ORDINAL SUMS IN INTERVAL-VALUED FUZZY SET THEORY

2005 ◽  
Vol 01 (02) ◽  
pp. 243-259 ◽  
Author(s):  
GLAD DESCHRIJVER
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Membership Degree ◽  
Triangular Norms ◽  
Ordinal Sum ◽  
Sum Theorem ◽  
The Universe ◽  
Interval Valued

Interval-valued fuzzy sets form an extension of fuzzy sets which assign to each element of the universe a closed subinterval of the unit interval. This interval approximates the "real", but unknown, membership degree. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. A method for constructing t-norms that satisfy the residuation principle is by using the ordinal sum theorem. In this paper, we construct the ordinal sum of t-norms on [Formula: see text], where [Formula: see text] is the underlying lattice of interval-valued fuzzy set theory, in such a way that if the summands satisfy the residuation principle, then the ordinal sum does too.

SMETS-MAGREZ AXIOMS FOR R-IMPLICATORS IN INTERVAL-VALUED AND INTUITIONISTIC FUZZY SET THEORY

2005 ◽  
Vol 13 (04) ◽  
pp. 453-464 ◽  
Author(s):  
GLAD DESCHRIJVER ◽  
ETIENNE E. KERRE
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Intuitionistic Fuzzy Set ◽  
Membership Degree ◽  
Triangular Norms ◽  
Intuitionistic Fuzzy ◽  
Special Lattice ◽  
Interval Valued

Interval-valued fuzzy sets constitute an extension of fuzzy sets which give an interval approximating the "real" (but unknown) membership degree. Interval-valued fuzzy sets are equivalent to intuitionistic fuzzy sets in the sense of Atanassov which give both a membership degree and a non-membership degree, whose sum must be smaller than or equal to 1. Both are equivalent to L-fuzzy sets w.r.t. a special lattice L*. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In a previous paper5 we gave a construction for t-norms on L* satisfying the residuation principle which are not t-representable. In this paper we investigate the Smets-Magrez axioms and some other properties for the residual implicator generated by such t-norms.

CLASSES OF INTUITIONISTIC FUZZY T-NORMS SATISFYING THE RESIDUATION PRINCIPLE

2003 ◽  
Vol 11 (06) ◽  
pp. 691-709 ◽  
Author(s):  
GLAD DESCHRIJVER ◽  
ETIENNE E. KERRE
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Intuitionistic Fuzzy Sets ◽  
Membership Degree ◽  
Important Class ◽  
Triangular Norms ◽  
Intuitionistic Fuzzy

Intuitionistic fuzzy sets constitute an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a non-membership degree. The only constraint on those two degrees is that the sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In the fuzzy case a t-norm satisfies the residuation principle if and only if it is left-continuous. Deschrijver, Cornelis and Kerre proved that for intuitionistic fuzzy t-norms the equivalence between the residuation principle and intuitionistic fuzzy left-continuity only holds for t-representable t-norms.1 In this paper we construct particular subclasses of intuitionistic fuzzy t-norms that satisfy the residuation principle but that are not t-representable and we show that a continuous intuitionistic fuzzy t-norm [Formula: see text] satisfying the residuation principle is t-representable if and only if [Formula: see text].

scholarly journals REPRESENTABILITY IN INTERVAL-VALUED FUZZY SET THEORY

2007 ◽  
Vol 15 (03) ◽  
pp. 345-361 ◽  
Author(s):  
GLAD DESCHRIJVER ◽  
CHRIS CORNELIS
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Theoretical Approach ◽  
Fuzzy Set Theory ◽  
Logical Connectives ◽  
Representation Method ◽  
Application Developers ◽  
Interval Valued

Interval-valued fuzzy set theory is an increasingly popular extension of fuzzy set theory where traditional [0,1]-valued membership degrees are replaced by intervals in [0,1] that approximate the (unknown) membership degrees. To construct suitable graded logical connectives in this extended setting, it is both natural and appropriate to "reuse" ingredients from classical fuzzy set theory. In this paper, we compare different ways of representing operations on interval-valued fuzzy sets by corresponding operations on fuzzy sets, study their intuitive semantics, and relate them to an existing, purely order-theoretical approach. Our approach reveals, amongst others, that subtle differences in the representation method can have a major impact on the properties satisfied by the generated operations, and that contrary to popular perception, interval-valued fuzzy set theory hardly corresponds to a mere twofold application of fuzzy set theory. In this way, by making the mathematical machinery behind the interval-valued fuzzy set model fully transparent, we aim to foster new avenues for its exploitation by offering application developers a much more powerful and elaborate mathematical toolbox than existed before.

