Dual extension principle and dual fuzzification using lower level sets: representation theorems

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally, by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functions at each point of discontinuity, where one-sided limits exist.

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Characterization of matrix classes involving some sets of sequences of fuzzy numbers

Filomat ◽

10.2298/fil1719101d ◽

2017 ◽

Vol 31 (19) ◽

pp. 6101-6112 ◽

Cited By ~ 2

Author(s):

Hemen Dutta ◽

Jyotishmaan Gogoi

Keyword(s):

Level Sets ◽

Fuzzy Numbers ◽

Extension Principle ◽

New Approach ◽

Matrix Classes ◽

Sequences Of Fuzzy Numbers

In 1996, M. Stojakovic and Z. Stojakovic examined the convergence of a sequence of fuzzy numbers via Zadeh?s Extension Principle, which is quite difficult for practical use. In this paper, we utilize the notion ?-level sets to deal with convergence and summable related notions and adopted a relatively new approach to characterize matrix classes involving some sets of single sequences of fuzzy numbers. The approach is expected to be useful in dealing with characterization of several other matrix classes involving different kinds of sets of sequences of fuzzy numbers, single or multiple

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On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions

Journal of Function Spaces ◽

10.1155/2015/870179 ◽

2015 ◽

Vol 2015 ◽

pp. 1-10

Author(s):

Uğur Kadak ◽

Hakan Efe

Keyword(s):

Power Series ◽

Uniform Convergence ◽

Membership Function ◽

Level Sets ◽

Radius Of Convergence ◽

Extension Principle ◽

Membership Functions ◽

Function Series ◽

Convergence Of Sequences ◽

The Relationship

The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, we introduce the power series with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.

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Some New Sets of Sequences of Fuzzy Numbers with Respect to the Partial Metric

The Scientific World JOURNAL ◽

10.1155/2015/735703 ◽

2015 ◽

Vol 2015 ◽

pp. 1-10

Author(s):

Uğur Kadak ◽

Muharrem Ozluk

Keyword(s):

Bounded Variation ◽

Level Sets ◽

Fuzzy Numbers ◽

Metric Spaces ◽

Partial Ordering ◽

Extension Principle ◽

Partial Metric ◽

Partial Metric Spaces ◽

Sequences Of Fuzzy Numbers ◽

Null Series

In this paper, we essentially deal with Köthe-Toeplitz duals of fuzzy level sets defined using a partial metric. Since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct some classical notions. In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some of their properties like completeness and duality. Finally, we obtain the Köthe-Toeplitz duals of fuzzy level sets with respect to the partial metric based on a partial ordering.

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Level Sets, the Representation Theorem and the Extension Principle for Interval Valued Fuzzy Sets

Lecture Notes in Electrical Engineering - Electrical Power Systems and Computers ◽

10.1007/978-3-642-21747-0_110 ◽

2011 ◽

pp. 853-860

Author(s):

Hongmei Ju

Keyword(s):

Fuzzy Sets ◽

Representation Theorem ◽

Level Sets ◽

Extension Principle ◽

Interval Valued

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Inner Product of Fuzzy Vectors

10.21203/rs.3.rs-486220/v1 ◽

2021 ◽

Author(s):

Hsien-Chung Wu

Keyword(s):

Potential Application ◽

Level Sets ◽

Optimization Problems ◽

Linear Optimization ◽

Decomposition Theorem ◽

Inner Product ◽

Extension Principle ◽

Membership Functions ◽

Linear Optimization Problems ◽

Fuzzy Interval

Abstract The inner product of vectors of non-normal fuzzy intervals will be studied in this paper by using the extension principle and the form of decomposition theorem. The membership functions of inner product will be different with respect to these two different methodologies. Since the non-normal fuzzy interval is more general than the normal fuzzy interval, the corresponding membership functions will become more complicated. Therefore, we shall establish their relationship including the equivalence and fuzziness based on the a-level sets. The potential application of inner product of fuzzy vectors is to study the fuzzy linear optimization problems.

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Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures

Information Sciences ◽

10.1016/j.ins.2008.05.022 ◽

2008 ◽

Vol 178 (18) ◽

pp. 3565-3576 ◽

Cited By ~ 29

Author(s):

Ronald R. Yager

Keyword(s):

Fuzzy Sets ◽

Level Sets ◽

Extension Principle ◽

Uncertainty Measures ◽

Interval Valued

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Arithmetics of Vectors of Fuzzy Sets

Mathematics ◽

10.3390/math8091614 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1614

Author(s):

Hsien-Chung Wu

Keyword(s):

Fuzzy Sets ◽

Level Sets ◽

Decomposition Theorem ◽

Extension Principle ◽

Membership Functions ◽

Arithmetic Operations ◽

Mild Conditions ◽

Different Types ◽

Scalar Products ◽

Α Level

The arithmetic operations of fuzzy sets are completely different from the arithmetic operations of vectors of fuzzy sets. In this paper, the arithmetic operations of vectors of fuzzy intervals are studied by using the extension principle and a form of decomposition theorem. These two different methodologies lead to the different types of membership functions. We establish their equivalences under some mild conditions. On the other hand, the α-level sets of addition, difference and scalar products of vectors of fuzzy intervals are also studied, which will be useful for the different usage in applications.

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