Fundamentals of Fuzzy Systems Theory

Everything is a matter of a degree.

(Australian Minister of Defence, 1908)

Fuzzy sets theory and fuzzy logic were introduced in 1965 by Lotfi A. Zadeh (Zadeh, 1965) but in a similar way as neural and evolutionary computation the theory of fuzzy sets, fuzzy logic, and fuzzy models and systems become popular only in the 1980s after the works of E. Mamdani from Imperial College, London (Mamdani and Asilian, 1975) on fuzzy controllers and T. Takagi and M. Sugeno from Japan (Takagi and Sugeno, 1985) on fuzzy modelling. It is somewhat similar to the delay between the first publications on a single perceptron in 1946 and the more wide use of neural networks in the 1980s and 1990s after the seminal works of Werbos (1974) and Rumelhart and McClelland (1986). In a very similar way, genetic algorithms (GA) pioneered by Holland (1975) were popularised only in 1990s after the much more practical book by Goldberg (1989) was published.

4.1 Fuzzy Sets

A fuzzy set is an extension of the normal set, with the main difference that an object can belong partially to the fuzzy set, instead of the binary choice that is used for the traditional (crisp, nonfuzzy) sets that limits the analysis to the following two options only:

a. to belong to a set ();
b. not to belong to the set ().

If the membership (belonging) of the jth object to the ith set is denoted by ν the ...

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