Maciej Wygralak

Rough sets and fuzzy sets—some remarks on interrelations

Abstract This short paper refers to two remarks given in [3] and disputes them. Any rough set can be expressed by means of a membership function U → {0, 0.5, 1}. We show that this representation can be extended to union and intersection.

Cardinalities of fuzzy sets

1. Triangular Operations and Negations (Allegro).- 1.1. Triangular Norms and Conorms.- 1.2. Negations.- 1.3. Associated Triangular Operations.- 1.4. Archimedean Triangular Operations.- 1.5. Induced Negations and Complementary Triangular Operations.- 1.6. Implications Induced by Triangular Norms.- 2. Fuzzy Sets (Andante spianato).- 2.1. The Concept of a Fuzzy Set.- 2.2. Operations on Fuzzy Sets.- 2.3. Generalized Operations.- 2.4. Other Elements of the Language of Fuzzy Sets.- 2.5. Towards Cardinalities of Fuzzy Sets.- 3. Scalar Cardinalities of Fuzzy Sets (Scherzo).- 3.1. An Axiomatic Viewpoint.- 3.2. Cardinality Patterns.- 3.3. Valuation Property and Subadditivity.- 3.4. Cartesian Product Rule and Complementarity.- 3.5. On the Fulfilment of a Group of the Properties.- 3.5.1. VAL and CART.- 3.5.2. CART and COMP.- 3.5.3. VAL and COMP.- 3.5.4. VAL, CART and COMP.- 4. Generalized Cardinals with Triangular Norms (Rondeau a la polonaise).- 4.1. Generalized FGCounts.- 4.1.1. The Corresponding Equipotency Relation.- 4.1.2. Inequalities.- 4.1.3. Arithmetical Operations.- 4.1.3.1. Addition.- 4.1.3.2. Subtraction.- 4.1.3.3. Multiplication.- 4.1.3.4. Division.- 4.1.3.5. Exponentiation.- 4.1.4. Some Derivative Concepts of Cardinality.- 4.2. Generalized FLCounts.- 4.2.1. Equipotencies and Inequalities.- 4.2.2. Addition and Other Arithmetical Operations.- 4.3. Generalized FECounts.- 4.3.1. The Height of a Generalized FECount.- 4.3.2. Singular Fuzzy Sets.- 4.3.3. Equipotencies, Inequalities and Arithmetical Questions.- List of Symbols.

An axiomatic approach to scalar cardinalities of fuzzy sets

Abstract 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.

Questions of cardinality of finite fuzzy sets

Abstract In this paper we focus our attention on finite fuzzy sets. A complete, simple and easily applicable cardinality theory for them is presented. Questions of equipotency and non-classically understood cardinal numbers of finite fuzzy sets are discussed in detail. Also, problems of arithmetical operations (addition, subtraction, multiplication, division, and exponentiation) on as well as ordering relation for those cardinals are carefully investigated.

Fuzzy sets with triangular norms and their cardinality theory

The most advanced and adequate approach to the question of cardinality of a fuzzy set seems to be that offering a fuzzy perception of cardinality. The resulting convex fuzzy sets of usual cardinals (of nonnegative integers, in the finite case) are then called generalized cardinal numbers. Three types of them are of special interest and importance, namely FGCounts, FLCounts, and FECounts. In this paper, first, we show that their original forms are suitable only for fuzzy sets with the classical min and max operations. Second, we propose an appropriate generalization to fuzzy sets with triangular norms and conorms. Further, we investigate the resulting generalized FGCounts, FLCounts, and FECounts from the viewpoint of the corresponding equipotency relations, inequalities, and arithmetical operations.

Generalized cardinal numbers and operations on them

Abstract In this paper we present a cardinality theory for so-called vaguely defined objects which are mild generalizations of fuzzy sets, obtained by introducing lower and upper approximations of the membership functions. The theory makes use of the sentential calculus in the infinite-valued Łukasiewicz logic, and can be applied to fuzzy sets with arbitrary supports, (some kinds of) twofold fuzzy sets, partial sets, etc. The notion of equipotency of vaguely defined objects is studied in detail. The resulting generalized cardinal numbers are convenient tools for describing the powers of vaguely defined objects. In the second part of the paper, basic operations on the generalized cardinals are defined and carefully investigated. Similarities and anomalies in comparison with the classical arithmetic of the usual cardinals are indicated.

A Bipolar View on Medical Diagnosis in OvaExpert System

In the paper we present OvaExpert - a unique tool for supporting gynecologists in the diagnosis of ovarian tumor, combining classical diagnostic scales with modern methods of machine learning and soft computing. A distinguishing feature of the system is its comprehensiveness, which makes it usable at any stage of a diagnostic process. We gather all the results and solutions making up the system, some of which were described in our other publications, to provide an overall picture of OvaExpert and its capabilities. A special attention is paid to a property of supporting uncertainty modeling and processing, that is an essential part of the system.

On triangular norm-based generalized cardinals and singular fuzzy sets

There are three basic types of triangular norm-based generalized cardinals of fuzzy sets, namely generalized FGCounts, FLCounts and FECounts. All of them are convex fuzzy sets of usual cardinal numbers. Our attention will be focused on generalized FECounts. If nonstrict Archimedean triangular norms are involved, generalized FECounts of many fuzzy sets become the zero function. Those singular fuzzy sets, being totally dissimilar to any set of any cardinality, and their properties are the subject of this paper. A relationship between singularity and fuzziness measures suited to fuzzy sets with complements induced by a strict negation will also be discussed.

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.

Triangular Operations, Negations, and Scalar Cardinality of a Fuzzy Set

Scalar approaches to cardinality of a fuzzy set are very simple and convenient, which justifies their frequent use in many areas of applications instead of more advanced and adequate forms such as fuzzy cardinals. On the other hand, theoretical investigations of scalar cardinalities in the hitherto existing subject literature are rather occasional and fragmentary, and lacking in closer references to triangular norms and conorms. This paper is an attempt at filling that gap by constructing an axiomatized theory of scalar cardinality for fuzzy sets with triangular norms and conorms. It brings together all standard scalar approaches, including the so-called sigma-counts and p-powers, and offers infinitely many new alternative options.