scholarly journals Some comments on level sets of fuzzy sets

2008 ◽  
Author(s):  
Carol L. Walker ◽  
Elbert A. Walker ◽  
Ronald R. Yager
Keyword(s):  
Fuzzy Sets ◽  
Level Sets

scholarly journals Convexity and level sets for interval-valued fuzzy sets

2022 ◽  
Author(s):  
Pedro Huidobro ◽  
Pedro Alonso ◽  
Vladimír Janiš ◽  
Susana Montes
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Level Sets ◽  
Fuzzy Optimization ◽  
Interval Valued

AbstractConvexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-valued fuzzy sets in order to be able to cover some wider area of imprecision. We show some of its interesting properties, and study the preservation under the intersection and the cutworthy property. Finally, we applied convexity to decision-making problems.

scholarly journals The resolution theorems for intuitionistic fuzzy sets with sheet-level sets

2019 ◽  
Vol 25 (2) ◽  
pp. 20-24
Author(s):  
Arif Bal ◽  
◽  
Gökhan Çuvalcıoğlu ◽  
Gül Tümen ◽  
◽  
...  
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy

scholarly journals Sheet level sets and the representation theorem for intuitionistic fuzzy sets

2019 ◽  
Vol 25 (2) ◽  
pp. 15-19
Author(s):  
Gökhan Çuvalcıoğlu ◽  
◽  
Sinem Tarsuslu ◽  
Emine Demirbaş ◽  
◽  
...  
Keyword(s):  
Fuzzy Sets ◽  
Representation Theorem ◽  
Level Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy

Generating fuzzy sets from the families of sets

2021 ◽  
pp. 1-22
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Decision Makers ◽  
Membership Functions ◽  
Engineering Problems ◽  
Original Family ◽  
Mechanical Procedure ◽  
Subjective Bias ◽  
Α Level

The main purpose of this paper is to establish a mechanical procedure to determine the membership functions using the data collected from the economic and engineering problems. Determining the membership functions from the collected data may depend on the subjective viewpoint of decision makers. The mechanical procedure proposed in this paper can get rid of the subjective bias of decision makers. The concept of solid families is also proposed by regarding the sets in a family to be continuously varied. The desired fuzzy sets will be generated in the sense that its α-level sets will be identical to the sets of the original family. In order to achieve this purpose, any arbitrary families will be rearranged as the nested families by applying some suitable functions to the original families that are formulated from the collected data.

scholarly journals Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Mariusz Michta
Keyword(s):  
Differential Equations ◽  
Stochastic Processes ◽  
Fuzzy Sets ◽  
Stochastic Differential Equations ◽  
Integral Equations ◽  
Metric Space ◽  
Level Sets ◽  
Stochastic Integral ◽  
Stochastic Integral Equations ◽  
Fuzzy Solutions

The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms.

scholarly journals Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 11 ◽  
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Fuzzy Sets ◽  
Level Sets ◽  
Decomposition Theorem ◽  
Dual Decomposition ◽  
Α Level

The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets.

scholarly journals Arithmetics of Vectors of Fuzzy Sets

Mathematics ◽  
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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