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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

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 [13] 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 r-cut method. Bashar and Shirin [5] defined the square root of the membership function for a fuzzy set. Lamara et al. [6] introduced a new definition of the square root using the φ-function, and Amirfakhriana [7] suggested a method for numerically determining the root of a fuzzy number. In addition, several studies [815] proposed the square root of fuzzy real numbers and many examples were obtained using Zadeh’s extension principle. However, the square roots of fuzzy real numbers are handled by considering the square roots of the endpoints of each α-level set.

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 n-th root for fuzzy numbers.

Determining an α-level set is relatively straightforward for a given fuzzy number. However, determining the fuzzy number that satisfies a given α-level set remains challenging. The previous definitions of the square root of a fuzzy number are complex because they are not expressed as functions. To address the gap between a fuzzy number’s functional representation and its square root’s nonfunctional definition, Byun et al. [16] introduced the delta root. A distinction exists between the traditional method of defining the square root of a fuzzy number using the α-level set and the delta square root defined by Byun et al. [16]. Because fuzzy numbers are inherent functions, the delta square root expresses the square root of a fuzzy number in functional form. Byun et al. [16] demonstrated that the delta root is equivalent to the traditional square root and preserves the operational properties of the square roots for real numbers.

Research on the n-th root of fuzzy numbers has also been conducted by Lamara et al. [6] and Ibrahim et al. [17]. However, these studies used the α-level set approach, which is similar to traditional methods for handling square roots.

In this study, we investigate whether the delta cube root of a fuzzy number η is well defined when η ≥ 0̃ and under other conditions, and whether it exhibits properties similar to the cube root of a real number. There are differences between the square and cube roots in a real-number system. Although a quadratic equation with real coefficients does not consistently guarantee a real root, a cubic equation consistently has at least one real root. Moreover, the square root of a negative real number does not exist in real values, whereas the cubic root does. This distinction can also be applied to fuzzy numbers. Furthermore, we proved that the delta cube root retains the properties of the cube root in real numbers across various scenarios involving fuzzy numbers.

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 α-level set. This method enhances the clarity of the cube root concept and can handle various scenarios, particularly those involving negative numbers.

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 t ∈ ℝ, the cube root of t3 becomes t. Therefore, we aim to justify whether this also holds for the delta cube root of a fuzzy number. We consider the case that holds and provide some examples of cases that do not hold.

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 αlevel set.

Definition 1

A mapping η : ℝ → [0, 1] is called a fuzzy (real) number, whose α-level set is denoted by [η]α = {t : η(t) ≥ α}, if it satisfies the following conditions:

  • (i) There exists t0 such that η(t0) = 1.

  • (ii) For each α ∈ (0, 1], there exist real numbers ηα-ηα+ such that the α-level set [η]α is equal to the closed interval [ηα-,ηα+].

In this study, the notation for F(ℝ), F*(ℝ) and F(ℝ) follows that of [16]. F(ℝ) denotes the set of all the fuzzy (real) numbers. If ηF(ℝ) and η(t) = 0 whenever t < 0, then η is called a nonnegative fuzzy (real) number and F*(ℝ) denotes the set of all nonnegative fuzzy numbers. Each r ∈ ℝ can be considered a fuzzy number F(ℝ) denoted by

r˜(t)={1,t=r,0,tr,

Hence, ℝ can be embedded into F(ℝ).

The Zadeh [1] extension principle is as follows:

Definition 2

Let fuzzy sets A1, A2, ..., An be defined on the universes X1, X2, ..., Xn. The mapping for these particular input sets can now be defined as B = f(A1, A2, ..., An), where the membership function of image B is given by

μB(y)=maxy=f(x1,x2,,xn){min[μA1(x1),μA2(x2),,μAn(xn)]}.

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 μ(x) must have a maximum value of one. That is, there exists some x0 such that:


    x0such that μ(x0)=1

  • • Boundedness


    The membership function μ(x) must take values within the range [0, 1]. This ensures that:


    0μ(x)1         for all x

  • • Convexity


    The membership function must be continuous and convex. This property can be expressed as follows: for all x1, x2 ∈ ℝ and λ ∈ [0, 1],


    μ(λx1+(1-λ)x2)min(μ(x1),μ(x2))

  • • 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:

    μ(x) is continuous or piecewise continuous with a finite number of discontinuities.

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:

Definition 3

The arithmetic operations ⊕, ⊖, ⊗, and ⊘ on F(ℝ) × F(ℝ) are defined by

(ηγ)(t)=supt=x+y(min (η(x),γ(y))),(ηγ)(t)=supt=x-y(min (η(x),γ(y))),(ηγ)(t)=supt=xy(min (η(x),γ(y))),(ηγ)(t)=supt=x/y(min (η(x),γ(y))).

The square root of a fuzzy number is defined using α-level sets by Bashar and Shirin [5] and Hasankhani et al. [18] as follows:

Definition 4

For all α ∈ (0, 1] and ηF(ℝ) with η ≥ 0̃, we define

[η]α:=[ηα-,ηα+].

Given a fuzzy number, the delta root of a fuzzy number by Byun et al. [16] is defined as follows.

Definition 5

For ηF(ℝ) with η ≥ 0̃, the delta root of η(t) is given by

Δη(t)={η(t2),t0,0,t<0.

