## 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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2.1 Basic

2.1 Basic

**Definitions**A classical (crisp) set is normally**defined**as a collection of elements or objects x e X that can be finite, countable, or overcountable. Each single element can either belong to or not belong to a set A, ... Page 13

Often it is appropriate to consider those elements of the universe that have a nonzero degree of membership in a fuzzy set.

Often it is appropriate to consider those elements of the universe that have a nonzero degree of membership in a fuzzy set.

**Definition**2–2 The support of a fuzzy set A, S(A|},. FUZZY SETS-BASIC**DEFINITIONS**13. Page 14

**Definition**2–2 The support of a fuzzy set A, S(A|}, is the crisp set of all x e X such that pla(x) > 0. Example 2–2 Let us consider example 2–1a again: The support of S(Å) = {1,2,3,4,5,6}. The elements (types of houses) {7, 8, 9, ... Page 16

**Definition**2–5 For a finite fuzzy set A, the cardinality Al is**defined**as |A|=Xua(x) xeX in-Al - . . . . ~ 7 |A|= I, is called the relative cardinality of A. Obviously, the relative cardinality of a fuzzy set depends on the cardinality ... Page 17

**Definition**2–7 The membership function puff(x) of the union D = A U B is pointwise**defined**by pub(x) = max{ua(x), pub(x)}, x e X**Definition**2–8 The membership function of the complement of a normalized fuzzy set Á, |ltà(x) is**defined**by ...### What people are saying - Write a review

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### Contents

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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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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