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
A fuzzy (real) number is represented by a function from a set of real numbers ℝ to the interval [0, 1]. The concept of a fuzzy set was first introduced in 1965 by Zadeh [1].
Since then, various operations and properties of fuzzy numbers have been proposed by the authors of [1–3] to handle data that are not clearly expressed by real numbers or include imprecise information. The concepts of powers and roots of fuzzy numbers are frequently required to handle complex fuzzy numbers in fuzzy data. AbuAarqob et al. [4] employed the
Both square and cube roots are essential concepts in this field of research. Both these concepts are necessary when fuzzy numbers are used. As research has explored the square root of fuzzy numbers, it is important to discuss the cube root and the generalized concept of the
Determining an
Research on the
In this study, we investigate whether the delta cube root of a fuzzy number
By defining a delta cube root similarly to a delta root, we propose a generalized functional expression for the cube root of a fuzzy number. This study presents the delta cube root as a complete function that enables intuitive understanding and rigorous verification through graphical representation, in contrast to traditional approaches that compute the cube root case-by-case using the
The remainder of this paper is organized as follows. In Section 2, we provide the preliminaries of both the operations on fuzzy numbers and the definition of the delta root of a fuzzy number. In Section 3, we define the delta cube root of a fuzzy number and provide its properties using examples. In Section 4, we observe that the delta cube root of the cube of a nonnegative fuzzy number becomes the original fuzzy number. Furthermore, the cube of its delta cube root becomes itself. For real numbers, for any
In this section, we provide the basic definitions and properties of fuzzy numbers. In addition, we introduce the definition of the delta root of a fuzzy number, which is foundational for defining the delta cube root of a fuzzy number.
Xiao eand Zhu [3] proposed a fuzzy number with an
A mapping
(i) There exists
(ii) For each
In this study, the notation for
Hence, ℝ can be embedded into
The Zadeh [1] extension principle is as follows:
Let fuzzy sets
It is essential to implement the boundary conditions of the fuzzy sets to ensure that the membership functions are meaningful and aligned with the property characteristics of the real numbers. The boundary conditions for the fuzzy sets that must be satisfied are as follows:
• Normality
The membership function
• Boundedness
The membership function
• Convexity
The membership function must be continuous and convex. This property can be expressed as follows: for all
• Piecewise continuity
The membership function must be continuous or piecewise continuous. This implies that even if there are discontinuities, their number must be finite, ensuring the function remains manageable and meaningful:
These boundary conditions are critical for defining and applying fuzzy numbers. By implementing these conditions, we ensure that the membership functions used in this study provide meaningful and robust representations of fuzzy sets that align with real-number properties.
Kaieva and Seikkala [2] applied the Zadeh extension principle to arithmetic operations, as follows:
The arithmetic operations ⊕, ⊖, ⊗, and ⊘ on
The square root of a fuzzy number is defined using
For all
Given a fuzzy number, the delta root of a fuzzy number by Byun et al. [16] is defined as follows.
For
This is because a fuzzy number (see the definition of a fuzzy number) itself is a type of fuzzy set, and the delta root of a fuzzy number is a type of fuzzy number. Therefore, a delta root is a special type of fuzzy set.
Delta root is a concept proposed to define the fuzzy square root. Unlike the traditional approach, which starts from the
Let
Then
The two-membership functions in Example 1 are illustrated in Figure 1.
We introduce the following properties of the delta root proven by Byun et al. [16]:
Let
(i) For
(ii) It holds that
The delta cube root generalizes the cube root of a fuzzy number, thereby extending the concept established by the delta root. Unlike a square root, a cube root can have multiple values for real numbers, including negative numbers. Although a quadratic equation with real coefficients does not always guarantee a real root, a cubic equation consistently has at least one real root. This principle should be extended to fuzzy numbers, necessitating the delta cube root definition as a function. The delta cube root must account for various cases, including both nonnegative and negative fuzzy numbers. The delta cube root is defined as follows.
Traditionally, the cube root of a fuzzy number using the
For
When addressing the delta root, only the case where
Case 1.
Case 2.
Case 3.
Case 4.
Case 5.
In Case 1, the delta cube root of
Let
Subsequently, for
Case 3 is illustrated in Example 3.
Let
Then, for
Case 4 is described by Example 4.
Let
Then, for
Case 5 is described in Example 5.
Let
Then, for
Figures 2
More cases must be considered for delta cube roots than for delta roots. But the following properties hold for the delta cube root as they do for the delta root by Byun et al. [16].
Let
Let
Also in the same way,
To simplify the expression, we introduce a notation for the delta cube root of the
For
In the real number system, ℝ, for any
In this section, we first examine the delta cube root of the cube of a nonnegative fuzzy number and then extend the analysis to other cases. We also show that the delta cube root of the cube of a nonnegative or nonpositive fuzzy number is the fuzzy number.