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 α-level set perspective, the delta fuzzy represents the fuzzy square root as a fuzzy number as a function. The delta square root of a fuzzy number is also a fuzzy number and satisfies the properties of fuzzy sets. Fuzzy set operations are also well preserved. This methodology was extended to the cube roots and n-th roots, resulting in a delta cube root and a delta n-th root. Although fuzzy sets may not differ, one might notice a difference in the process of finding the n-th root because it does not start from the α-level set. However, the computed delta n-th root is also a fuzzy number that satisfies the properties of fuzzy sets. The following is an example of a fuzzy number and its delta root:

Example 1

Let ηF(ℝ) be defined by

η(t)={12t,0t<2,-12t+2,2t4,0,otherwise.

Then

Δη(t)={12t2,0t<2,-12t2+2,2t2,0,otherwise.

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]:

Theorem 1

Let η, γF(ℝ) be two fuzzy numbers. The following properties hold.

  • (i) For η, γ ≥ 0̃,


    ΔηΔγ=Δ(ηγ).

  • (ii) It holds that


    ηηθ˜Δ(ηη)=η.

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 α-level set is calculated through case-by-case or logical analyses. However, the delta cube root allows the computation of the cube root of a fuzzy number as a complete function that can be graphically represented. This approach enhances the intuitive understanding of the cube root and can clearly develop and rigorously verify logic across various cases, particularly when dealing with the cube roots of negative numbers.

Definition 6

For ηF(ℝ) and for any t, the delta cube root of η(t) is given by

Δ13η(t):=η(t3).

When addressing the delta root, only the case where η ≥ 0̃ was considered. However, for the delta cube root, we must examine the following five cases:

  • Case 1. η ≥ 0̃,

  • Case 2. η < 0̃,

  • Case 3. η1-0 and η1+0,

  • Case 4. η1->0 and ∃α ∈ (0, 1) such that ηα-<0,

  • Case 5. η1+<0 and ∃α ∈ (0, 1) such that ηα+>0.

In Case 1, the delta cube root of η is similar to the delta root. An example similar to Example 1 can be obtained by substituting t3 with t2: Here, we provide examples of the remaining four cases. Case 2 is illustrated in the following Example 2.

Example 2

Let ηF(ℝ) be defined by

η(t)={12t+2,-4t<-2,-12t,-2t0,0,otherwise.

Subsequently, for n = 3,

Δ13η(t)={12t3+2,-43t<-23,-12t3,-23t0,0,otherwise.

Case 3 is illustrated in Example 3.

Example 3

Let ηF(ℝ) be defined by

η(t)={13t+1,-3t<0,-13t+1,0t3,0,otherwise.

Then, for n = 3,

Δ13η(t)={13t3+1,-33t<0,-13t3+1,0t-33,0,otherwise.

Case 4 is described by Example 4.

Example 4

Let ηF(ℝ) be defined by

η(t)={13t+23,-2t<1,-13t+43,1t4,0,otherwise.

Then, for n = 3,

Δ13η(t)={13t3+23,-23t<1,-13t3+43,1t-43,0,otherwise.

Case 5 is described in Example 5.

Example 5

Let ηF(ℝ) be defined by

η(t)={13t+43,-4t<-1,-13t+23,-1t2,0,otherwise.

Then, for n = 3,

Δ13η(t)={13t3+43,-43t<-1,-13t3+23,-1t-23,0,otherwise.

Figures 25 show the membership functions of the delta cube roots in Examples 2–5, respectively.

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].

Theorem 2

Let η, γF(ℝ) be two fuzzy numbers. Then,

Δ13ηΔ13γ=Δ13(ηγ).

Proof. Given real numbers r, s ∈ ℝ, there are unique real numbers x, y ∈ ℝ such that x3 = r and y3 = s. Subsequently, for any t ∈ ℝ with rs = t3, the value of xy is uniquely determined as t from (xy)3 = x3y3 = t3. Therefore, we have

(Δ13ηΔ13γ)(t)=supxy=t(min(Δ13η(x),Δ13γ(y)))=supxy=t(min (η(x3),γ(y3)))=suprs=t3(min (η(r),γ(s)))=(ηγ)(t3)=Δ13(ηγ)(t).

Corollary 1

Let ηF(ℝ) be a fuzzy number. Then,

(Δ13ηΔ13ηΔ13η)(t)=Δ13(ηηη)(t).

Proof. The proof is similar to that of Theorem 2. So

(Δ13ηΔ13η)(t)=Δ13(ηη)(t).

Also in the same way,

(Δ13(ηη)Δ13η)(t)=Δ13((ηη)η)(t)=Δ13(ηηη)(t).

To simplify the expression, we introduce a notation for the delta cube root of the n-th power of a fuzzy number η.

Definition 7

For ηF(ℝ) and for any t, the delta cube root of the n-th power of η is

Δ13(ηn)(t):=Δ13(ηη)n-times(t).

In the real number system, ℝ, for any t ≥ 0, the square root of t2 is t when t > 0. Byun et al. [16] similarly demonstrate that for the delta root, Δ(ηη) = |η| holds for any ηF(ℝ), provided ηη ≥ 0̃.

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 η ≥ 0̃.

Theorem 3

For ηF(ℝ) with η ≥ 0̃ and for any t, the following holds.

Δ13(η3)(t)=η(t).

Proof. First, note the following:

Δ13(η3)(t)=supxyz=t3(min(η(x),η(y),η(z)))

from Definitions 2 and 5.

First, we consider the case in which t = 0.

For any (x, y, z) ∈ ℝ3 satisfying xyz = 0, at least one of x, y or z must be 0. Without a loss of generality, we assume x = 0. Then

η(0)=η(x)min (η(x),η(y),η(z))0=η(0),

which implies that η(0) = min (η(x), η(y), η(z)). Thus, for t = 0,

Δ13(η3)(t)=supxyz=t3(min(η(x),η(y),η(z)))=supxyz=0(min(η(x),η(y),η(z)))=η(0)=η(t).

