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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

Delta Cube Root as a General Concept of the Cube Root of a Fuzzy Number

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)

Received: November 28, 2023; Revised: August 7, 2024; Accepted: August 24, 2024

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

This work was supported by the National Research Foundation of Korea grant funded by the Korean government (MSIT) (No. RS-2024-00351610), the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (No. 2022R1I1A1A01072471), and the BK21 FOUR program through the National Research Foundation of Korea under the Department of Mathematics at Sogang University “Nurturing team for creative and convergent mathematical science talents.”

No potential conflict of interest relevant to this article was reported.

Ji-Hoon Hong received his B.S. degree in mathematics from Sogang University. He is a graduate of an integrated M.S. and Ph.D. program in the Department of Mathematics at Sogang University, Seoul, Korea. He is interested in soft computing, including fuzzy theory, genetic algorithms, and other metaheuristics. He also studied coding theory and artificial intelligence.

Jon-Lark Kim received his B.S. degree in mathematics from Pohang University of Science and Technology (POSTECH), finished the master’s course in mathematics from Seoul National University, and received his Ph.D. in mathematics from the University of Illinois at Chicago, in 1993, 1997, and 2002, respectively. He joined the University of Nebraska-Lincoln from 2002 to 2005 as a research assistant professor. He joined the University of Louisville between 2005 and 2012 as an assistant and association professor. Since 2012, he has been a professor at the Department of Mathematics at Sogang University, Seoul, Korea. His research interests include coding theory, cryptography, soft computing, and artificial intelligence.

Taechang Byun received his B.S. degree in mathematics from Seoul National University (SNU), finished the master’s course in mathematics from Seoul National University, and received his Ph.D. in Mathematics from the University of Oklahoma (OU), in 1999, 2002, and 2011, respectively. He joined Sejong University, Seoul, Korea, in 2014 and is currently a visiting professor in the Department of Mathematics and Statistics at Sejong University. His research interests include Riemannian geometry, Lie groups, differential topologies, and fuzzy mathematics.

Jin Hee Yoon received her B.S., M.S., and Ph.D. degrees in mathematics from Yonsei University, Korea. She is currently a faculty member of the Department of Mathematics and Statistics at Sejong University in Seoul, Korea. Her research interests include soft computing, fuzzy theories, intelligent systems, and machine learning. She is a board member of KIIS the Korean Institute of Intelligent Systems and has been working as an associate editor, guest editor, and editorial board member for several journals, including SCI. In addition, she is a co-representative of Korea for IFSA the International Fuzzy Systems Association and the vice chair of Communications and Education for the IEEE Fuzzy System Technical Committee.

Article

Original Article

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.

Delta Cube Root as a General Concept of the Cube Root of a Fuzzy Number

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)

Received: November 28, 2023; Revised: August 7, 2024; Accepted: August 24, 2024

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.

Abstract

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

Fig 1.

Figure 1.

The delta root of the fuzzy number in Example 1.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 2.

Figure 2.

Membership function of the Delta cube root in Example 2.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 3.

Figure 3.

Membership function of the delta cube root in Example 3.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 4.

Figure 4.

Membership function of the delta cube root in Example 4.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 5.

Figure 5.

Membership function of the delta cube root in Example 5.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 6.

Figure 6.

Membership function of Example 6.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 7.

Figure 7.

Membership function of Example 7.

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 8.

Figure 8.

The membership function of η(t) in Case (1-3).

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Fig 9.

Figure 9.

Membership function of η(t) in Case (2-3).

The International Journal of Fuzzy Logic and Intelligent Systems 2024; 24: 203-214https://doi.org/10.5391/IJFIS.2024.24.3.203

Table 1 . All possible cases of Case (1).

Subcase numberxyzt
(1-1)++++
(1-2)++
(1-3)++
(1-4)

Table 2 . All possible cases of Case (2).

Subcase numberxyzt
(2-1)++
(2-2)++
(2-3)++
(2-4)++

Table 3 . Possible cases for delta 5-th root in Remark 2.

x1x2x3x4x5tcase number
++++++(1)
++++(2)
++++(3)
++(4)
++(5)
(6)