To explore the delta cube root of the cube of a fuzzy number, we begin by considering the case where
For
from Definitions 2 and 5.
First, we consider the case in which
For any (
which implies that
Second, we consider the case in which
We observe the case where
To show
Without a loss of generality, we assume that
Case (1) If
Case (2) If
These two cases are presented in Tables 1 and 2.
As we must observe
However, by assumption, it holds that:
Thus,
holds for any
We prove that
Suppose we aim to solve the equation
Then we can denote
Example 6 is shown in Figure 6.
Suppose we need to solve the equation
Note that
Subsequently, we can denote
Example 7 is shown in Figure 7.
We now observe the delta cube root when
For
We also observe that the cube of the delta cube root of a nonnegative/nonpositive fuzzy number becomes itself.
For
where the first equality is given by Corollary 1.
Thus far, a fuzzy number
Hereafter, a general fuzzy number
We give examples on
1. SubCase (1-3): Define
Then by Definitions 2 and 5,
For
Consider a particular one (
Hence, the equation
2. SubCase (2-3): Define
Then by Definitions 2 and 5,
Note that, for
Consider a particular one (
Hence,
Therefore,
Consequently, it is trivial for
Naturally, we can consider the delta
(Observation about the delta 5-th root of 5-th power of fuzzy real number)
Notice that
by Definitions 2 and 5. The signs of
We can verify that
The main point is to set the value
First, let us consider the Cases (2) and (4) in Table 3. Because
Then consider the case of |
Second, consider the Cases (3) and (5) in Table 3. Because
For Case (3), let (
For an other odd
Therefore, Byun et al. [16] defined the delta root of a fuzzy number as a function that naturally extends certain properties of the square root of a real number ℝ to a fuzzy real number.
As a one-step generalization, we propose a delta-cube root. The most important difference between the cube root and square root of real numbers is that the former can be defined for any real number, whereas the latter cannot. The same phenomenon occurs for the delta cube root and the delta root of a fuzzy number. In this study, we define the delta cube root and examine the properties of the fuzzy number
For any real number
Our study introduces a novel perspective by extending the delta-root concept to the delta cube root, thereby providing a unified functional approach to the delta roots of previously unexplored fuzzy numbers. This generalization provides new insights and potential applications in fuzzy mathematics.
Further generalization of the delta
No potential conflict of interest relevant to this article was reported.
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) |
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
A fuzzy (real) number is represented by a function from a set of real numbers ℝ to the interval [0, 1]. The concept of a fuzzy set was first introduced in 1965 by Zadeh [1].
Since then, various operations and properties of fuzzy numbers have been proposed by the authors of [1–3] to handle data that are not clearly expressed by real numbers or include imprecise information. The concepts of powers and roots of fuzzy numbers are frequently required to handle complex fuzzy numbers in fuzzy data. AbuAarqob et al. [4] employed the
Both square and cube roots are essential concepts in this field of research. Both these concepts are necessary when fuzzy numbers are used. As research has explored the square root of fuzzy numbers, it is important to discuss the cube root and the generalized concept of the
Determining an
Research on the
In this study, we investigate whether the delta cube root of a fuzzy number
By defining a delta cube root similarly to a delta root, we propose a generalized functional expression for the cube root of a fuzzy number. This study presents the delta cube root as a complete function that enables intuitive understanding and rigorous verification through graphical representation, in contrast to traditional approaches that compute the cube root case-by-case using the
The remainder of this paper is organized as follows. In Section 2, we provide the preliminaries of both the operations on fuzzy numbers and the definition of the delta root of a fuzzy number. In Section 3, we define the delta cube root of a fuzzy number and provide its properties using examples. In Section 4, we observe that the delta cube root of the cube of a nonnegative fuzzy number becomes the original fuzzy number. Furthermore, the cube of its delta cube root becomes itself. For real numbers, for any
In this section, we provide the basic definitions and properties of fuzzy numbers. In addition, we introduce the definition of the delta root of a fuzzy number, which is foundational for defining the delta cube root of a fuzzy number.
Xiao eand Zhu [3] proposed a fuzzy number with an
A mapping
(i) There exists
(ii) For each
In this study, the notation for
Hence, ℝ can be embedded into
The Zadeh [1] extension principle is as follows:
Let fuzzy sets
It is essential to implement the boundary conditions of the fuzzy sets to ensure that the membership functions are meaningful and aligned with the property characteristics of the real numbers. The boundary conditions for the fuzzy sets that must be satisfied are as follows:
• Normality
The membership function
• Boundedness
The membership function
• Convexity
The membership function must be continuous and convex. This property can be expressed as follows: for all
• Piecewise continuity
The membership function must be continuous or piecewise continuous. This implies that even if there are discontinuities, their number must be finite, ensuring the function remains manageable and meaningful:
These boundary conditions are critical for defining and applying fuzzy numbers. By implementing these conditions, we ensure that the membership functions used in this study provide meaningful and robust representations of fuzzy sets that align with real-number properties.
