Some remarks on fuzzy Baire sets

G. Thangaraj,N. Raji 원광대학교 기초자연과학연구소 2023 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.26 No.2

In this paper, the means by which fuzzy Baire sets are obtained from fuzzy simply$^{\ast}$ open sets in fuzzy hyperconnected spaces are discussed. It is obtained that fuzzy Baire sets in fuzzy fraction dense spaces are not fuzzy dense sets. Also the conditions under which fuzzy Baire sets are generated from fuzzy nowhere dense sets, fuzzy dense and fuzzy open sets in fuzzy fraction dense and fuzzy DG$_\delta$-spaces are obtained. It is established that existence of a fuzzy co-$\sigma$-boundary set in fuzzy weakly Baire spaces ensures the existence of a pair of disjoint fuzzy Baire sets and fuzzy open sets in fuzzy hyperconnected and fuzzy nodef spaces are fuzzy Baire sets.

Normal fuzzy probability for generalized triangular fuzzy sets

Chul Kang(강철),Yong Sik Yun(윤용식) 한국지능시스템학회 2012 한국지능시스템학회논문지 Vol.22 No.2

확률공간 (Ω,￡,Ρ) 위에 정의된 퍼지집합을 퍼지이벤트라 한다. Zadeh는 확률 Ρ를 이용하여 퍼지이벤트 A에 대한 확률을 정의하였다. 우리는 일반화된 삼각퍼지집합을 정의하고 거기에 확장된 대수적 작용소를 적용하였다. 일반화된 삼각퍼지집합은 대칭적이지만 함숫값으로 1을 갖이 않을 수 있다. 두 개의 일반화된 삼각퍼지집합 A와 B에 대하여 A(+)B와 A(-)B는 일반화된 사다리꼴퍼지집합이 되었지만, A(·)B와 A(/)B는 일반화된 삼각퍼지집합도 되지 않았고 일반화된 사다리꼴퍼지집합도 되지 않았다. 그리고 정규분포를 이용하여 R 위에서 정규퍼지확률을 정의하였다. 그리고 일반화된 삼각퍼지집합에 대한 정규퍼지확률을 계산하였다. A fuzzy set A defined on a probability space (Ω,￡. P) is called a fuzzy event. Zadeh defines the probability of the fuzzy event A using the probability P. We generalized triangular fuzzy set and apply the extended algebraic operations to these fuzzy sets. A generalized triangular fuzzy set is symmetric and may not have value 1. For two generalized triangular fuzzy sets A and B, A(+)B and A(-)B become generalized trapezoidal fuzzy sets, but A(·)B and A(/)B need not to be a generalized triangular fuzzy set or a generalized trapezoidal fuzzy set. We define the normal fuzzy probability on R using the normal distribution. And we calculate the normal fuzzy probability for generalized triangular fuzzy sets.

구 퍼지집합에 기반을 둔 퍼지시스템 신뢰도 분석

김동혁,조상엽 한국지식정보기술학회 2023 한국지식정보기술학회 논문지 Vol.18 No.6

Since the fuzzy set proposed by Zadeh has been applied to evaluate the reliability of fuzzy systems, various fuzzy sets have been used to analyze the reliability of fuzzy systems. Expressing the reliability of the system as an accurate value is a difficult problem due to the ambiguity of the data. It is possible to overcome these problems with fuzzy sets. Therefore, fuzzy sets provide a way to appropriately express the reliability of inaccurate data that occurs in the real world. In fuzzy sets, reliability is expressed as a real number, which is the degree of membership. In intuitionistic fuzzy sets, reliability is expressed as an interval using positive and negative membership degrees. In the Pythagorean fuzzy sets, reliability is expressed by expressing the positive and negative membership degrees as squares, respectively. Therefore, it is possible to solve the problem where the sum of the positive and negative membership degrees exceeds 1. In picture fuzzy sets, reliability is expressed based on positive membership, neutral membership, and negative membership. Since the picture fuzzy set uses a neutral degree of membership, it becomes possible to express uncertainty that is neither positive nor negative. In the spherical fuzzy sets, reliability is expressed by expressing the positive membership degree, neutral membership degree, and negative membership degree as squares, respectively. Therefore, it is possible to solve the problem in which the sum of the positive membership degree, the neutral membership degree, and the negative membership degree exceeds 1. And since the spherical fuzzy set can express all the properties of conventional fuzzy sets, it can be used in various environments. Therefore, it becomes possible to express the reliability of fuzzy systems in various environments using sphere fuzzy sets, thereby enabling more flexible reliability analysis.