Second, we consider the case in which t < 0. Thus, η(t) = 0 holds because η ≥ 0̃. For any (x, y, z) ∈ ℝ satisfying xyz = t3, at least one x, y, z should be negative. Without a loss of generality, we assume that x < 0. Then 0 ≥ min(η(x), η(y), η(z)) ≤ η(x) = 0. Hence min(η(x), η(y), η(z)) = 0. Therefore, Eq. (29) states that Δ13(η3)(t)=0=η(t),

We observe the case where t > 0. Eq. (29) yields

Δ13(η3)(t)max (min(η(t),η(t),η(t)),min(η(t),η(-t),η(-t)))=η(t).

To show Δ13(η3)(t)η(t) by contradiction, we assume that Δ13(η3)(t)>η(t). Then from Eq. (29), we find x, y, z ∈ ℝ satisfies

xyz=t3,x2+y2+z20,and min(η(x),η(y),η(z))>η(t).

Without a loss of generality, we assume that xyz. Subsequently, we considered two cases.

  • Case (1) If xy > 0 then zt3 > 0.

  • Case (2) If xy < 0 then zt3 < 0.

These two cases are presented in Tables 1 and 2.

As we must observe t > 0, we do not need to consider SubCase (1-2), (1-4), (2-1), and (2-3) in Table 1. Moreover, SubCase (2-2) and (2-4) contradict the assumption xyz. Therefore, we must verify SubCase (1-1) and (1-3). For SubCase (1-1), t satisfies 0 < xytz or 0 < xtyz (because xyz = t3 does not hold for either t < x or z < t). In the case of 0 < xytz, for the first time assuming that η(x) < η(y). If η(x) ≤ η(z), for α = η(x), we obtain

t[x,z][ηα-,ηα+].

However, by assumption, it holds that:

0η(t)<min(η(x),η(y),η(z))=η(x)=α.

Thus, t[ηα-,ηα+] are contradictory. If η(x) ≥ η(z), then for α = η(z), a similar argument contradicts. However, we assume η(x) > η(y). We assume that ytz. If η(y) ≤ η(z), for α = η(y), we can use a similar argument to obtain a contradiction: Also for η(y) ≥ η(z), a similar argument also offers a contradiction. Moreover, if we assume η(x) = η(y) for α = η(x) = η(y), we can easily obtain the same contradiction. Starting with 0 < xtyz, we can obtain a contradiction by using a method similar to that described above. For SubCase (1-3), t satisfies xy < 0 < tz. Here, η(x) = η(y) = 0 from η ≥ 0̃. This implies that 0 = η(x) ≤ η(t). However, this result contradicts the assumption η(t) < min(η(x), η(y), η(z)) = η(x). Thus, for t > 0, Δ13(η3)(t)=η(t) holds true for SusbCase (1-1) and (1-3). Therefore, for both Cases (1) and (2). Hence,

Δ13(η3)(t)=η(t)

holds for any t ∈ ℝ when η ≥ 0̃.

We prove that Δ13(η3)(t)=η(t) holds true when η ≥ 0̃. Examples 3 are given as Examples 6 and 7.

Example 6

Suppose we aim to solve the equation x3(t) = η(t) where ηF(ℝ) is defined by η(t)={13t,0t<3,-13t+2,3t6,0,otherwise.

Then we can denote x(t)=Δ13η(t)=η(t3), so

x(t)={13t3,0t<33,-13t3+2,33t63,0,otherwise.

Example 6 is shown in Figure 6.

Example 7

Suppose we need to solve the equation x3(t) = (η ⊕ 2̃)(t) where ηF(ℝ) is defined by η(t)={13t,0t<3,-13t+2,3t6,0,otherwise.

Note that

(η2˜)(t)=η(t-2)={13t-23,2t<5,-13t+83,5t8,0,otherwise.

Subsequently, we can denote x(t)=Δ13(η2˜)(t)=(η2˜)(t3), so

x(t)={13t3-23,23t<53,-13t3+83,53t2,0,otherwise.

Example 7 is shown in Figure 7.

We now observe the delta cube root when η is not constrained by η ≥ 0̃. Without a loss of generality, we assume that xyz. In Table 1, SubCase (1-2), (2-1), (2-2), and (2-4) are illogical. Therefore, we must consider SubCase (1-1), (1-3), (1-4), and (2-3). From Theorem 3, which is the result for SubCase (1-1), we can easily obtain Corollary 2 for SubCase (1-4).

Corollary 2

For ηF(ℝ) with η ≤ 0̃ and for any t, we have

Δ13(η3)(t)=η(t).

Proof. For ηF(ℝ) with η ≤ 0̃, let γ = −η, which yields γ(s) = η(−s) for any s ∈ ℝ: We demonstrate that γ3 = −η3 from γ=-η=-1˜η. We note γ ≥ 0̃. By Theorem 3, Δ13(γ3)(s)=γ(s) holds true for any s ∈ ℝ. Thus.

Δ13(η3)(t)=(η3)(t3)=(-η3)(-t3)=γ3((-t)3)=Δ13(γ3)(-t)=γ(-t)=η(-(-t))=η(t).

We also observe that the cube of the delta cube root of a nonnegative/nonpositive fuzzy number becomes itself.

