## 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. |

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Page vi

8.2.1 Fuzzy Relations and Fuzzy Graphs Fuzzy Relations on Sets and Fuzzy Sets Compositions of Fuzzy Relations

8.2.1 Fuzzy Relations and Fuzzy Graphs Fuzzy Relations on Sets and Fuzzy Sets Compositions of Fuzzy Relations

**Properties**of the Min-Max Composition Fuzzy Graphs Special Fuzzy Relations Fuzzy Analysis Fuzzy Functions on Fuzzy Sets ... Page 25

Figure 3–1 indicates the difference between the appearance of fuzzy sets and probabilistic sets [Hirota 1981, p. 33]. Of course, the mathematical

Figure 3–1 indicates the difference between the appearance of fuzzy sets and probabilistic sets [Hirota 1981, p. 33]. Of course, the mathematical

**properties**of probabilistic sets differ from those of fuzzy sets ... Page 29

Adaptability ranges from uniquely defined (for example, nonadaptable) concepts via parameterized “families” of operators to general classes of operators that satisfy certain

Adaptability ranges from uniquely defined (for example, nonadaptable) concepts via parameterized “families” of operators to general classes of operators that satisfy certain

**properties**. We shall investigate the two basic classes of ... Page 31

These

These

**properties**are formulated with the following conditions: 1. s(1, 1) = 1; s(pla(x), 0) = S(0, pla(x)) = pla(x), x e X 2. s(HA(x), Hä(x)) < s(pic(x), HB(x)) if [14(x) < pla(x) and puff(x) < |15(x) (monotonicity) 3. s(plå(x), ...Page 37

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### Contents

9 | |

11 | |

16 | |

22 | |

29 | |

Criteria for Selecting Appropriate Aggregation Operators | 43 |

The Extension Principle and Applications | 54 |

Special Extended Operations | 61 |

Applicationoriented Modeling of Uncertainty | 111 |

Linguistic Variables | 140 |

Fuzzy Data Bases and Queries | 265 |

Decision Making in Fuzzy Environments | 329 |

Applications of Fuzzy Sets in Engineering and Management | 371 |

Empirical Research in Fuzzy Set Theory | 443 |

Future Perspectives | 477 |

181 | 485 |

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### Common terms and phrases

aggregation algorithm analysis applications approach appropriate approximately areas assignment assume base called chapter classical clustering compute concepts considered constraints contains corresponding crisp criteria customers decision defined definition degree of membership depends described determine discussed distribution domain elements engineering example exist expert systems expressed extension Figure fuzzy control fuzzy numbers fuzzy set theory given goal human important indicate inference input instance integral interpreted intersection interval knowledge linguistic variable logic mathematical mean measure membership function methods normally objective objective function observed obtain operators optimal positive possible probability problem programming properties provides reasoning relation representing require respect rules scale shown shows similarity situation solution space specific statement structure suggested t-norms Table tion true truth uncertainty values Zadeh