International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(3): 203-214
Published online September 25, 2024
https://doi.org/10.5391/IJFIS.2024.24.3.203
© The Korean Institute of Intelligent Systems
Ji-Hoon Hong1, Jon-Lark Kim1, Taechang Byun2, and Jin Hee Yoon2
1Department of Mathematics, Sogang University, Seoul, Korea.
2Department of Mathematics and Statistics, Sejong University, Seoul, Korea.
Correspondence to :
Jin Hee Yoon (jin9135@sejong.ac.kr)
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.
Since Zadeh introduced fuzzy sets, various operations for fuzzy numbers, including power and roots, have been proposed. Both square and cube roots are essential in fields that use numbers, including fuzzy numbers. Byun et al. (in Soft Computing, vol. 26, pp. 4163-4169, 2022) introduced the delta root for the square root of a fuzzy number. This study extends this concept by proposing a delta-cube root, offering a functional approach that maintains the integrity of α-level sets and aligns them with Zadeh’s extension principle. Additionally, we introduce the delta n-th root, which generalizes both the delta and delta cube roots, thus broadening the scope of operations on fuzzy numbers.
Keywords: Fuzzy real number, Delta root, Cube root, Delta cube root
No potential conflict of interest relevant to this article was reported.
International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(3): 203-214
Published online September 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.3.203
Copyright © The Korean Institute of Intelligent Systems.
Ji-Hoon Hong1, Jon-Lark Kim1, Taechang Byun2, and Jin Hee Yoon2
1Department of Mathematics, Sogang University, Seoul, Korea.
2Department of Mathematics and Statistics, Sejong University, Seoul, Korea.
Correspondence to:Jin Hee Yoon (jin9135@sejong.ac.kr)
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.
Since Zadeh introduced fuzzy sets, various operations for fuzzy numbers, including power and roots, have been proposed. Both square and cube roots are essential in fields that use numbers, including fuzzy numbers. Byun et al. (in Soft Computing, vol. 26, pp. 4163-4169, 2022) introduced the delta root for the square root of a fuzzy number. This study extends this concept by proposing a delta-cube root, offering a functional approach that maintains the integrity of α-level sets and aligns them with Zadeh’s extension principle. Additionally, we introduce the delta n-th root, which generalizes both the delta and delta cube roots, thus broadening the scope of operations on fuzzy numbers.
Keywords: Fuzzy real number, Delta root, Cube root, Delta cube root
The delta root of the fuzzy number in Example 1.
Membership function of the Delta cube root in Example 2.
Membership function of the delta cube root in Example 3.
Membership function of the delta cube root in Example 4.
Membership function of the delta cube root in Example 5.
Membership function of Example 6.
Membership function of Example 7.
The membership function of
Membership function of
Table 1 . All possible cases of Case (1).
Subcase number | ||||
---|---|---|---|---|
(1-1) | + | + | + | + |
(1-2) | + | + | − | − |
(1-3) | − | − | + | + |
(1-4) | − | − | − | − |
Table 2 . All possible cases of Case (2).
Subcase number | ||||
---|---|---|---|---|
(2-1) | + | − | + | − |
(2-2) | + | − | − | + |
(2-3) | − | + | + | − |
(2-4) | − | + | − | + |
Table 3 . Possible cases for delta 5-th root in Remark 2.
case number | ||||||
---|---|---|---|---|---|---|
+ | + | + | + | + | + | (1) |
− | + | + | + | + | − | (2) |
− | − | + | + | + | + | (3) |
− | − | − | + | + | − | (4) |
− | − | − | − | + | + | (5) |
− | − | − | − | − | − | (6) |
The delta root of the fuzzy number in Example 1.
|@|~(^,^)~|@|Membership function of the Delta cube root in Example 2.
|@|~(^,^)~|@|Membership function of the delta cube root in Example 3.
|@|~(^,^)~|@|Membership function of the delta cube root in Example 4.
|@|~(^,^)~|@|Membership function of the delta cube root in Example 5.
|@|~(^,^)~|@|Membership function of Example 6.
|@|~(^,^)~|@|Membership function of Example 7.
|@|~(^,^)~|@|The membership function of
Membership function of