## 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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**exist**in the field are widely scattered over many areas and in many journals . Existing books are edited volumes containing specialized contri- butions or monographs that focus only on specific areas of fuzzy sets , such as pattern ... Page 6

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**exist**. Fuzziness has so far not been defined uniquely semantically , and probably never will be . It will mean different things , depending on the application area and the way it is measured . In the meantime , numerous authors have ... Page 16

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**exist**. 2.2 Basic Set - Theoretic Operations for Fuzzy Sets The membership function is obviously the crucial component of a fuzzy set . It is therefore not surprising that operations with fuzzy sets are defined via their mem- bership ... Page 25

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**exists**an E e B that satisfies P ( E ) = 1 and μ ( x , ∞ ) ≤ μg ( x , ∞ ) for all ∞ = E. ( Q , B , P ) is called the parameter space . One of the main advantages of the notion of probabilistic sets in modeling fuzzy and stochastic ... Page 33

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**exists**with -1 μã‚ ̧ ( x ) = ƒ ( ƒ ̃1 ( μÃ ( x ) ) + ƒ ̄1 ( μμ ( x ) ) ) If ƒ is a rational function in μÃ ( x ) and μ¿ ( x ) , then the only possible operator is that shown in definition 3-14 . ( For y = 1 , this reduces to the ...### 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 Ã 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 µÃ(x µµ(x