplicationsThe primary purpose of this book is to provide the reader with a comprehensive coverage of theoretical foundations of fuzzy set theory and fuzzy logic, as well as a broad overview of the increasingly important applications of these novel areas of mathematics. Although it is written as a text for a course at the graduate or upper division undergraduate level, the book is also suitable for self-study and for industry-oriented courses of continuing education. No previous knowledge of fuzzy set theory and fuzzy logic is required for understanding the material covered in the book. Although knowledge of basic ideas of classical (nonfuzzy) set theory and classical (two-valued) logic is useful, fundamentals of these subject areas are briefly overviewed in the book. In addition, basic ideas of neural networks, genetic algorithms, and rough sets are also explained. This makes the book virtually self-contained. Throughout the book, many examples are used to illustrate concepts, methods, and generic applications as they are introduced. Each chapter is followed by a set of exercises, which are intended to enhance readers' understanding of the material presented in the chapter. Extensive and carefully selected bibliography, together with bibliographical notes at the end of each chapter and a bibliographical subject index, is an invaluable resource for further study of fuzzy theory and applications. |
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Results 1-3 of 59
Page 115
... solution of the equation . - Let X = [ x1 , x2 ] . Then , [ a1 + x1 , a2 + x2 ] = [ b1 , b2 ] follows immediately from the equation . This results in two ordinary equations of real numbers , - a1 + x1 = b1 , a2 + x2 = b2 , -- whose solution ...
... solution of the equation . - Let X = [ x1 , x2 ] . Then , [ a1 + x1 , a2 + x2 ] = [ b1 , b2 ] follows immediately from the equation . This results in two ordinary equations of real numbers , - a1 + x1 = b1 , a2 + x2 = b2 , -- whose solution ...
Page 157
... solution set S ( Q , r ) . Employing the partial ordering on P , let an element p of S ( Q , r ) be called a maximal solution of ( 6.9 ) if , for all p Є S ( Q , r ) , p ≥ p implies p = = p ; if , for all p Є S ( Q , r ) , p ≤ ê ...
... solution set S ( Q , r ) . Employing the partial ordering on P , let an element p of S ( Q , r ) be called a maximal solution of ( 6.9 ) if , for all p Є S ( Q , r ) , p ≥ p implies p = = p ; if , for all p Є S ( Q , r ) , p ≤ ê ...
Page 166
... solution of ( 6.16 ) for the given P and R. 6.6 APPROXIMATE SOLUTIONS It is quite common that fuzzy relation equations of the general form PQ = R , ( 6.19 ) in which Q and R are given , have no solutions for P. Since a solution is ...
... solution of ( 6.16 ) for the given P and R. 6.6 APPROXIMATE SOLUTIONS It is quite common that fuzzy relation equations of the general form PQ = R , ( 6.19 ) in which Q and R are given , have no solutions for P. Since a solution is ...
Contents
THEORY | 1 |
FUZZY SETS VERSUS CRISP SETS | 35 |
OPERATIONS ON FUZZY SETS | 50 |
Copyright | |
20 other sections not shown
Common terms and phrases
a-cuts a₁ applications of fuzzy Approximate Reasoning Assume Axiom b₁ basic assignment binary relations characterized chromosomes classical concept crisp sets Cybernetics defuzzification degree denote determine engineering evidence theory example expert systems expressed finite focal elements formula fuzzified fuzzy complement fuzzy control fuzzy implications fuzzy intersection fuzzy logic fuzzy measure fuzzy measure theory fuzzy numbers fuzzy proposition fuzzy relation equations fuzzy set theory Fuzzy Systems fuzzy union genetic algorithms given Hence IEEE Trans inference rules input Intern linguistic terms log₂ mathematical matrix membership functions membership grade method neural networks nonspecificity obtain output parameters pattern recognition possibilistic possibility distribution possibility theory probability theory problem properties r₁ real numbers relevant represented respectively satisfies Sets and Systems shown in Fig solution standard fuzzy subsets t-conorms t-norms Theorem truth values types uncertainty universal set variable Yager Zadeh