Corollary 3

For η with either η ≤ 0̃ or η ≥ 0̃, we have

(Δ13η)3=η.

Proof. This follows because

(Δ13η)3=Δ13(η3)=η,

where the first equality is given by Corollary 1.

Thus far, a fuzzy number ηF(ℝ) with nonzero values only for SubCase either (1-1) or (1-4), has been investigated, and it has been shown that equation Δ13(η3)(t)=η(t) holds in Theorem 3 and Corollary 2, respectively.

Hereafter, a general fuzzy number ηF(ℝ) with nonzero values, even for SubCase either (1-3) or (2-3) must be dealt with. Remark 1 provides examples of η related to SubCase (1-3) and (2-3), where equation Δ13(η3)(t)=η(t) no longer holds.

Remark 1

We give examples on ηF(ℝ), related to SubCase (1-3) and (2-3), where the equation Δ13(η3)(t)=η(t) does not hold anymore.

  • 1. SubCase (1-3): Define η(t) by


    η(t)={t+9,-9t<-8,-111t+311,-8t3,0,otherwise.

    Then by Definitions 2 and 5,


    Δ13(η3)(t)=supxyz=t3(min (η(x),η(y),η(z))).

    For t = 2,


    Δ13(η3)(2)=supxyz=8(min (η(x),η(y),η(z))).

    Consider a particular one (x0, y0, z0) = (−8,−1, 1) among the elements (x, y, z) ∈ ℝ3 such that xyz = 8. Note that (x0, y0, z0) satisfies the condition for SubCase (1-3). Notice that Figure 8 is the membership function of η(t). In this case, min (η(x0), η(y0), η(z0))= η(z0) since η(x0) ≥ η(y0) ≥ η(z0). And η(2) < η(z0). Thus


    Δ13(η3)(2)=supxyz=8(min (η(x),η(y),η(z)))min (η(x0),η(y0),η(z0))=η(z0)>η(2).

    Hence, the equation Δ13(η3)(t)=η(t) does not hold.

  • 2. SubCase (2-3): Define η(t) by


    η(t)={111t+3,-3t<8,-t+9,8t9,0,otherwise.

    Then by Definitions 2 and 5,


    Δ13(η3)(t)=supxyz=t3(min (η(x),η(y),η(z))).

    Note that, for t = −2,


    Δ13(η3)(-2)=supxyz=-8(min (η(x),η(y),η(z))).

    Consider a particular one (x0, y0, z0) = (−1, 1, 8) among the elements (x, y, z) ∈ ℝ3 such that xyz = −8. Note that (x0, y0, z0) satisfies the condition for SubCase (2-3). Notice that Figure 9. is the membership function of η(t). Here, min (η(x0), η(y0), η(z0)) = η(z0) since η(x0) ≥ η(y0) ≥ η(z0). Furthermore, η(−2) < η(z0). Thus,

    Δ13(η3)(-2)=supxyz=-8(min (η(x),η(y),η(z)))min (η(x0),η(y0),η(z0))=η(z0)>η(-2).

    Hence, Δ13(η3)(t)=η(t) does not hold.

Therefore, Δ13(η3)(t)=η(t) does not hold for the Cases (1-3) and (2-3).

Consequently, it is trivial for η = 0. For cases in which η > 0̃, (1-1) is feasible, and for cases in which η < 0̃, (1-4) is feasible. This is consistent with the results in real numbers, where the cube root of 8 has only one real value, 2, and the cube root of −8 has only one real value −2.

Naturally, we can consider the delta n-th root for any n ∈ ℕ, which generalizes the delta and delta cube roots. When n is an even number, we can easily verify that we can define the delta nth root only when η ≥ 0̃. However, what we need to verify increases if n is odd. The main challenge with n increasing verifying the delta n-th root of n-th power of a fuzzy number. The number of variables increases as n increases because it produces Δ1n(ηn)(t)=supx1xn=tn(min (η(x1),,η(xn))). Although it is difficult to consider all cases, we introduce the following remarks to provide some insight.

Remark 2

(Observation about the delta 5-th root of 5-th power of fuzzy real number)

Notice that

Δ15(η5)(t)=supx1x2x3x4x5=t5(min (η(x1),η(x2),η(x3),η(x4),η(x5))),

by Definitions 2 and 5. The signs of x1x2x3 must be the same as those of x4x5. Similar to the arguments in the proof of Theorem 3, we assume that x1x2x3x4x5. The cases in Table 3.

We can verify that Δ15(η5)(t)=η(t) holds for the Cases (1) and (6) in Table 3 with similar arguments with ones in the proofs of Theorem 3 and Corollary 2. For the Cases (2) and (4) in Table 3, we suggest a way to make an example that does not satisfy Δ15(η5)(t)=η(t).

The main point is to set the value t which is strictly larger or smaller than the other variables x1, ..., x5.

First, let us consider the Cases (2) and (4) in Table 3. Because t < 0, t must be less than x1. Then, select any t that satisfies |t| > 1; for example, t = −2.

Then consider the case of |x5| = |t5| and |xi| = 1 for i = 1, 2, 3, 4. For the Case (2), let (x1, x2, x3, x4, x5) = (−1, 1, 1, 1, 32) and t = −2 for example. This enables us to create a fuzzy number that strictly increases from t to x5, as in the case of Figure 9. Finally, we verify that the equation Δ15(η5)(t)=η(t) does not hold.

Second, consider the Cases (3) and (5) in Table 3. Because t > 0, t must be greater than x5 and |t| > 1, we select t = 2. Consider the case where |x1| = |t5| and |xi| = 1 for i = 2, 3, 4, 5.

