Fuzzy Sets

Fuzzy sets provide a basis for fuzzy modeling. A fuzzy set is an extension of a classical set. In classical set theory an element either belongs or does not belong to a set. In a fuzzy set an element belongs gradually to a set. The fuzzy set theory traditionally uses the Greek letter p! [0, 1] as the membership function. The value between 0 and 1 describes the degree of membership of the element to the set A. A fuzzy set is defined by

Every element xeX gradually belongs to the set A. The value pA(x) = 0 means that x=Al and the value pA(x) = 1 means that xeA. All values strictly between 0 and 1 characterize the fuzzy members. In other words, if the membership values are restricted to 0 and 1, the fuzzy set is a classical set.

Another consequence of this definition is that an element x can belong to different fuzzy sets. For example, a biologist may express the quality of nutrition of crops for the crane (grus grus) as in Table 1.

Table 1 Nutrition quality of crops for a crane fuzzy set

Good

Medium

Bad

Maize

0.2

0.8

0

Rye

0.4

0.6

0

Wheat

0.6

0.4

0

Rape

0

0.1

0.9

Barley

0.6

0.4

0

This estimation could be part of a model of the habitat quality for the crane. Maize with a membership value of 0.8 belongs to the class 'medium nutrition quality' and with 0.2 to the class 'good nutrition quality' for example. This description can be used to express uncertainty or used in discussing it with other experts.

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