Fuzzy union

The fuzzy union, that is, the result of the operation A or B is denoted by U 5, and is interpreted as a fuzzy subset D of set U, where the membership function fulfills the condition... 

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 union The union of two fuzzy sets is the reverse of their intersection. That is, the fuzzy union is the largest membership value of the element in either set. The fuzzy union for forming the union of fuzzy sets A and B on the universe of discourse X can be given as ... 

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 ... 

Note the symbol + is not an addition in the normal algebraie sense, but in fuzzy arithmetie denotes a union operation. 

Union The union of two fuzzy sets A and B eorresponds to the Boolean OR funetion and is given by... 

Find the union and interseetion of fuzzy set low temperature L and medium temperature M shown in Figure 10.4. Find also the eomplement of fuzzy set M. Using equation (10.2) the fuzzy sets for = 11 are... 

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. 

Consider, for example, two disjoint events, A and B, defined in terms of adjoining intervals of real numbers, as shown in Figure 5a. Due to the effect of measurement errors, observations at the sharp boundary between events A and B are totally meaningless and should be completely discounted. Moreover, observations in the close neighborhood of this boundary (within the reach of measurement errors) are not fully credible and should be discounted according to some discount-rate function, as illustrated in Fig. 5a. When the same measurements are taken for the union A JB, as shown in Fig. 5b, the discount-rate function is not applicable. Hence, the same observations produce more evidence for the single event A JB than for the two disjoint events A and B. The evidential support for y4 U B is thus not equal to the sum of the evidential supports for A and B. That is, the additivity requirement of probability measures is violated the correct measure is in this case superadditive. Alternatively, an appropriate granulation can be used to define probabilities on fuzzy events. ... 

If the data set X is comprised of two classes, the cluster structure of X may be described by two disjoint fuzzy sets >4j and A 2 whose union is X. Each fuzzy set corresponds to a class (or cluster) of samples in X. The disjointness condition of 1, 2 minimal separation condition for the... 

Two common sets are unified by aggregating all elements that belong to at least one of the two sets into one set. The union of fuzzy sets results from calculating the maximum of their membership functions ... 

Figure 8.22 Intersection (a) and union (b) of two fuzzy sets and the cardinality of a fuzzy set (c).

The union of ail the sets w. r is then obtained by taking the maximum memberships which occur for any product element v) j. rfj from all the sets included in the union. For this problem, i le calculation has been performed on the computer and two typical results are shown in Fig. 6.9. This shows resulting fuzzy sets for the impact of surface excavation S and the total impact P. 

Similarly, OR is used to evaluate the disjunction of rule antecedents, which is implemented by the classical fuzzy operation union in fuzzy logic systems. Consider fuzzy rule 2 ... 

Inference Engine Inference Engine maps input type-2 fuzzy sets into output type-2 fuzzy sets by applying the consequent part where this process of mapping from the antecedent part into the consequent part is interpreted as a type-2 fuzzy implication which needs computations of union and intersection of type-2 fuzzy sets. The inference engine in Mamdani system maps the input fuzzy sets into the output fuzzy sets then the defuzzifier converts them to crisp outputs. The rules in Mamdani model have fuzzy sets in both the antecedent part and the consequent part. For example, the /th rule in a Mamdani rule base can be described as follows ... 

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

Commonly, within the arithmetic interval approach, judgments combination is performed by the classical mean, union or intersection operators. The mean and the intersection operators lead to a reduction of the aggregated interval, namely some values that sources of information consider as possible are ignored. Otherwise, by the union operator the aggregated interval expresses the highest level of uncertainty of the aggregated judgments. In the classic fuzzy approach, each interval information... 

The second layer contains one node for each one fiizzy IF. .. THEN rale. Each rule node performs a connective operation between rule antecedents (the premise or IF part). Usually the minimum or dot product is used as AND intersection. The union or OR is usually performed by using the maximum operator. The firing strength of the fuzzy rales can be computed according to ... 

In (16-38), ak represents the degree of activation of the membership function of the output for rule k. Third, all activated conclusions, for the whole set of rules are accumulated using the set union operation this result corresponds to the fuzzy output (see Fig. 16.21). The procedure described above is often called fuzzy reasoning. Finally, the control output, which is u(t) (see Fig. 16.22), is obtained by applying a defuzzication method. 

For the union of fuzzy sets A and B the maximum of both of the membership functions is taken (Figure 2d) ... 

The extension of classical logic to fuzzy logic may be performed by interpreting the intersection as fuzzy logical AND and the union as fuzzy logical OR . The fuzzy set A can be interpreted as the statement the element x belongs to set A is more or less true , cf. Zimmermann. ... 

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

The defuzzification process relies on the approach to calculating the union of the fuzzy terms of all the consequents. Further, the center of gravity of the union is calculated to be the final output correction. 

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