On fuzzy maximal, minimal and mean open sets

A. Swaminathan,S. Sivaraja 한국전산응용수학회 2022 Journal of Applied and Pure Mathematics Vol.4 No.1

We have observed that there exist certain fuzzy topological spaces with no fuzzy minimal open sets. This observation motivates us to investigate fuzzy topological spaces with neither fuzzy minimal open sets nor fuzzy maximal open sets. We have observed if such fuzzy topological spaces exist and if it is connected are not fuzzy cut-point spaces. We also study and characterize certain properties of fuzzy mean open sets in fuzzy T_1-connected fuzzy topological spaces.

q-rung 인접쌍 퍼지집합을 이용한 퍼지시스템 신뢰도 분석

조상엽 한국지식정보기술학회 2020 한국지식정보기술학회 논문지 Vol.15 No.6

Fuzzy set theory introduced by Zadeh has been very successful in dealing with vagueness and uncertainty in various fields. In the fuzzy set, each element of universe belongs to the fuzzy concept with a degree of membership in the unit interval [0, 1]. In order to overcome the problem of fuzzy sets expressing the degree of membership as only one real number, various extensions of fuzzy sets have been developed by many researchers: interval-valued fuzzy sets, intuitionistic fuzzy sets, vague sets, neutrosophic sets, hesitant fuzzy sets, Pythagorean fuzzy sets, orthopair fuzzy sets, etc. In interval-valued fuzzy sets proposed by Tursen, the degree of membership is expressed as a closed subinterval of [0, 1]. Intuitionistic fuzzy sets introduced by Atanassov allow us to represent the degree of membership as truth degree of membership and falsity degree of membership, and the sum of them is limited to 1. Gau et al. also explained vague sets that describe the degree of membership as subinterval. Bustince et al. has proved that these sets are mathematically equivalent to intuitionistic fuzzy sets. In neutrosophic sets proposed by Smarandache, the degree of membership is consisted of truth degree of membership, indeterminacy degree of membership, and falsity degree of membership, and then the indeterminacy is quantified explicitly. Torra introduced hesitant fuzzy sets in which the degree of membership is described by a set of possible values. In the Pythagorean fuzzy set proposed by Yager et al., to solve the problem that the sum of the truth degree of membership and the falsity is greater than one, each degree of membership is squared so that the sum of them is one or less. Orthpair fuzzy set proposed by Yager allow us to express the degree of membership as the q-th power of truth degree of membership and the q-th power of falsity degree of membership. The sum of them is bounded by one. These sets are called the q-rung orthopair fuzzy sets(q-ROFSs). If q = 1, q-ROFSs degenerates to an intuitionistic fuzzy sets and if q = 2, to a Pythagorean fuzzy sets. In this paper, we propose a method for calculating the reliability of fuzzy systems using q-ROFSs which are the generalization of the degree of membership expressed as intervals. Since this method uses the q-ROFS with generalized intervals, it is possible to calculate the reliability of systems more flexibly than the other approaches.

On fuzzy sets having the fuzzy Baire property in fuzzy topological spaces

G. Thangaraj,N. Raji 원광대학교 기초자연과학연구소 2021 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.21 No.2

In this paper, fuzzy sets having the property of fuzzy Baire in the fuzzy topological spaces are introduced by means of fuzzy first category sets. The conditions for the existence of fuzzy Baireness and fuzzy resolvability of fuzzy topological spaces are established by means of fuzzy sets having the property of fuzzy Baire. It is established that there are no fuzzy sets having the property of fuzzy Baire in fuzzy hyperconnected spaces and fuzzy sets having the property of fuzzy Baire in fuzzy Baire spaces are fuzzy second category sets.