Kaieva and Seikkala [2] applied the Zadeh extension principle to arithmetic operations, as follows:
The arithmetic operations ⊕, ⊖, ⊗, and ⊘ on
The square root of a fuzzy number is defined using
For all
Given a fuzzy number, the delta root of a fuzzy number by Byun et al. [16] is defined as follows.
For
This is because a fuzzy number (see the definition of a fuzzy number) itself is a type of fuzzy set, and the delta root of a fuzzy number is a type of fuzzy number. Therefore, a delta root is a special type of fuzzy set.
Delta root is a concept proposed to define the fuzzy square root. Unlike the traditional approach, which starts from the
Let
Then
The two-membership functions in Example 1 are illustrated in Figure 1.
We introduce the following properties of the delta root proven by Byun et al. [16]:
Let
(i) For
(ii) It holds that
The delta cube root generalizes the cube root of a fuzzy number, thereby extending the concept established by the delta root. Unlike a square root, a cube root can have multiple values for real numbers, including negative numbers. Although a quadratic equation with real coefficients does not always guarantee a real root, a cubic equation consistently has at least one real root. This principle should be extended to fuzzy numbers, necessitating the delta cube root definition as a function. The delta cube root must account for various cases, including both nonnegative and negative fuzzy numbers. The delta cube root is defined as follows.
Traditionally, the cube root of a fuzzy number using the
For
When addressing the delta root, only the case where
Case 1.
Case 2.
Case 3.
Case 4.
Case 5.
In Case 1, the delta cube root of
Let
Subsequently, for
Case 3 is illustrated in Example 3.
Let
Then, for
Case 4 is described by Example 4.
Let
Then, for
Case 5 is described in Example 5.
Let
Then, for
Figures 2
More cases must be considered for delta cube roots than for delta roots. But the following properties hold for the delta cube root as they do for the delta root by Byun et al. [16].
Let
Let
Also in the same way,
To simplify the expression, we introduce a notation for the delta cube root of the
For
In the real number system, ℝ, for any
In this section, we first examine the delta cube root of the cube of a nonnegative fuzzy number and then extend the analysis to other cases. We also show that the delta cube root of the cube of a nonnegative or nonpositive fuzzy number is the fuzzy number.
To explore the delta cube root of the cube of a fuzzy number, we begin by considering the case where
For
from Definitions 2 and 5.
First, we consider the case in which
For any (
which implies that
Second, we consider the case in which
We observe the case where
To show
Without a loss of generality, we assume that
Case (1) If
Case (2) If
These two cases are presented in Tables 1 and 2.
As we must observe
However, by assumption, it holds that:
Thus,
holds for any
We prove that
Suppose we aim to solve the equation
Then we can denote
Example 6 is shown in Figure 6.
Suppose we need to solve the equation
Note that
Subsequently, we can denote
Example 7 is shown in Figure 7.
We now observe the delta cube root when
For
We also observe that the cube of the delta cube root of a nonnegative/nonpositive fuzzy number becomes itself.
For
where the first equality is given by Corollary 1.
Thus far, a fuzzy number
Hereafter, a general fuzzy number
We give examples on
1. SubCase (1-3): Define
Then by Definitions 2 and 5,
For
Consider a particular one (
Hence, the equation
2. SubCase (2-3): Define
Then by Definitions 2 and 5,
Note that, for
Consider a particular one (
Hence,
Therefore,
Consequently, it is trivial for
Naturally, we can consider the delta
(Observation about the delta 5-th root of 5-th power of fuzzy real number)
Notice that
by Definitions 2 and 5. The signs of
We can verify that
The main point is to set the value
First, let us consider the Cases (2) and (4) in Table 3. Because
Then consider the case of |
Second, consider the Cases (3) and (5) in Table 3. Because
For Case (3), let (
For an other odd
Therefore, Byun et al. [16] defined the delta root of a fuzzy number as a function that naturally extends certain properties of the square root of a real number ℝ to a fuzzy real number.
As a one-step generalization, we propose a delta-cube root. The most important difference between the cube root and square root of real numbers is that the former can be defined for any real number, whereas the latter cannot. The same phenomenon occurs for the delta cube root and the delta root of a fuzzy number. In this study, we define the delta cube root and examine the properties of the fuzzy number
For any real number
Our study introduces a novel perspective by extending the delta-root concept to the delta cube root, thereby providing a unified functional approach to the delta roots of previously unexplored fuzzy numbers. This generalization provides new insights and potential applications in fuzzy mathematics.
Further generalization of the delta
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