Fuzzy complement

Fuzzy complements, intersections, and unions have been characterized and studied on axiomatic grounds. Efficient procedures are now available by which various classes of functions can be generated, each of which covers the whole recognized semantic range of the respective operation. In addition, averaging operations for fuzzy sets, which have no counterparts for crisp sets, have also been investigated in this way. This rather theoretical subject, which is beyond the scope of this overview, is thoroughly covered in ref. 18. 

Any fuzzy power set with the subsethood relation is a lattice, in which the standard fuzzy intersection and union play the roles of the meet and the join, respectively. The lattice is distributive and complemented under the standard fuzzy complement. Contrary to the Boolean lattice, which is associated with classical power sets, it does not satisfy the law of the excluded middle and the law of contradiction. Such a lattice is usually called a DeMorgan lattice. 

In fuzzy logic, operators such as AND, OR, and NOT are implemented by fuzzy intersection or conjunction (AND), fuzzy union or disjunction (OR), and fuzzy complement (NOT). There are various ways to define these operators, but commonly, AND, OR, and NOT logic operators are implemented by the min, max, and complement operators. The fuzzy truth, T, of a complex sentence is evaluated in this way ... 

Fuzzy set operations are a generalization of crisp set operations, each of which is a fuzzy set operation. Infuzzy logic, three operations, including fuzzy complement, fuzzy intersection and fuzzy union, are the most commonly used. Let fuzzy sets ... 

Fuzzy complement The complement of a fuzzy set is the opposite of the set in question. The fuzzy complement of fuzzy sets can be represented as... 

Complement The eomplement of fuzzy set A eorresponds to the Boolean NOT funetion and is given by... 

CMETS 150 Coatings, fuzzy 151 Complement 171 Concanavalin A 145 Contact area, hydrophobic 159... 

A fuzzy generalization of the complement is obtained as the result of operation not A, denoted by A, interpreted as a fuzzy subset A of set... 

Moments provide useful information about the layout or shape of an image subset that contrasts with its complement they can be computed without the need to segment the subset explicitly from the rest of the image. (Many of the geometric properties defined in Section XII can also be defined for fuzzy image subsets that have not been explicitly segmented.)... 

The three basic operators for fuzzy sets are complement, union, and intersection ... 

The achieved results complement the set of approaches to the indirect observation of a technical condition. The approaches using purely a regression analysis and fuzzy logic, see e.g. Koucky Valis (2011), Valis et al. (2012), have been applied so far. Following the conclusions of modelling with the Wiener process, the results of previous approaches might be completed when searching for ... 

For fuzzy sets the complement has the membership function (Figure 2a, solid line) ... 

Figure 2 Set-theoretic operations (complement, intersection, and union) on fuzzy sets...

In more general terms, fuzzy intersection is defined by fuzzy AND operator, fuzzy union is defined by fuzzy OR operator, and complement by fuzzy NOT operator. All properties of crisp set are also applicable for fuzzy sets except for the exeluded-middle laws. In fuzzy set theory, the union of fuzzy set with its complement does not yield the universe and the intersection of fuzzy set and its complement is not null. This difference is shown below ... 

Fuzzy rules are generated based on available historical data, experience and complemented by expert knowledge. Where possible, logbooks are analysed for casualty and accident reports to develop the following rules ... 

The traditional FMEA, the fuzzy rule based method and the grey theory approach may complement each other to produce a risk ranking with confidence. 

Examples of the s plications of fiizzy logic in combination with an AHP are provided by Ayag (2005) and Kwong Bai (2002). Oz goglu Ozdagoglu (2007) provide a comparison between an AHP and a fuzzy AHP. They conclude that many decisions in complex business situations are made in an environment of uncertainty, which benefits fiom the utilization of fiizzy AHP. However, crisp and fiizzy AHPs do not oppose but complement each other since the degree of uncertainty determines the use of the particular method. 

---