International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 399-406
Published online December 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.4.399
© The Korean Institute of Intelligent Systems
V. Chandiran1 and G. Thangaraj2
1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India
2Department of Mathematics, Thiruvalluvar University, Vellore, India
Correspondence to :
V. Chandiran (profvcmaths@gmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper establishes that pairwise fuzzy semi σ-Baire spaces, pairwise fuzzy semi Baire with pairwise fuzzy semi P-spaces, pairwise fuzzy semi hyperconnected with pairwise fuzzy semi P-spaces are pairwise fuzzy semi Volterra spaces. Furthermore, it establishes that pairwise fuzzy semi σ-second category spaces, pairwise fuzzy semi strongly irresolvable, and pairwise fuzzy semi second category with pairwise fuzzy semi P-(resp. submaximal) spaces are pairwise fuzzy semi weakly Volterra spaces. The conditions for fuzzy bitopological spaces to become pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces, pairwise fuzzy semi Volterra spaces to become fuzzy pairwise semi Baire spaces, and vice versa are discussed. In addition, the conditions for the pairwise fuzzy semi second category using the pairwise fuzzy semi P-(resp. submaximal) spaces to become pairwise fuzzy semi weakly Volterra spaces, pairwise fuzzy semi almost P-spaces to become pairwise fuzzy semi weakly Volterra spaces are investigated.
Keywords: Pairwise fuzzy semi Baire space, Pairwise fuzzy semi P-space, Pairwise fuzzy semi almost P-space, Pairwise fuzzy semi almost GP-space, Pairwise fuzzy semi Volterra space, Pairwise fuzzy semi weakly Volterra space
Zadeh [1] introduced the fundamental concept of fuzzy sets in 1965, forming the backbone of fuzzy mathematics. In 1968, Chang [2] introduced fuzzy topological spaces. Fuzzy semi-open sets and fuzzy semi-continuous mapping was studied by Azad [3]. In 1989, Kandil and El-Shafee [4] introduced fuzzy bitopological spaces. The concept of pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces were introduced and studied in [5, 6]. Kareem and Kumara [7] introduced a CPU scheduling algorithm based on fuzzy logic, designed to facilitate more nuanced decisions in complex situations. The multi-objective optimization approach focused on by Alburaikan et al. [8] provided the best solution in cooperative continuous static games. Fuzzy-logic support systems were studied by Kamel and El-Mougi [9], which addresses uncertainty and imprecision in the medical field. The aim of this paper is to discuss the inter-relations between pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces and other fuzzy bitopological spaces, such as pairwise fuzzy semi hyperconnected spaces, pairwise fuzzy semi Baire spaces, pairwise fuzzy semi
In this section, we review some basic notions and results that are used in the subsequent sections. In this paper, indices
(1) 0
(2) If
(3) If
(
(
(1) (
(2)
(3)
Then
Proof. Suppose that the pfsnd set
Proof. Now
Then,
Proof. Because (
Proof. Suppose that every pfsnd set is a pfs
Proof. Let (
Proof. Let the pfsr sets (
Proof. Let the pfsfc sets
Then,
Proof. Let (
Proof. Let (
Proof. Let (
Proof. Let (
Proof. Let (
Proof. Let (
Clearly,
Proof. Let (
Proof. Let (
Proof. Let
Proof. Let
Proof. Let (
This paper successfully explores the inter-relationships between fuzzy bitopological spaces and various types of fuzzy bitopological spaces such as pairwise fuzzy semi Baire space, pairwise fuzzy semi
Future research should aim to broaden the classification of fuzzy bitopological spaces and their inter-relations beyond those covered in this paper, particularly by incorporating additional space types and new properties. Investigating the practical applications of pairwise fuzzy semi Volterra spaces in different fields such as decision-making, data analysis, and optimization problems. Exploring the connections between the fuzzy bitopological spaces and other fields such as machine learning and network theory.
No potential conflict of interest relevant to this article was reported.
No potential conflict of interest relevant to this article was reported.
The relationship between pairwise fuzzy semi Baire spaces and pairwise fuzzy semi Volterra spaces.
The relationship between pairwise fuzzy semi Baire, pairwise fuzzy semi hyperconnected, pairwise fuzzy semi
The relationship between pairwise fuzzy semi strongly irresolvable, pairwise fuzzy semi second category, pairwise fuzzy semi submaximal, pairwise fuzzy semi
The relationship between pairwise fuzzy semi
E-mail: g.thangaraj@rediffmail.com
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 399-406
Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.399
Copyright © The Korean Institute of Intelligent Systems.
