International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 251-258
Published online September 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.3.251
© The Korean Institute of Intelligent Systems
Islam M. Taha^{1,2}
^{1}Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt
^{2}Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
Correspondence to :
Islam M. Taha (imtaha2010@yahoo.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.
In this study, a new form of fuzzy r-minimal structure [28] called a fuzzy soft r-minimal structure, is defined, which is an extension of the fuzzy soft topology introduced by Aygunoglu et al. [26]. In addition, the concepts of fuzzy soft r-minimal continuity and fuzzy soft r-minimal compactness are introduced in fuzzy soft r-minimal spaces. Some interesting properties and characterizations are also discussed. Finally, several types of fuzzy soft r-minimal compactness are defined, and the relationships between them are characterized.
Keywords: Fuzzy soft set, Fuzzy soft topology, Fuzzy soft r-minimal structure, Fuzzy soft r-minimal space, Fuzzy soft r-minimal continuity, Fuzzy soft r-minimal compactness
In 1999, Molodtsov [1] proposed a completely new concept called a soft set theory to model uncertainty, which associates a set with a set of parameters. Using a soft set, we mean a pair (
The concept of a fuzzy
The aim of this study is to introduce and study the concepts of a fuzzy soft
Throughout this paper,
A fuzzy soft set
A fuzzy soft point
A mapping
Then, the pair (
All definitions and properties of fuzzy soft sets and fuzzy soft topologies are found in [2,4,12,16,24,26].
In this section, a new form of a fuzzy
Let
Let (
Let (
Let (
(1)
(2)
The following implications hold:
In general, the converses are not true.
Let
Then,
Let (
Let (
Let (
(1)
(2)
(3) If
(4)
(5)
(6)
(7)
We prove (5) only because the others are obvious. For
Hence, we obtain
Let (X, M̃) and (Y, M̃^{*}) be an
Let
(1)
(2)
(3)
(4)
(5)
(6) If for every fuzzy soft point
Then, (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇒ (6).
(1) ⇔ (2) Obvious.
(2) ⇒ (3) For all
Hence
(3) ⇔ (4) Let
(4) ⇔ (5) From (7) of Theorem 3.1.
(5) ⇒ (6) Let
Let
(1)
(2)
(3)
The proof is obvious.
Let
Let (
(1) From Theorem 3.1, it is sufficient to show that if
(2) From Theorem 3.1, it is sufficient to show that if
Let
(1)
(2)
(3)
(4)
(5)
The proof follows from Theorems 3.2 and 3.3.
Let
(1) fuzzy soft
(2) fuzzy soft
Let
(1)
(2)
(3)
Then, (1) ⇒ (2) ⇔ (3).
(1) ⇒ (2) For
Hence,
(2) ⇔ (3) For
Hence, we obtain (3). Similarly, we obtain (3) ⇒ (2).
Let
From Theorem 3.4(2), the proof is obvious.
Let
(1)
(2)
(3)
The proof follows from Theorems 3.3 and 3.5.
Let
(1)
(2)
(3)
Then, (1) ⇒ (2) ⇔ (3).
(1) ⇒ (2) For
Hence,
(2) ⇔ (3) For
In this section, several types of fuzzy soft
Let (
Let
Let
Let (
Let (
Obvious.
Let
We define the fuzzy soft
Then,
Let
Let
Hence,
Let (
Let (
For any
Let
We define the fuzzy soft
Then,
Let
Let
From Theorems 3.2, and 3.4, it follows that
Hence
Many researchers have studied fuzzy soft set theory, which is easily applied to many problems having uncertainties from social life. In this paper, we introduced and explored a new form of fuzzy
No potential conflicts of interest relevant to this article are reported.
E-mail: imtaha2010@yahoo.com
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 251-258
Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.251
Copyright © The Korean Institute of Intelligent Systems.