For Case (3), let (x1, x2, x3, x4, x5) = (−32,−1, 1, 1, 1) and t = 2 for an example. This enables us to create a fuzzy real number that strictly decreases from x1 to t, as shown in Figure 8. In addition, we can check whether the equation Δ15(η5)(t)=η(t) holds.

For an other odd n such as n = 7, 9, ..., we will be able to prove that the equation Δ1n(ηn)(t)=η(t) holds in the case of either η ≥ 0̃ or η ≤ 0̃. By following arguments similar to those in Remark 2, we can create examples such that Δ1n(ηn)(t)=η(t) does not hold when the fuzzy real number ηF(ℝ) has a nonzero value in each of the negative and positive domains.

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 η.

For any real number t ∈ ℝ, the root of its cube (t3)13 always becomes t. We investigate whether this property also applies to fuzzy numbers. Consequently, this holds in the case of either a nonnegative fuzzy real number or a nonpositive number. However, we identified cases and examples in which this property did not hold.

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 n-th root of a fuzzy number becomes complex as n increases, owing to the growing number of variables to consider. To determine whether the n-th power of the delta n-th root of a fuzzy number η becomes η, we presented a method for dividing the cases for consideration. By defining the fuzzy n-th root using a functional definition, we apply the n-th root concept to fuzzy numbers as if they were real numbers. Consequently, fuzzy numbers can be applied to a wider and more diverse range of fields. These generalizations open up opportunities for further applications, emphasizing the growing scope and impact of our research on operations involving fuzzy numbers.

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.”
Fig. 1.

The delta root of the fuzzy number in Example 1.


Fig. 2.

Membership function of the Delta cube root in Example 2.


Fig. 3.

Membership function of the delta cube root in Example 3.


Fig. 4.

Membership function of the delta cube root in Example 4.


Fig. 5.

Membership function of the delta cube root in Example 5.


Fig. 6.

Membership function of Example 6.


Fig. 7.

Membership function of Example 7.


Fig. 8.

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


Fig. 9.

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


Table. 1.

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

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

Table. 2.

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

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

Table. 3.

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

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

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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

1. Introduction

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 [13] 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 r-cut method. Bashar and Shirin [5] defined the square root of the membership function for a fuzzy set. Lamara et al. [6] introduced a new definition of the square root using the φ-function, and Amirfakhriana [7] suggested a method for numerically determining the root of a fuzzy number. In addition, several studies [815] proposed the square root of fuzzy real numbers and many examples were obtained using Zadeh’s extension principle. However, the square roots of fuzzy real numbers are handled by considering the square roots of the endpoints of each α-level set.

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 n-th root for fuzzy numbers.

Determining an α-level set is relatively straightforward for a given fuzzy number. However, determining the fuzzy number that satisfies a given α-level set remains challenging. The previous definitions of the square root of a fuzzy number are complex because they are not expressed as functions. To address the gap between a fuzzy number’s functional representation and its square root’s nonfunctional definition, Byun et al. [16] introduced the delta root. A distinction exists between the traditional method of defining the square root of a fuzzy number using the α-level set and the delta square root defined by Byun et al. [16]. Because fuzzy numbers are inherent functions, the delta square root expresses the square root of a fuzzy number in functional form. Byun et al. [16] demonstrated that the delta root is equivalent to the traditional square root and preserves the operational properties of the square roots for real numbers.

Research on the n-th root of fuzzy numbers has also been conducted by Lamara et al. [6] and Ibrahim et al. [17]. However, these studies used the α-level set approach, which is similar to traditional methods for handling square roots.

In this study, we investigate whether the delta cube root of a fuzzy number η is well defined when η ≥ 0̃ and under other conditions, and whether it exhibits properties similar to the cube root of a real number. There are differences between the square and cube roots in a real-number system. Although a quadratic equation with real coefficients does not consistently guarantee a real root, a cubic equation consistently has at least one real root. Moreover, the square root of a negative real number does not exist in real values, whereas the cubic root does. This distinction can also be applied to fuzzy numbers. Furthermore, we proved that the delta cube root retains the properties of the cube root in real numbers across various scenarios involving fuzzy numbers.

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 α-level set. This method enhances the clarity of the cube root concept and can handle various scenarios, particularly those involving negative numbers.

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 t ∈ ℝ, the cube root of t3 becomes t. Therefore, we aim to justify whether this also holds for the delta cube root of a fuzzy number. We consider the case that holds and provide some examples of cases that do not hold.

2. Preliminaries

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 αlevel set.

Definition 1

A mapping η : ℝ → [0, 1] is called a fuzzy (real) number, whose α-level set is denoted by [η]α = {t : η(t) ≥ α}, if it satisfies the following conditions:

  • (i) There exists t0 such that η(t0) = 1.

  • (ii) For each α ∈ (0, 1], there exist real numbers ηα-ηα+ such that the α-level set [η]α is equal to the closed interval [ηα-,ηα+].

In this study, the notation for F(ℝ), F*(ℝ) and F(ℝ) follows that of [16]. F(ℝ) denotes the set of all the fuzzy (real) numbers. If ηF(ℝ) and η(t) = 0 whenever t < 0, then η is called a nonnegative fuzzy (real) number and F*(ℝ) denotes the set of all nonnegative fuzzy numbers. Each r ∈ ℝ can be considered a fuzzy number F(ℝ) denoted by

r˜(t)={1,t=r,0,tr,

Hence, ℝ can be embedded into F(ℝ).