BI-IDEALS OF ORDERED SEMIGROUPS BASED ON THE INTERVAL-VALUED FUZZY POINT

2016 ◽  
Vol 78 (2) ◽  
Author(s):  
Hidayat Ullah Khan ◽  
Nor Haniza Sarmin ◽  
Asghar Khan ◽  
Faiz Muhammad Khan
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Real World ◽  
Fuzzy Set Theory ◽  
Fuzzy Subset ◽  
Ordered Semigroups ◽  
Fuzzy Point ◽  
Interval Valued ◽  
Real World Problems

Interval-valued fuzzy set theory (advanced generalization of Zadeh's fuzzy sets) is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, we introduce the notion of generalized quasi-coincident with () relation of an interval-valued fuzzy point with an interval-valued fuzzy set. In fact, this new concept is a more generalized form of quasi-coincident with relation of an interval-valued fuzzy point with an interval-valued fuzzy set. Applying this newly defined idea, the notion of an interval-valued -fuzzy bi-ideal is introduced. Moreover, some characterizations of interval-valued -fuzzy bi-ideals are described. It is shown that an interval-valued -fuzzy bi-ideal is an interval-valued fuzzy bi-ideal by imposing a condition on interval-valued fuzzy subset. Finally, the concept of implication-based interval-valued fuzzy bi-ideals, characterizations of an interval-valued fuzzy bi-ideal and an interval-valued -fuzzy bi-ideal are considered. 

Modelling Numerical and Spatial Uncertainty in GrayscaleImage Capture Using Fuzzy Set Theory

2009 ◽  
Vol 13 (5) ◽  
pp. 529-536 ◽  
Author(s):  
Mike Nachtegae ◽  
◽  
Peter Sussner ◽  
Tom Mélange ◽  
Etienne E. Kerre ◽  
...  
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Intuitionistic Fuzzy Sets ◽  
Spatial Uncertainty ◽  
Image Model ◽  
Image Capture ◽  
Morphological Theory ◽  
Interval Valued

In this paper, we will discuss interval-valued and intuitionistic fuzzy sets as a model for grayscale images, taking into account the uncertainty regarding the measured grayscale values, which in some cases is also related to the uncertainty regarding the spatial position of an object in an image. We will demonstrate the practical potential of this image model by introducing an interval-valued morphological theory and by illustrating its application with some examples. The results show that the uncertainty that is present during the image capture not only can be modelled, but can also be propagated such that the information regarding the uncertainty is never lost.

scholarly journals An Improved Correlation Coefficient of Intuitionistic Fuzzy Sets

2019 ◽  
Vol 28 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Han-Liang Huang ◽  
Yuting Guo
Keyword(s):  
Fuzzy Sets ◽  
Correlation Coefficient ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Medical Diagnosis ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

Abstract The intuitionistic fuzzy set is a useful tool to deal with vagueness and uncertainty. Correlation coefficient of the intuitionistic fuzzy sets is an important measure in intuitionistic fuzzy set theory and has great practical potential in a variety of areas, such as decision making, medical diagnosis, pattern recognition, etc. In this paper, an improved correlation coefficient of the intuitionistic fuzzy sets is defined, and it can overcome some drawbacks of the existing ones. The properties of this correlation coefficient are discussed. Then, the generalization of the coefficient of interval-valued intuitionistic fuzzy sets is also introduced. Finally, two examples about the application of the proposed correlation coefficient of the intuitionistic fuzzy sets in medical diagnosis and clustering are shown to illustrate the advantages over the existing methods.

scholarly journals A method for classification of red, blue, and green galaxies using fuzzy set theory

2020 ◽  
Vol 499 (1) ◽  
pp. L31-L35
Author(s):  
Biswajit Pandey
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Sloan Digital Sky Survey ◽  
Sky Survey ◽  
Blue Galaxies ◽  
Boundary Sets ◽  
Red Galaxies

ABSTRACT Red and blue galaxies are traditionally classified using some specific cuts in colour or other galaxy properties, which are supported by empirical arguments. The vagueness associated with such cuts are likely to introduce a significant contamination in these samples. Fuzzy sets are vague boundary sets that can efficiently capture the classification uncertainty in the absence of any precise boundary. We propose a method for classification of galaxies according to their colours using fuzzy set theory. We use data from the Sloan Digital Sky Survey (SDSS) to construct a fuzzy set for red galaxies with its members having different degrees of ‘redness’. We show that the fuzzy sets for the blue and green galaxies can be obtained from it using different fuzzy operations. We also explore the possibility of using fuzzy relation to study the relationship between different galaxy properties and discuss its strengths and limitations.

Fuzzy Logic in the Broad Sense

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Human Reasoning ◽  
Human Beings ◽  
Real Numbers ◽  
New Directions ◽  
Gradual Development

The chapter begins by introducing the important and useful distinction between the research agendas of fuzzy logic in the narrow and the broad senses. The chapter deals with the latter agenda, whose ultimate goal is to employ intuitive fuzzy set theory for emulating commonsense human reasoning in natural language and other unique capabilities of human beings. Restricting to standard fuzzy sets, whose membership degrees are real numbers in the unit interval [0,1], the chapter describes how this broad agenda has become increasingly specific via the gradual development of standard fuzzy set theory and the associated fuzzy logic. An overview of currently recognized nonstandard fuzzy sets, which open various new directions in fuzzy logic, is presented in the last section of this chapter.

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