Maciej Wygralak

• Ph.D., D.Sc., Assoc. Professor
• Head of Department at Adam Mickiewicz University in Poznań

A concept of general IF-sets, i.e. “intuitionistic” fuzzy sets according to Atanassov, with triangular norm-based hesitation degrees is introduced and developed. That concept is used to construct flexible algorithms of group decision making which involve relative scalar cardinalities defined by means of generalized sigma counts of fuzzy sets. Two c...

We present a general, triangular norm-based approach to hesitation degrees related to I-fuzzy sets, Atanassov’s intuitionistic fuzzy sets. Our main aim will be a closer look at hesitation degrees generated by the three basic t-norms. We will give some illustrative examples showing a true practical sense of those degrees and emphasizing that the nat...

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 s...

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 r...

This article presents a new approach to making decisions when information, possibly incomplete, is provided by many sources. The proposed method is based on IVFS scalar cardinality (sigma f-count). First a general algorithm is introduced, and next an application in supporting medical decisions in ovarian tumor differentiation (based on multiple dia...

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 main topic of this paper is the notion of relative cardinality for interval-valued fuzzy sets - its definition, properties and computation. First we define relative cardinality for interval-valued fuzzy sets following the concept of uncertainty modelling given by Mendel's Wavy-Slice Representation Theorem. We expand on previous approaches by co...

The main topic of this paper is the notion of relative cardinality for interval-valued fuzzy sets – its definition, properties and computation. First we define relative cardinality for interval-valued fuzzy sets following the concept of uncertainty modelling given by Mendel's Wavy-Slice Representation Theorem. We expand on previous approaches by co...

This paper deals with intelligent counting, i.e. counting performed under imprecision, fuzziness of information about the objects of counting. Formally, this collapses to counting in fuzzy sets. We will show that the presented methods of intelligent counting are human-consistent, and reflect and formalize real, human counting procedures. Other appl...

Objectives The aim of this study was to externally validate the diagnostic performance of the International Ovarian Tumor Analysis logistic regression models (LR1 and LR2, 2005) and other popular prognostic models including the Timmerman logistic regression model (1999), the Alcazar model (2003), the risk of malignancy index (RMI, 1990), and the ri...

In this paper we consider applications of bipolarity in modelling problems encountered in ovarian tumor diagnosis. We focus on imprecision of data obtained by a gynaecologist during examinations. We also present a wide range of predictive diagnostic models and propose a new idea for their improvement.

Using the notion of a vaguely defined object, we systematize and unify different existing approaches to vagueness and its mathematical representations, including fuzzy sets and derived concepts. Moreover, a new, approximative approach to vaguely defined objects will be introduced and investigated.

This paper deals with interval-valued fuzzy sets and I-fuzzy sets, Atanassov's intuitionistic fuzzy sets. They are treated in a systematic way as two, formally equivalent, natural extensions of methods of representing incomplete knowledge about sets. We define and investigate triangular norm-based areas of uncertainty of interval-valued fuzzy...

In this paper, we propose a generalization of the definition of an IF-set, an intuitionistic fuzzy set, and related hesitation degrees. We flexibilize the original method of computing these values by the use of triangular norms. Next, we present its application to group decision making problems.

This paper presents a variety of cardinality concepts offered by cardinality theory of triangular norm-based fuzzy sets, including the scalar and the "fuzzy" streams.

This paper concerns the theory of intui- tionistic fuzzy sets according to Atanassov. If triangular norms, especially nonstrict Archimedean ones, are used, we propose a revision and a flexibilizing generalization of some fundamental notions and constructions of that theory. Its application to group de- cision making is outlined.

This chapter presents a theory of generalized FGCounts, FLCounts and FE-Counts which have been introduced in Section 2.5 as generalized cardinals of fuzzy sets with triangular operations. We shall use the notation and terminology established in that section as well as in Chapter 1. Among other questions, the following key issues will be discussed:...

Cardinality, one of the most basic characteristics of a fuzzy set, is a notion having many applications. One of them is elementary probability theory of imprecise events. Fuzzy and nonfuzzy approaches to probabilities of events like “a ball drawn at random from an urn containing balls of various sizes is large” do require an appropriate notion of t...

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,...

Fuzzy sets with triangular norms and their cardinalities understood as convex fuzzy sets pf usual cardinal numbers are subject of this paper. It appears that if nonstrict Archimedean triangular norms are involved, some fuzzy sets become totally dissimilar to any set of any cardinality. In other words, they become singular with respect to cardinalit...

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 cardin...

The “fuzzy” approach to the question of cardinality of a fuzzy set offers a very adequate and complete cardinal information in the form of a convex fuzzy set of ordinary cardinal numbers (of nonnegative integers, in the finite case). The existing studies of that approach, however, are restricted to cardinalities of fuzzy sets with the classical min...

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, divisio...

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...

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 i...

We present a general, complete and easily applicable nonclassical cardinality theory which makes use of the infinite-valued Łukasiewicz logic. The theory pertains to vaguely defined objects, understood as objects that are separated from a universe by means of arbitrary sharp or vague properties, including subdefinite sets (incompletely known sets,...

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 arbitrar...

In this paper we show some unconventional techniques of applying mathematics. More precisely, making use of the ukasiewicz logic we build a nonclassical cardinality theory for vaguely defined objects which are more general constructions than sets. Our attention is focused on questions related to inequalities and comparisons. They seem to be essenti...

In this paper we look, at metrics and norms through many-valued logics, especially through the infinite-valued Lukasiewicz logic. It appears that the roots of those notions and constructions are just in many-valued logics rather than in the classical two-valued logic. This fact and some interpretations following therefrom can stimulate new approach...

In this paper a general theory of power for hardly characterizable objects as well as related generalized cardinal numbers are presented. The attention is focused on the questions of order. ukasiewicz logic is used as a supporting logic. The theory refers both to fuzzy sets and twofold fuzzy sets, partial sets, rough sets, etc.

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.

The purpose of this paper is to present a solution of the still open problem how to describe the cardinality of a rough set. A via-fuzzy-sets approach is proposed. Its main properties are briefly discussed.

The purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality. The proposed idea will be compared with several approaches which were earlier offered in ‘fuzzy’ literature. Using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy...

In this paper we introduce a new definition of fuzzy inclusion and 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....