The Zadeh [1] extension principle is as follows:

Definition 2

Let fuzzy sets A1, A2, ..., An be defined on the universes X1, X2, ..., Xn. The mapping for these particular input sets can now be defined as B = f(A1, A2, ..., An), where the membership function of image B is given by

μB(y)=maxy=f(x1,x2,,xn){min[μA1(x1),μA2(x2),,μAn(xn)]}.

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 μ(x) must have a maximum value of one. That is, there exists some x0 such that:


    x0such that μ(x0)=1

  • • Boundedness


    The membership function μ(x) must take values within the range [0, 1]. This ensures that:


    0μ(x)1         for all x

  • • Convexity


    The membership function must be continuous and convex. This property can be expressed as follows: for all x1, x2 ∈ ℝ and λ ∈ [0, 1],


    μ(λx1+(1-λ)x2)min(μ(x1),μ(x2))

  • • 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:

    μ(x) is continuous or piecewise continuous with a finite number of discontinuities.

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:

Definition 3

The arithmetic operations ⊕, ⊖, ⊗, and ⊘ on F(ℝ) × F(ℝ) are defined by

(ηγ)(t)=supt=x+y(min (η(x),γ(y))),(ηγ)(t)=supt=x-y(min (η(x),γ(y))),(ηγ)(t)=supt=xy(min (η(x),γ(y))),(ηγ)(t)=supt=x/y(min (η(x),γ(y))).

The square root of a fuzzy number is defined using α-level sets by Bashar and Shirin [5] and Hasankhani et al. [18] as follows:

Definition 4

For all α ∈ (0, 1] and ηF(ℝ) with η ≥ 0̃, we define

[η]α:=[ηα-,ηα+].

Given a fuzzy number, the delta root of a fuzzy number by Byun et al. [16] is defined as follows.

Definition 5

For ηF(ℝ) with η ≥ 0̃, the delta root of η(t) is given by

Δη(t)={η(t2),t0,0,t<0.

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 α-level set perspective, the delta fuzzy represents the fuzzy square root as a fuzzy number as a function. The delta square root of a fuzzy number is also a fuzzy number and satisfies the properties of fuzzy sets. Fuzzy set operations are also well preserved. This methodology was extended to the cube roots and n-th roots, resulting in a delta cube root and a delta n-th root. Although fuzzy sets may not differ, one might notice a difference in the process of finding the n-th root because it does not start from the α-level set. However, the computed delta n-th root is also a fuzzy number that satisfies the properties of fuzzy sets. The following is an example of a fuzzy number and its delta root:

Example 1

Let ηF(ℝ) be defined by

η(t)={12t,0t<2,-12t+2,2t4,0,otherwise.

Then

Δη(t)={12t2,0t<2,-12t2+2,2t2,0,otherwise.

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]:

Theorem 1

Let η, γF(ℝ) be two fuzzy numbers. The following properties hold.

  • (i) For η, γ ≥ 0̃,


    ΔηΔγ=Δ(ηγ).

  • (ii) It holds that


    ηηθ˜Δ(ηη)=η.

3. The Delta Cube Root of a Fuzzy Number

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 α-level set is calculated through case-by-case or logical analyses. However, the delta cube root allows the computation of the cube root of a fuzzy number as a complete function that can be graphically represented. This approach enhances the intuitive understanding of the cube root and can clearly develop and rigorously verify logic across various cases, particularly when dealing with the cube roots of negative numbers.

Definition 6

For ηF(ℝ) and for any t, the delta cube root of η(t) is given by

Δ13η(t):=η(t3).

When addressing the delta root, only the case where η ≥ 0̃ was considered. However, for the delta cube root, we must examine the following five cases:

  • Case 1. η ≥ 0̃,

  • Case 2. η < 0̃,

  • Case 3. η1-0 and η1+0,

  • Case 4. η1->0 and ∃α ∈ (0, 1) such that ηα-<0,

  • Case 5. η1+<0 and ∃α ∈ (0, 1) such that ηα+>0.

In Case 1, the delta cube root of η is similar to the delta root. An example similar to Example 1 can be obtained by substituting t3 with t2: Here, we provide examples of the remaining four cases. Case 2 is illustrated in the following Example 2.

Example 2

Let ηF(ℝ) be defined by

η(t)={12t+2,-4t<-2,-12t,-2t0,0,otherwise.

Subsequently, for n = 3,

Δ13η(t)={12t3+2,-43t<-23,-12t3,-23t0,0,otherwise.

Case 3 is illustrated in Example 3.

Example 3

Let ηF(ℝ) be defined by

η(t)={13t+1,-3t<0,-13t+1,0t3,0,otherwise.

Then, for n = 3,

Δ13η(t)={13t3+1,-33t<0,-13t3+1,0t-33,0,otherwise.

Case 4 is described by Example 4.

Example 4

Let ηF(ℝ) be defined by

η(t)={13t+23,-2t<1,-13t+43,1t4,0,otherwise.

Then, for n = 3,

Δ13η(t)={13t3+23,-23t<1,-13t3+43,1t-43,0,otherwise.

Case 5 is described in Example 5.

Example 5

Let ηF(ℝ) be defined by

η(t)={13t+43,-4t<-1,-13t+23,-1t2,0,otherwise.

Then, for n = 3,

Δ13η(t)={13t3+43,-43t<-1,-13t3+23,-1t-23,0,otherwise.

Figures 25 show the membership functions of the delta cube roots in Examples 2–5, respectively.

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].

