Cardinality Of Fuzzy Set Research Articles

On Ralescu's cardinality of fuzzy sets

We provide a direct formula for Ralescu's scalar cardinality. Unlike the original, iterative definition, the formula reveals intuitive shortcomings of this concept of cardinality. These are apparent from examples and reflected formally in that, as we show, the concept violates one of the axioms of cardinality of fuzzy sets. In addition, we provide a relationship of this concept to Ralescu's concept of fuzzy cardinality which unveils a tight link between the two concepts and points out another counterintuitive property of the concept of scalar cardinality. We argue that the discussed concept of fuzzy cardinality represents an interesting proposition, suggest its geometric interpretation, and provide preliminary observations as a basis for future considerations.

Cardinality of Fuzzy Sets and Accumulation of Small Membership

Cardinality of Fuzzy Sets and Accumulation of Small Membership

Avoiding flatness in factoring ordinal data

Factorization of classical, two-valued Boolean data became a widely studied topic in the past decade due to its role in analyzing relational data as well as its significance for other fields. Recently, various extensions to factorization of ordinal data, or data with graded (fuzzy) attributes, have been proposed. We identify and describe a fundamental problem regarding quality of factors, which is non-existent in the Boolean case, but naturally appears in the more general setting of ordinal data. As we demonstrate, the problem gets more significant with growing size of the factorized data. We analyze the problem, propose a method to alleviate it, and evaluate experimentally our solution to the problem. We also provide a discussion regarding ramifications of our findings for the concept of cardinality of fuzzy sets.

A Bag Theoretic Approach towards the Count of an Intuitionistic Fuzzy Set

The cardinality of fuzzy sets was introduced by DeLuca and termini, Zadeh and Tripathy et al, where the first one is a basic one, the second one is based on fuzzy numbers and the final one introduces a bag theoretic approach. The only approach to find the cardinality of an intuitionistic fuzzy set is due to Tripathy et al. In this paper, we introduce a bag theoretic approach to find the cardinality of intuitionistic fuzzy set, which extends the corresponding definition of fuzzy sets introduced by Tripathy et al. In fact three types of intuitionistic fuzzy counts are introduced and we also establish several properties of these count functions.

An Intuitionistic fuzzy count and cardinality of Intuitionistic fuzzy sets

The notion of Intuitonistic fuzzy sets was introduced by Atanassov [1] as an extension of the concept of fuzzy sets introduced by Zadeh such that it is applicable to more real life situations. In order to measure the cardinality of fuzzy sets several attempts have been made [4,6,8]. However, there are no such measures for intuitionistic fuzzy sets. In this paper we define the sigma count and relative sigma count for intuitionistic fuzzy sets and establish their properties. Also, we illustrate the genration of quantification rules.

On Cardinality of Fuzzy Sets

In this article, we would like to revisit and comment on the widely used definition of cardinality of fuzzy sets. For this purpose we have given a brief description of the history of development of fuzzy cardinality. In the process, we can find that the existing definition fails to give a proper cardinality while dealing with complementation of fuzzy sets. So there arises the need of defining the cardinality in a different manner. Here a new definition of cardinality is proposed which is rooted in the definition of complementation of fuzzy sets on the basis of reference function. This definition of cardinality will inevitably play an important role in any problem area that involves complementation. Further, some important results are proven with the help of the proposed definition and it is found that these properties are somewhat analogus to those obtained with the help of the existing definition.

On the representation of cardinalities of interval-valued fuzzy sets: The valuation property

In the previous work, we developed an axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralak's axiomatic theory of the scalar cardinality of fuzzy sets. Cardinality was defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞). We showed that the scalar cardinality of each interval-valued fuzzy set can be characterized using an appropriate mapping called a cardinality pattern. Moreover, we found some basic conditions under which the valuation property, the subadditivity property, the complementarity rule and the Cartesian product rule are satisfied using different cardinality patterns, t-norms, t-conorms and negations on the lattice LI (the underlying lattice of interval-valued fuzzy set theory). This paper is the first in a series that further investigates the proposed theory, providing a description of cardinality patterns, t-norms, t-conorms and negations satisfying the properties mentioned above. This paper focuses on the valuation property.