Islam M. Taha^{1,2}
^{1}Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt
^{2}Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
Correspondence to:Islam M. Taha (imtaha2010@yahoo.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.
In this study, a new form of fuzzy r-minimal structure [28] called a fuzzy soft r-minimal structure, is defined, which is an extension of the fuzzy soft topology introduced by Aygunoglu et al. [26]. In addition, the concepts of fuzzy soft r-minimal continuity and fuzzy soft r-minimal compactness are introduced in fuzzy soft r-minimal spaces. Some interesting properties and characterizations are also discussed. Finally, several types of fuzzy soft r-minimal compactness are defined, and the relationships between them are characterized.
Keywords: Fuzzy soft set, Fuzzy soft topology, Fuzzy soft r-minimal structure, Fuzzy soft r-minimal space, Fuzzy soft r-minimal continuity, Fuzzy soft r-minimal compactness
In 1999, Molodtsov [1] proposed a completely new concept called a soft set theory to model uncertainty, which associates a set with a set of parameters. Using a soft set, we mean a pair (
The concept of a fuzzy
The aim of this study is to introduce and study the concepts of a fuzzy soft
Throughout this paper,
A fuzzy soft set
A fuzzy soft point
A mapping
Then, the pair (
All definitions and properties of fuzzy soft sets and fuzzy soft topologies are found in [2,4,12,16,24,26].
In this section, a new form of a fuzzy
Let
Let (
Let (
Let (
(1)
(2)
The following implications hold:
In general, the converses are not true.
Let
Then,
Let (
Let (
Let (
(1)
(2)
(3) If
(4)
(5)
(6)
(7)
We prove (5) only because the others are obvious. For
Hence, we obtain
Let (X, M̃) and (Y, M̃^{*}) be an
Let
(1)
(2)
(3)
(4)
(5)
(6) If for every fuzzy soft point
Then, (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇒ (6).
(1) ⇔ (2) Obvious.
(2) ⇒ (3) For all
Hence
(3) ⇔ (4) Let
(4) ⇔ (5) From (7) of Theorem 3.1.
(5) ⇒ (6) Let
Let
(1)
(2)
(3)
The proof is obvious.
Let
Let (
(1) From Theorem 3.1, it is sufficient to show that if
(2) From Theorem 3.1, it is sufficient to show that if
Let
(1)
(2)
(3)
(4)
(5)
The proof follows from Theorems 3.2 and 3.3.
Let
(1) fuzzy soft
(2) fuzzy soft
Let
(1)
(2)
(3)
Then, (1) ⇒ (2) ⇔ (3).
(1) ⇒ (2) For
Hence,
(2) ⇔ (3) For
Hence, we obtain (3). Similarly, we obtain (3) ⇒ (2).
Let
From Theorem 3.4(2), the proof is obvious.
Let
(1)
(2)
(3)
The proof follows from Theorems 3.3 and 3.5.
Let
(1)
(2)
(3)
Then, (1) ⇒ (2) ⇔ (3).
(1) ⇒ (2) For
Hence,
(2) ⇔ (3) For
In this section, several types of fuzzy soft
Let (
Let
Let
Let (
Let (
Obvious.
Let
We define the fuzzy soft
Then,
Let
Let
Hence,
Let (
Let (
For any
Let
We define the fuzzy soft
Then,
Let
Let
From Theorems 3.2, and 3.4, it follows that
Hence
Many researchers have studied fuzzy soft set theory, which is easily applied to many problems having uncertainties from social life. In this paper, we introduced and explored a new form of fuzzy
Muhammad Ihsan, Atiqe Ur Rahman, Muhammad Saeed, and Hamiden Abd El-Wahed Khalifa
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 233-242 https://doi.org/10.5391/IJFIS.2021.21.3.233Jiří Močkoř
International Journal of Fuzzy Logic and Intelligent Systems 2020; 20(4): 298-315 https://doi.org/10.5391/IJFIS.2020.20.4.298