Fuzzy set theory is based on the idea that vague notions as “big”, “near”, “hold” can be modelled by “fuzzy subsets”. The idea of a fuzzy subset of a set is the following: each element , there is a number such that is the “probability” that is in .
To formally define a fuzzy set, let us first recall a well-known fact about subsets: a subset of a set corresponds uniquely to the characteristic function , such that iff . So if one were to replace with the the closed unit interval , one obtains a fuzzy subset:
A fuzzy subset of a set is a map from into the interval .
More precisely, the interval is considered as a complete lattice with an involution . We call fuzzy subset of any element of the direct power . Whereas there are subsets of , there are fuzzy subsets of .
The join and meet operations in the complete lattice are named union and intersection, respectively. The operation induced by the involution is called complement. This means that if and are two fuzzy subsets, then the fuzzy subsets , are defined by the equations
It is also possible to consider any lattice instead of . In such a case we call -subset of any element of the direct power and the union and the intersection are defined by setting
where and denote the join and the meet operations in , respectively. In the case an order reversing function is defined in , the complement of is defined by setting
Fuzzy set theory is devoted mainly to applications. The main success is perhaps fuzzy control.
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|Last modified on||2013-03-22 16:34:54|
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