On nonstrict Archimedean triangular norms, Hamming distances, and cardinalities of fuzzy sets

This work concerns nonstrict Archimedean triangular norms and cardinalities of fuzzy sets. Values of those t-norms are expressed in the language of Hamming distances between some fuzzy sets. We apply this optics to two important types of cardinalities, namely generalized FGCounts and generalized sigma counts. Finally, we prove that the well-known relationship between the FGCount and the sigma count of a fuzzy set is a limit property of generalized FGCounts involving Yager t-norms. © 2009 Wiley Periodicals, Inc.

Extended fuzzy grouping queries in relational databases and derivation rules

There are few researches on fuzzy queries of relational databases involving the fuzzy conditions or relative linguistic quantifiers in "having" clause.Methods based on fuzzy set theory to these queries were presented here.The fuzzy conditions in "having" clause were translated to general SQL queries on the basis of-cut of membership function.Therefore,the tuples can be filtered by the RDBMS,which improves the efficiency of fuzzy queries.Nonfuzzy cardinality of fuzzy set was used to evaluate the quantified "having" clause,so the computation and the result became simple.

Measures of general fuzzy rough sets on a probabilistic space

Fuzzy rough set is a generalization of crisp rough set, which deals with both fuzziness and vagueness in data. The measures of fuzzy rough sets aim to dig its numeral characters in order to analyze data effectively. In this paper we first develop a method to compute the cardinality of fuzzy set on a probabilistic space, and then propose a real number valued function for each approximation operator of the general fuzzy rough sets on a probabilistic space to measure its approximate accuracy. The functions of lower and upper approximation operators are natural generalizations of the belief function and plausibility function in Dempster–Shafer theory of evidence, respectively. By using these functions, accuracy measure, roughness degree, dependency function, entropy and conditional entropy of general fuzzy rough set are proposed, and the relative reduction of fuzzy decision system is also developed by using the dependency function and characterized by the conditional entropy. At last, these measure functions for approximation operators are characterized by axiomatic approaches.

Subsethood, entropy, and cardinality for interval-valued fuzzy sets—An algebraic derivation

In this paper a unified formulation of subsethood, entropy, and cardinality for interval-valued fuzzy sets (IVFSs) is presented. An axiomatic skeleton for subsethood measures in the interval-valued fuzzy setting is proposed, in order for subsethood to reduce to an entropy measure. By exploiting the equivalence between the structures of IVFSs and Atanassov's intuitionistic fuzzy sets (A-IFSs), the notion of average possible cardinality is presented and its connection to least and biggest cardinalities, proposed in [E. Szmidt, J. Kacprzyk, Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems 118 (2001) 467–477], is established both algebraically and geometrically. A relation with the cardinality of fuzzy sets (FSs) is also demonstrated. Moreover, the entropy-subsethood and interval-valued fuzzy entropy theorems are stated and algebraically proved, which generalize the work of Kosko [Fuzzy entropy and conditioning, Inform. Sci. 40(2) (1986) 165–174; Fuzziness vs. probability, International Journal of General Systems 17(2–3) (1990) 211–240; Neural Networks and Fuzzy Systems, Prentice-Hall International, Englewood Cliffs, NJ, 1992; Intuitionistic Fuzzy Sets: Theory and Applications, Vol. 35 of Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 1999] for FSs. Finally, connections of the proposed subsethood and entropy measures for IVFSs with corresponding definitions for FSs and A-IFSs are provided.

On the cardinalities of interval-valued fuzzy sets

Wygralak has developed an axiomatic theory of scalar cardinalities of fuzzy sets with finite support which covers as particular cases all standard (historical) concepts of the scalar cardinality. In this paper we present a possible extension of this theory to interval-valued fuzzy set theory. We introduce the cardinality of interval-valued fuzzy sets as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [ 0 , + ∞ [ . We study some properties of these cardinalities using t-norms, t-conorms and negations on the lattice L I (the underlying lattice of interval-valued fuzzy set theory). In particular, the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule will be discussed.

Meta-theorems on inequalities for scalar fuzzy set cardinalities

We present meta-theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality. The conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection. In particular, this conjunctor should fulfil a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations can be avoided. This is illustrated in the demonstration of the Łukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the Jaccard coefficient, or equivalently, the triangle inequality of the corresponding dissimilarity measures.