피타고라스 퍼지집합에 기반을 둔 퍼지시스템 신뢰도 분석

조상엽 한국지식정보기술학회 2019 한국지식정보기술학회 논문지 Vol.14 No.4

Reliability models play an important role when we design the engineering systems. In conventional reliability model we attempts to describe the values for the reliability of the systems with accurateness. But in real world it is often difficult to get the these exact values. To overcome these problem the fuzzy set theory is used in the reliability model for engineering systems. In the fuzzy sets, the reliability is represented by a real number as the degree of membership of the fuzzy set. ∈ . In the interval valued fuzzy sets, the reliability is described by an interval as the degree of membership of the interval valued fuzzy sets. 0 ≤ ≤ ≤ 1, ⊆ . In the intuitionistic fuzzy sets to express the belief in a belief systems, the reliability is represented by the degree of true membership and degree of falsity membership . , ∈ , 0 ≤ + ≤ 1. In the neurotrophic sets that can represent indeterminacy the reliability is represented as a true membership value , an indeterminacy membership value , and a false membership value . In the multicriteria decision making, the decision maker may or may not provide a degree of satisfying the criteria . In this case, the preference for can be 0 ≤ ≰ 1, which is difficult to process with the conventional fuzzy sets. In this paper, we propose a method to evaluate the reliability of decision making system using Pythagorean fuzzy sets which can be used to solve this problem.

구간값 피타고라스 퍼지집합에 기반을 둔 퍼지 공학시스템 신뢰도 분석

조상엽 한국지식정보기술학회 2023 한국지식정보기술학회 논문지 Vol.18 No.6

The reliability of the engineering systems operating in the real world has an important meaning. Such reliability becomes difficult to obtain an accurate reliability due to inaccurate data, wrong manipulation, etc. To solve this problems, the fuzzy sets have been used. There are several fuzzy sets used to calculate reliability. In the fuzzy sets, it is expressed as a real number , which is the belongingness degree of the fuzzy set. ∈ . In the intuitionistic fuzzy sets, the belongingness degree of the fuzzy set is expressed as , and the non-belongingness degree is described as respectively. , ∈ , 0 ≤ + ≤ 1. In the interval-valued intuitionistic fuzzy sets, the belongingness degree of the fuzzy set is expressed as [, ] and the non-belongingness degree is represented as [, ] respectively. , , , ∈ , + ≤ 1, 0 ≤ + ≤ 1, 0 ≤ + ≤ 1. In the Pythagorean fuzzy sets, the belongingness degree in a fuzzy set is expressed as , and the non-belongingness degree is expressed as . , ∈ , 0 ≤ + ≤ 1. In this paper, we propose a method for calculating the reliability of the fuzzy engineering systems using the interval-valued Pythagorean fuzzy sets that can express flexibility in the Pythagorean fuzzy sets.

픽쳐 퍼지집합을 이용한 퍼지시스템 신뢰도 분석

조상엽 한국지식정보기술학회 2018 한국지식정보기술학회 논문지 Vol.13 No.5

Reliability analysis is the important discipline of reliability engineering. In conventional reliability analysis, the reliability of the components of a system is represented as exact values. Obtaining these data under changing environment conditions is often difficult. Hence fuzzy set theory is used to analyze the fuzzy system reliability, where the reliabilities of the components of a system are represented by fuzzy sets. There are various types of fuzzy sets used to evaluate the reliability of the systems such as the fuzzy sets, interval valued fuzzy sets, intutionistic fuzzy sets, picture fuzzy sets. In the fuzzy sets, the degree of membership is represented as a real number. In the interval valued fuzzy sets, the degree of membership is represented as an interval [, ], where is the minimum degree of membership and is the maximum degree of membership. [, ] ⊆ . In the intuitionistic fuzzy sets, the degree of membership consist of and , where is the degree of membership and is the degree of non-membership. , ∈ . In the picture fuzzy sets, the degree of membership consist of , , and , where is called the degree of positive membership, is called the degree of neutral membership, and is called the degree of negative membership. , , ∈ . In this paper we propose the way to analyze the fuzzy system reliability based on the picture fuzzy sets. The picture fuzzy sets have the capability of representing the positive, negative, neutral, and refusal situation. Therefore the picture fuzzy sets become more flexible to describe the reliabilities than the other methods.

$\alpha$-$b$-regularity in a fuzzy topological space

Anjana Bhattacharyya 원광대학교 기초자연과학연구소 2024 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.27 No.1

This paper deals with a new type of fuzzy separation axiom, viz., fuzzy $\alpha$-$b$-regular space by introducing fuzzy $\alpha$-$b$-open set as a basic tool. This newly defined class of sets is strictly larger than that of fuzzy open set as well as fuzzy preopen set, fuzzy semiopen set, fuzzy $\alpha$-open set and fuzzy $\beta$-open set. Also, we introduce new type of fuzzy compact space and a strong form of fuzzy $T_{2}$-space. However, three different types of functions are introduced and studied. Also the mutual relationships of these functions are established. Lastly some applications of these functions on the spaces introduced here are established.