In classical set theory the membership of elements in relation to a set is assessed in binary terms according to a crisp condition. An element either belongs or does not belong to the set, the boundary of the set is crisp. As a further development of classical set theory, fuzzy set theory permits the gradual assessment of the membership of elements in relation to a set; this is described with the aid of a membership function. A fuzzy set is defined as follows (see also Fig. 1):
If X represents a fundamental set and x are the elements of this fundamental set, to be assessed according to an (lexical or informal) uncertain proposition and assigned to a subset A of X, the set
A fuzzy function may be explained by extending the definition of a classical function. A classical function is a single-valued mapping of the elements t from the fundamental set T onto the elements x of the fundamental set X . It may be denoted by
The fuzzy function () may thus also be interpreted as being a set of fuzzy results or fuzzy functional values F(X) belonging to specified F(T)
- the time coordinate τ
- the spartial coordinate θ
- and additional parameters φ
Figure 2: Fuzzy process
Bunch parameter representation
A multi-dimensional fuzzy function (t) may be formulated depending on fuzzy bunch parameters and crisp arguments t
x(t) = x(s, t) with µ(x(t)) = µ(s) is obtained.
Figure 3: Fuzzy field with bunch parameter representation
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