## 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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**interpreted**as the fraction of elements of X being in Ã , weighted by their degrees of membership in Ã . For infinite X , the cardinal- ity is defined by | Ã ] = ↓ ÂμÃ ( x ) dx . Of course , | Ã¦ does not always exist . 2.2 Basic Set ... Page 17

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**interpreted**as " not large type of house , " is ¢ Ẽ = { ( 1 , 1 ) , ( 2 , 1 ) , ( 3 , .8 ) , ( 4 , .6 ) , ( 5 , .4 ) , ( 6 , .2 ) , ( 9 , 1 ) , ( 10 , 1 ) } Example 2-7 Let us assume that A = " x is considerable larger than 10 , ” and B ... Page 18

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**interpreting**the intersection as " logical and , " the union as " logical or , " and the fuzzy set A as the statement " The element x belongs to set A , ” which can be accepted as more or less true . It is very instruc- tive to follow ... Page 19

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**interpreted**as the values of the membership functions of the inter- section and union , respectively , of S and T. We are now looking for two real - valued functions ƒ and g such that μS and T = ƒ ( μs , μT ) μsor T = g ( μs , HT ) ... Page 29

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**interpreted**as the logical " and , " was modeled as the min - operator and the union ,**interpreted**as " or , " as the max- operator . Other operators have also been suggested . These suggestions vary with respect to the generality or ...### 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