V. Chandiran1 and G. Thangaraj2
1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India
2Department of Mathematics, Thiruvalluvar University, Vellore, India
Correspondence to:V. Chandiran (profvcmaths@gmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper establishes that pairwise fuzzy semi σ-Baire spaces, pairwise fuzzy semi Baire with pairwise fuzzy semi P-spaces, pairwise fuzzy semi hyperconnected with pairwise fuzzy semi P-spaces are pairwise fuzzy semi Volterra spaces. Furthermore, it establishes that pairwise fuzzy semi σ-second category spaces, pairwise fuzzy semi strongly irresolvable, and pairwise fuzzy semi second category with pairwise fuzzy semi P-(resp. submaximal) spaces are pairwise fuzzy semi weakly Volterra spaces. The conditions for fuzzy bitopological spaces to become pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces, pairwise fuzzy semi Volterra spaces to become fuzzy pairwise semi Baire spaces, and vice versa are discussed. In addition, the conditions for the pairwise fuzzy semi second category using the pairwise fuzzy semi P-(resp. submaximal) spaces to become pairwise fuzzy semi weakly Volterra spaces, pairwise fuzzy semi almost P-spaces to become pairwise fuzzy semi weakly Volterra spaces are investigated.
Keywords: Pairwise fuzzy semi Baire space, Pairwise fuzzy semi P-space, Pairwise fuzzy semi almost P-space, Pairwise fuzzy semi almost GP-space, Pairwise fuzzy semi Volterra space, Pairwise fuzzy semi weakly Volterra space
Zadeh [1] introduced the fundamental concept of fuzzy sets in 1965, forming the backbone of fuzzy mathematics. In 1968, Chang [2] introduced fuzzy topological spaces. Fuzzy semi-open sets and fuzzy semi-continuous mapping was studied by Azad [3]. In 1989, Kandil and El-Shafee [4] introduced fuzzy bitopological spaces. The concept of pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces were introduced and studied in [5, 6]. Kareem and Kumara [7] introduced a CPU scheduling algorithm based on fuzzy logic, designed to facilitate more nuanced decisions in complex situations. The multi-objective optimization approach focused on by Alburaikan et al. [8] provided the best solution in cooperative continuous static games. Fuzzy-logic support systems were studied by Kamel and El-Mougi [9], which addresses uncertainty and imprecision in the medical field. The aim of this paper is to discuss the inter-relations between pairwise fuzzy semi Volterra (resp. weakly Volterra) spaces and other fuzzy bitopological spaces, such as pairwise fuzzy semi hyperconnected spaces, pairwise fuzzy semi Baire spaces, pairwise fuzzy semi
In this section, we review some basic notions and results that are used in the subsequent sections. In this paper, indices
(1) 0
(2) If
(3) If
(
(
(1) (
(2)
(3)
Then
Proof. Suppose that the pfsnd set
Proof. Now
Then,
Proof. Because (
Proof. Suppose that every pfsnd set is a pfs
Proof. Let (
Proof. Let the pfsr sets (
Proof. Let the pfsfc sets
Then,
Proof. Let (
Proof. Let (
Proof. Let (
Proof. Let (
Proof. Let (
Proof. Let (
Clearly,
Proof. Let (
Proof. Let (
Proof. Let
Proof. Let
Proof. Let (
This paper successfully explores the inter-relationships between fuzzy bitopological spaces and various types of fuzzy bitopological spaces such as pairwise fuzzy semi Baire space, pairwise fuzzy semi
Future research should aim to broaden the classification of fuzzy bitopological spaces and their inter-relations beyond those covered in this paper, particularly by incorporating additional space types and new properties. Investigating the practical applications of pairwise fuzzy semi Volterra spaces in different fields such as decision-making, data analysis, and optimization problems. Exploring the connections between the fuzzy bitopological spaces and other fields such as machine learning and network theory.
No potential conflict of interest relevant to this article was reported.
The relationship between pairwise fuzzy semi Baire spaces and pairwise fuzzy semi Volterra spaces.
The relationship between pairwise fuzzy semi Baire, pairwise fuzzy semi hyperconnected, pairwise fuzzy semi
The relationship between pairwise fuzzy semi strongly irresolvable, pairwise fuzzy semi second category, pairwise fuzzy semi submaximal, pairwise fuzzy semi
The relationship between pairwise fuzzy semi
The relationship between pairwise fuzzy semi Baire spaces and pairwise fuzzy semi Volterra spaces.
|@|~(^,^)~|@|The relationship between pairwise fuzzy semi Baire, pairwise fuzzy semi hyperconnected, pairwise fuzzy semi
The relationship between pairwise fuzzy semi strongly irresolvable, pairwise fuzzy semi second category, pairwise fuzzy semi submaximal, pairwise fuzzy semi
The relationship between pairwise fuzzy semi