## 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

Results 1-5 of 87

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**example**. 384 385 386 386 387 388 388 392 392 394 394 399 Figure 15-17 Structure of OPAL . 402 Figure 15-18 Figure 15-19 Fuzzy sets for the ratio in the " if " part of the rules .**Example**of an FMS [ Hartley 1984 , p . 194 ] . 404 405 ... Page 12

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**Example**2-1b Ã = " real numbers considerably larger than 10 " Ã = { ( x , μμ ( x ) ) | x € X } where ( 0 , x ≤10 μ2 ( x ) = { ( ( 1+ ( x − 10 ) −2 ) ̄` ' , x > 10**Example**2-1c Ã = " real numbers close to 10 " Ã = { ( x , μ¿ ( x ) ... Page 13

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**Example**2-1d X 10 15 n Ã = μÃ ( x1 ) / x ; + μ¿ ( x2 ) / x2 ... .. = 1⁄2μã ( xi ) / xi Σμα i = 1 or [ Hx ( x ) / x Ã = " integers close to 10 " Ã = 0.1 / 7 + 0.5 / 8 + 0.8 / 9 + 1 / 10 + 0.8 / 11 + 0.5 / 12 + 0.1 / 13**Example**2 - le A ... Page 14

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**Example**2-2 Let us consider**example**2-1a again : The support of S ( Ã ) = { 1 , 2 , 3 , 4 , 5 , 6 } . The elements ( types of houses ) { 7 , 8 , 9 , 10 } are not part of the support of A ! A more general and even more useful notion is ... Page 16

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**Example**2-5 For the fuzzy set " comfortable type of house for a four - person family " from**example**2-1a , the cardinality is | Ã | = .2 + .5 + .8 + 1 + .7 + .3 = 3.5 Its relative cardinality is 3.5 || Ã || = 0.35 10 The relative ...### 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