.

0 (hodnocen0 x )

(4) Půjčeno:4x 

BK

3rd ed.

Boca Raton : Chapman & Hall/CRC, c2006

x, 430 s. ; 24 cm

ISBN 1-58488-526-2 (váz.) ISBN !978-1-58488-526-9 (chyb.)

Obsahuje bibliografii na s. 387-399 a rejstřík

000177660

Contents // 1 The Concept of Fuzziness 1 // 1.1 Examples... 1 // 1.2 Mathematical modeling... 2 // 1.3 Some operations on fuzzy sets... 6 // 1.4 Fuzziness as uncertainty... 11 // 1.5 Exercises ... 14 // 2 Some Algebra of Fuzzy Sets 17 // 2.1 Boolean algebras and lattices... 17 // 2.2 Equivalence relations and partitions... 23 // 2.3 Composing mappings... 27 // 2.4 Isomorphisms and homomorphisms... 29 // 2.5 Alpha-cuts... 32 // 2.6 Images of alpha-level sets... 34 // 2.7 Exercises ... 36 // 3 Fuzzy Quantities 45 // 3.1 Fuzzy quantities ... 45 // 3.2 Fuzzy numbers... 52 // 3.3 Fuzzy intervals... 55 // 3.4 Exercises ... 56 // 4 Logical Aspects of Fuzzy Sets 59 // 4.1 Classical two-valued logic... 60 // 4.2 A three-valued logic... 64 // 4.3 Fuzzy logic ... 65 // 4.4 Fuzzy and Lukasiewicz logics... 66 // 4.5 Interval-valued fuzzy logic... 68 // 4.6 Canonical forms ... 70 // 4.7 Notes on probabilistic logic... 74 // 4.8 Exercises ... 76 // vii // CONTENTS // viii // 5 Basic Connectives 81 // 5.1 t-norms... 81 // 5.2 Generators of t-norms... 85 // 5.3 Isomorphisms of t-norms... 93 // 5.4 Negations... 98 // 5.5 Nilpotent t-norms and negations...102 // 5.6 t-conorms...106 // 5.7 De Morgan systems...109 // 5.7.1 Strict De Morgan systems...109 // 5.7.2 Nilpotent De Morgan systems...113 // 5.7.3 Nonuniqueness of negations in strict // De Morgan systems ...116 // 5.8 Groups and t-norms...118 // 5.8.1 The normalizer of M+ ...119 // 5.8.2 Families of strict t-norms...122 // 5.8.3 Families of

nilpotent t-norms...126 // 5.9 Interval-valued fuzzy sets ...127 // 5.9.1 t-norms on interval-valued fuzzy sets...128 // 5.9.2 Negations and t-conorms ...130 // 5.10 Type-2 fuzzy sets...134 // 5.10.1 Pointwise operations and convolutions...134 // 5.10.2 Type-2 fuzzy sets...135 // 5.10.3 The algebra (Map(J, /), U, ?,* , 0,1)...136 // 5.10.4 Two order relations ...143 // 5.10.5 Subalgebras of type-2 fuzzy sets ...145 // 5.10.6 Convolutions using product...153 // 5.10.7 T-norms for type-2 fuzzy sets...157 // 5.10.8 Comments...163 // 5.11 Exercises ...163 // 6 Additional Topics on Connectives 171 // 6.1 Fuzzy implications...171 // 6.2 Averaging operators...177 // 6.2.1 Averaging operators and negations...180 // 6.2.2 Averaging operators and nilpotent t-norms ... 184 // 6.2.3 De Morgan systems with averaging operators ... 187 // 6.3 Powers of t-norms ...190 // 6.4 Sensitivity of connectives ...194 // 6.5 Copulas and t-norms...197 // 6.6 Exercises ...200 // CONTENTS // ix // 7 Fuzzy Relations 207 // 7.1 Definitions and examples...207 // 7.2 Binary fuzzy relations...208 // 7.3 Operations on fuzzy relations...212 // 7.4 Fuzzy partitions ...214 // 7.5 Fuzzy relations as Chu spaces...215 // 7.6 Approximate reasoning...217 // 7.7 Approximate reasoning in expert systems...220 // 7.7.1 Fuzzy syllogisms...226 // 7.7.2 Truth qualification...226 // 7.7.3 Probability qualification...226 // 7.7.4 Possibility qualification ...227 // 7.8 A simple form of generalized modus ponens...227 // 7.9 The

compositional rule of inference...229 // 7.10 Exercises ...230 // 8 Universal Approximation 235 // 8.1 Fuzzy rule bases ...235 // 8.2 Design methodologies ...238 // 8.3 Some mathematical background...240 // 8.4 Approximation capability...242 // 8.5 Exercises ...247 // 9 Possibility Theory 251 // 9.1 Probability and uncertainty...251 // 9.2 Random sets ...254 // 9.3 Possibility measures...256 // 9.3.1 Measures of noncompactness...261 // 9.3.2 Fractal dimensions...262 // 9.3.3 Information measures ...263 // 9.4 Exercises ...267 // 10 Partial Knowledge 271 // 10.1 Motivation ...271 // 10.2 Belief functions and incidence algebras...274 // 10.3 Monotonicity...278 // 10.4 Beliefs, densities, and allocations...282 // 10.5 Belief functions on infinite sets...287 // 10.5.1 Inner measures and belief functions...288 // 10.5.2 Possibility measures and belief functions...289 // 10.6 Note on Möbius transforms // of set-functions...292 // 10.7 Reasoning with belief functions...293 // ? CONTENTS // 10.8 Decision making using belief functions...295 // 10.8.1 A minimax viewpoint...296 // 10.8.2 An expected-value approach...297 // 10.8.3 Maximum entropy principle...298 // 10.9 Rough sets ...302 // 10.9.1 An example...305 // 10.9.2 The structure oilZ...307 // 10.10 Conditional events...309 // 10.11 Exercises...311 // 11 Fuzzy Measures 319 // 11.1 Motivation and definitions...319 // 11.2 Fuzzy measures and lower probabilities...321 // 11.3 Fuzzy measures in other areas...326 // 11.3.1 Capacities...326

// 11.3.2 Measures and dimensions...328 // 11.3.3 Game theory...330 // 11.4 Conditional fuzzy measures...331 // 11.5 Exercises ...336 // 12 The Choquet Integral 341 // 12.1 The Lebesgue integral...341 // 12.2 The Sugeno integral...343 // 12.3 The Choquet integral...348 // 12.3.1 Motivation...348 // 12.3.2 Foundations...352 // 12.3.3 Radon-Nikodym derivatives...359 // 12.3.4 Multicriteria decisions with Choquet integrals . . . 363 // 12.4 Exercises ...365 // 13 Ftizzy Modeling and Control 371 // 13.1 Motivation for fuzzy control...371 // 13.2 The methodology of fuzzy control...374 // 13.3 Optimal fuzzy control...381 // 13.4 An analysis of fuzzy control techniques...382 // 13.5 Exercises ...385 // Bibliography 387 // Answers to Selected Exercises 401 // Index // 425