## Fuzzy Set Theory—and Its ApplicationsSince its inception, the theory of fuzzy sets has advanced in a variety of ways and in many disciplines. Applications of fuzzy technology can be found in artificial intelligence, computer science, control engineering, decision theory, expert systems, logic, management science, operations research, robotics, and others. Theoretical advances have been made in many directions. The primary goal of Fuzzy Set Theory - and its Applications, Fourth Edition is to provide a textbook for courses in fuzzy set theory, and a book that can be used as an introduction. To balance the character of a textbook with the dynamic nature of this research, many useful references have been added to develop a deeper understanding for the interested reader. Fuzzy Set Theory - and its Applications, Fourth Edition updates the research agenda with chapters on possibility theory, fuzzy logic and approximate reasoning, expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. Chapters have been updated and extended exercises are included. |

### From inside the book

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Page vi

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**Interval**100 FFRR88 88888 71 71 76 79 83 86 93 93 95 99 7.3.2 Integration of a ( Crisp ) Real - Valued Function over a Fuzzy**Interval**103 7.4 Fuzzy Differentiation 107 8 Uncertainty Modeling 111 8.1 Application - oriented Modeling of ... Page ix

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**interval**. 104 Figure 8-1 Uncertainty as situational property . 113 Figure 8-2 Probability of a fuzzy event . 134 Figure 9-1 Linguistic variable “ Age ” . 143 Figure 9-2 Linguistic variable “ Probability ” . 144 Figure 9-3 Linguistic ... Page xii

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**interval**for measurement . 458 Figure 16-2 Subject 34 , " Old Man " . 460 Figure 16-3 Subject 58 , " Very Old Man " . 461 Figure 16-4 Subject 5 , " Very Young Man " . 461 Figure 16-5 Subject 15 , " Very Young Man ” . 462 Figure 16-6 ... Page 19

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**interval**[ 0 , 1 ] , which are mutually distributive ( see ( vi ) ) with respect to one another . 1. με 1 μ . = με με Λ Λ με ν με = με να με 2. ( μs ^ μT ) ^ μv = μs ^ ( μT ^ μU ) ( με ν μπ ) ν μυ = με ν ( με ν μυ ) 3. μs ^ ( μT V μv ) ... Page 25

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**interval**[ 0 , 1 ] is a poset ( partially ordered set ) , the fuzzy set in definition 2-1 is a special L - fuzzy set . Further attempts at representing vague and uncertain data with different types of fuzzy sets were made by Atanassov ...### Contents

1 | |

8 | |

22 | |

4 | 44 |

The Extension Principle and Applications | 54 |

Fuzzy Relations on Sets and Fuzzy Sets | 71 |

3 | 82 |

7 | 88 |

Applications of Fuzzy Set Theory | 139 |

3 | 154 |

4 | 160 |

5 | 169 |

Fuzzy Sets and Expert Systems | 185 |

Fuzzy Control | 223 |

Fuzzy Data Bases and Queries | 265 |

Decision Making in Fuzzy Environments | 329 |

3 | 95 |

4 | 105 |

2 | 122 |

4 | 131 |

Applications of Fuzzy Sets in Engineering and Management | 371 |

Empirical Research in Fuzzy Set Theory | 443 |

Future Perspectives | 477 |

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### Common terms and phrases

a-level aggregation algebraic algorithm applications of fuzzy approach approximately areas base basic Bezdek chapter classical computational concepts considered constraints crisp criteria customers data analysis DataEngine decision defined definition defuzzification degree of membership described determine domain Dubois and Prade elements engineering example expert systems feature formal Fril fuzzy c-means fuzzy clustering fuzzy control fuzzy control systems fuzzy function fuzzy graph fuzzy logic fuzzy measures fuzzy numbers fuzzy relation fuzzy set Ć fuzzy set theory goal inference inference engine input integral intersection interval linear programming linguistic variable Mamdani mathematical measure of fuzziness membership function methods min-operator objective function operators optimal parameters possibility distribution probability probability theory problem properties respect rules scale level scheduling semantic solution structure Sugeno t-conorms t-norms Table tion trajectories truth tables truth values uncertainty vector x₁ Yager Zadeh Zimmermann µĆ(x µµ(x