GENERALIZED RELATIVE CARDINALITIES OF FUZZY SETS

The subject of this paper is a generalized approach to relative scalar cardinalities of fuzzy sets. In the main part of the paper, we discuss basic properties of triangular norm-based relative cardinality such as valuation property, the cartesian product rule and complementary rule. Examples of cardinalities satisfying those properties are also given.

Transitivity-preserving fuzzification schemes for cardinality-based similarity measures

A family of fuzzification schemes is proposed that can be used to transform cardinality-based similarity measures for ordinary sets into similarity measures for fuzzy sets in a finite universe. The family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets. In particular, the fuzzy set intersections are pointwisely generated by Frank t-norms. The fuzzification schemes are applied to a variety of previously studied rational cardinality-based similarity measures for ordinary sets and it is demonstrated that transitivity is preserved in the fuzzification process.

A probabilistic definition of a nonconvex fuzzy cardinality

The existing methods to assess the cardinality of a fuzzy set with finite support are intended to preserve the properties of classical cardinality. In particular, the main objective of researchers in this area has been to ensure the convexity of fuzzy cardinalities, in order to preserve some properties based on the addition of cardinalities, such as the additivity property. We have found that in order to solve many real-world problems, such as the induction of fuzzy rules in Data Mining, convex cardinalities are not always appropriate. In this paper, we propose a possibilistic and a probabilistic cardinality of a fuzzy set with finite support. These cardinalities are not convex in general, but they are most suitable for solving problems and, contrary to the generalizing opinion, they are found to be more intuitive for humans. Their suitability relies mainly on the fact that they assume dependency among objects with respect to the property “to be in a fuzzy set”. The cardinality measures are generalized to relative ones among pairs of fuzzy sets. We also introduce a definition of the entropy of a fuzzy set by using one of our probabilistic measures. Finally, a fuzzy ranking of the cardinality of fuzzy sets is proposed, and a definition of graded equipotency is introduced.

Characterization of aggregate fuzzy membership functions using Saaty's eigenvalue approach

This paper describes and extends Saaty's eigenvalue approach to fuzzy membership determination. A genetic algorithm-based procedure is adopted to minimize the failure rates in fuzzy membership determination using Saaty's eigenvalue approach. The proposed method is then extended to develop an aggregate fuzzy membership function using multiple decision-maker environment. A theoretical framework for understanding the magnitude of failures with the increase in the cardinality of fuzzy sets is provided. Scope and purpose Several researchers have shown that characterizing fuzzy memberships functions using the AHP lead to certain failures. We illustrate how a genetic algorithm-based procedure can be used to lower such failures.

An axiomatic approach to scalar cardinalities of fuzzy sets

We present an axiomatic approach to scalar cardinalities of fuzzy sets which is based on a system of three simple postulates. A characterization theorem for those cardinalities is given. The infinite family of possible scalar cardinalities of a fuzzy set generated by the postulates contains as particular cases all standard concepts of scalar cardinality, e.g. the sigma count of a fuzzy set, the cardinality of its core or support, and the cardinality of its t-level set.

Fuzzy cardinality based evaluation of quantified sentences

Quantified statements are used in the resolution of a great variety of problems. Several methods have been proposed to evaluate statements of types I and II. The objective of this paper is to study these methods, by comparing and generalizing them. In order to do so, we propose a set of properties that must be fulfilled by any method of evaluation of quantified statements, we discuss some existing methods from this point of view and we describe a general approach for the evaluation of quantified statements based on the fuzzy cardinality and fuzzy relative cardinality of fuzzy sets. In addition, we discuss some concrete methods derived from the mentioned approach. These new methods fulfill all the properties proposed and, in some cases, they provide an interpretation or generalization of existing methods.

On the Best Scalar Approximation of Cardinality of a Fuzzy Set

It seems that a suitably constructed fuzzy set of natural numbers does form the most complete and adequate description of cardinality of a finite fuzzy set. Nevertheless, in many applications, one needs a simple scalar approximation (evaluation) of that cardinality. Usually, the well-known, but very imperfect concept of sigma-count of a fuzzy set is then used. The aim of this paper is to find the best approximation of cardinality of a finite fuzzy set by a single natural number.