Equality of Fuzzy Set
The Equality of fuzzy sets corresponds to the Lukasiewicz t-norm. In this paper we study the notion of (∗, δ)-equality, a concept which generalizes the Eequality to the case of the fuzzy set theory based on an arbitrary continuous t-norm ∗. We investigate the robustness of some fuzzy implication operators in terms of (∗, δ)-equality.
Equality between fuzzy subsets. The main idea is that we evaluate a fuzzy equality not only by one value, as was done up to the present, but by two ones, i.e. by a pair of numbers from the closed unit interval. Basic properties of the proposed fuzzy inclusion and equality are given. Using the new notion of fuzzy equality we next define fuzzy operations for fuzzy subsets. These operations are also characterized by a pair of numbers. All fundamental laws for fuzzy operations are listed. Some connections between łLukasiewicz logic and introduced definitions of equality of fuzzy sets are also described.
Cardinality of Fuzzy Set
The problem of counting the number of elements in a fuzzy set plays a central role in the theories of fuzzy probability and in the definition of linguistic quantifiers [1,2]. In Ref. [3] Dubois and Prade provide a very extensive survey of the various approaches to obtaining the cardinality of a fuzzy set. In Ref. [3] they note that the two basic approaches to measuring the cardinality of a fuzzy set are the scalar method, which essentially adds the membership grades, and the fuzzy integer approach, which represents the cardinality as a fuzzy number rather than a scalar. In Ref. [3] they also note a number of drawbacks associated with each of these methods. In this paper we suggest an alternate approach to representing the cardinality (or count) of a fuzzy subset in terms of bags [4]. Since bags provide a natural structure for representing “set-like” objects in which a count of the number of elements is of relevence, they would seem to be of possible use to represent counts of sets. Because the bag structure is not as yet very commonly used, we begin with an introduction to this topic. More information on bags can be found in Ref. [4] as well as in Refs [5,6].
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