Theorem 2

Let η, γF(ℝ) be two fuzzy numbers. Then,

Δ13ηΔ13γ=Δ13(ηγ).

Proof. Given real numbers r, s ∈ ℝ, there are unique real numbers x, y ∈ ℝ such that x3 = r and y3 = s. Subsequently, for any t ∈ ℝ with rs = t3, the value of xy is uniquely determined as t from (xy)3 = x3y3 = t3. Therefore, we have

(Δ13ηΔ13γ)(t)=supxy=t(min(Δ13η(x),Δ13γ(y)))=supxy=t(min (η(x3),γ(y3)))=suprs=t3(min (η(r),γ(s)))=(ηγ)(t3)=Δ13(ηγ)(t).

Corollary 1

Let ηF(ℝ) be a fuzzy number. Then,

(Δ13ηΔ13ηΔ13η)(t)=Δ13(ηηη)(t).

Proof. The proof is similar to that of Theorem 2. So

(Δ13ηΔ13η)(t)=Δ13(ηη)(t).

Also in the same way,

(Δ13(ηη)Δ13η)(t)=Δ13((ηη)η)(t)=Δ13(ηηη)(t).

To simplify the expression, we introduce a notation for the delta cube root of the n-th power of a fuzzy number η.

Definition 7

For ηF(ℝ) and for any t, the delta cube root of the n-th power of η is

Δ13(ηn)(t):=Δ13(ηη)n-times(t).

4. The Delta Cube Root of the Cube of a Fuzzy Number

In the real number system, ℝ, for any t ≥ 0, the square root of t2 is t when t > 0. Byun et al. [16] similarly demonstrate that for the delta root, Δ(ηη) = |η| holds for any ηF(ℝ), provided ηη ≥ 0̃.

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 η ≥ 0̃.

Theorem 3

For ηF(ℝ) with η ≥ 0̃ and for any t, the following holds.

Δ13(η3)(t)=η(t).

Proof. First, note the following:

Δ13(η3)(t)=supxyz=t3(min(η(x),η(y),η(z)))

from Definitions 2 and 5.

First, we consider the case in which t = 0.

For any (x, y, z) ∈ ℝ3 satisfying xyz = 0, at least one of x, y or z must be 0. Without a loss of generality, we assume x = 0. Then

η(0)=η(x)min (η(x),η(y),η(z))0=η(0),

which implies that η(0) = min (η(x), η(y), η(z)). Thus, for t = 0,

Δ13(η3)(t)=supxyz=t3(min(η(x),η(y),η(z)))=supxyz=0(min(η(x),η(y),η(z)))=η(0)=η(t).

Second, we consider the case in which t < 0. Thus, η(t) = 0 holds because η ≥ 0̃. For any (x, y, z) ∈ ℝ satisfying xyz = t3, at least one x, y, z should be negative. Without a loss of generality, we assume that x < 0. Then 0 ≥ min(η(x), η(y), η(z)) ≤ η(x) = 0. Hence min(η(x), η(y), η(z)) = 0. Therefore, Eq. (29) states that Δ13(η3)(t)=0=η(t),

We observe the case where t > 0. Eq. (29) yields

Δ13(η3)(t)max (min(η(t),η(t),η(t)),min(η(t),η(-t),η(-t)))=η(t).

To show Δ13(η3)(t)η(t) by contradiction, we assume that Δ13(η3)(t)>η(t). Then from Eq. (29), we find x, y, z ∈ ℝ satisfies

xyz=t3,x2+y2+z20,and min(η(x),η(y),η(z))>η(t).

Without a loss of generality, we assume that xyz. Subsequently, we considered two cases.

  • Case (1) If xy > 0 then zt3 > 0.

  • Case (2) If xy < 0 then zt3 < 0.

These two cases are presented in Tables 1 and 2.

As we must observe t > 0, we do not need to consider SubCase (1-2), (1-4), (2-1), and (2-3) in Table 1. Moreover, SubCase (2-2) and (2-4) contradict the assumption xyz. Therefore, we must verify SubCase (1-1) and (1-3). For SubCase (1-1), t satisfies 0 < xytz or 0 < xtyz (because xyz = t3 does not hold for either t < x or z < t). In the case of 0 < xytz, for the first time assuming that η(x) < η(y). If η(x) ≤ η(z), for α = η(x), we obtain

t[x,z][ηα-,ηα+].

However, by assumption, it holds that:

0η(t)<min(η(x),η(y),η(z))=η(x)=α.

Thus, t[ηα-,ηα+] are contradictory. If η(x) ≥ η(z), then for α = η(z), a similar argument contradicts. However, we assume η(x) > η(y). We assume that ytz. If η(y) ≤ η(z), for α = η(y), we can use a similar argument to obtain a contradiction: Also for η(y) ≥ η(z), a similar argument also offers a contradiction. Moreover, if we assume η(x) = η(y) for α = η(x) = η(y), we can easily obtain the same contradiction. Starting with 0 < xtyz, we can obtain a contradiction by using a method similar to that described above. For SubCase (1-3), t satisfies xy < 0 < tz. Here, η(x) = η(y) = 0 from η ≥ 0̃. This implies that 0 = η(x) ≤ η(t). However, this result contradicts the assumption η(t) < min(η(x), η(y), η(z)) = η(x). Thus, for t > 0, Δ13(η3)(t)=η(t) holds true for SusbCase (1-1) and (1-3). Therefore, for both Cases (1) and (2). Hence,

Δ13(η3)(t)=η(t)

holds for any t ∈ ℝ when η ≥ 0̃.

We prove that Δ13(η3)(t)=η(t) holds true when η ≥ 0̃. Examples 3 are given as Examples 6 and 7.

Example 6

Suppose we aim to solve the equation x3(t) = η(t) where ηF(ℝ) is defined by η(t)={13t,0t<3,-13t+2,3t6,0,otherwise.

Then we can denote x(t)=Δ13η(t)=η(t3), so

x(t)={13t3,0t<33,-13t3+2,33t63,0,otherwise.

Example 6 is shown in Figure 6.

Example 7

Suppose we need to solve the equation x3(t) = (η ⊕ 2̃)(t) where ηF(ℝ) is defined by η(t)={13t,0t<3,-13t+2,3t6,0,otherwise.

Note that

(η2˜)(t)=η(t-2)={13t-23,2t<5,-13t+83,5t8,0,otherwise.

Subsequently, we can denote x(t)=Δ13(η2˜)(t)=(η2˜)(t3), so

x(t)={13t3-23,23t<53,-13t3+83,53t2,0,otherwise.

Example 7 is shown in Figure 7.

We now observe the delta cube root when η is not constrained by η ≥ 0̃. Without a loss of generality, we assume that xyz. In Table 1, SubCase (1-2), (2-1), (2-2), and (2-4) are illogical. Therefore, we must consider SubCase (1-1), (1-3), (1-4), and (2-3). From Theorem 3, which is the result for SubCase (1-1), we can easily obtain Corollary 2 for SubCase (1-4).

Corollary 2

For ηF(ℝ) with η ≤ 0̃ and for any t, we have

Δ13(η3)(t)=η(t).

Proof. For ηF(ℝ) with η ≤ 0̃, let γ = −η, which yields γ(s) = η(−s) for any s ∈ ℝ: We demonstrate that γ3 = −η3 from γ=-η=-1˜η. We note γ ≥ 0̃. By Theorem 3, Δ13(γ3)(s)=γ(s) holds true for any s ∈ ℝ. Thus.

Δ13(η3)(t)=(η3)(t3)=(-η3)(-t3)=γ3((-t)3)=Δ13(γ3)(-t)=γ(-t)=η(-(-t))=η(t).

We also observe that the cube of the delta cube root of a nonnegative/nonpositive fuzzy number becomes itself.

Corollary 3

For η with either η ≤ 0̃ or η ≥ 0̃, we have

(Δ13η)3=η.

Proof. This follows because

(Δ13η)3=Δ13(η3)=η,

where the first equality is given by Corollary 1.

Thus far, a fuzzy number ηF(ℝ) with nonzero values only for SubCase either (1-1) or (1-4), has been investigated, and it has been shown that equation Δ13(η3)(t)=η(t) holds in Theorem 3 and Corollary 2, respectively.

Hereafter, a general fuzzy number ηF(ℝ) with nonzero values, even for SubCase either (1-3) or (2-3) must be dealt with. Remark 1 provides examples of η related to SubCase (1-3) and (2-3), where equation Δ13(η3)(t)=η(t) no longer holds.

Remark 1

We give examples on ηF(ℝ), related to SubCase (1-3) and (2-3), where the equation Δ13(η3)(t)=η(t) does not hold anymore.

  • 1. SubCase (1-3): Define η(t) by


    η(t)={t+9,-9t<-8,-111t+311,-8t3,0,otherwise.

    Then by Definitions 2 and 5,


    Δ13(η3)(t)=supxyz=t3(min (η(x),η(y),η(z))).

    For t = 2,


    Δ13(η3)(2)=supxyz=8(min (η(x),η(y),η(z))).

    Consider a particular one (x0, y0, z0) = (−8,−1, 1) among the elements (x, y, z) ∈ ℝ3 such that xyz = 8. Note that (x0, y0, z0) satisfies the condition for SubCase (1-3). Notice that Figure 8 is the membership function of η(t). In this case, min (η(x0), η(y0), η(z0))= η(z0) since η(x0) ≥ η(y0) ≥ η(z0). And η(2) < η(z0). Thus


    Δ13(η3)(2)=supxyz=8(min (η(x),η(y),η(z)))min (η(x0),η(y0),η(z0))=η(z0)>η(2).

    Hence, the equation Δ13(η3)(t)=η(t) does not hold.

  • 2. SubCase (2-3): Define η(t) by


    η(t)={111t+3,-3t<8,-t+9,8t9,0,otherwise.

    Then by Definitions 2 and 5,


    Δ13(η3)(t)=supxyz=t3(min (η(x),η(y),η(z))).

    Note that, for t = −2,


    Δ13(η3)(-2)=supxyz=-8(min (η(x),η(y),η(z))).

    Consider a particular one (x0, y0, z0) = (−1, 1, 8) among the elements (x, y, z) ∈ ℝ3 such that xyz = −8. Note that (x0, y0, z0) satisfies the condition for SubCase (2-3). Notice that Figure 9. is the membership function of η(t). Here, min (η(x0), η(y0), η(z0)) = η(z0) since η(x0) ≥ η(y0) ≥ η(z0). Furthermore, η(−2) < η(z0). Thus,

    Δ13(η3)(-2)=supxyz=-8(min (η(x),η(y),η(z)))min (η(x0),η(y0),η(z0))=η(z0)>η(-2).

    Hence, Δ13(η3)(t)=η(t) does not hold.

Therefore, Δ13(η3)(t)=η(t) does not hold for the Cases (1-3) and (2-3).

Consequently, it is trivial for η = 0. For cases in which η > 0̃, (1-1) is feasible, and for cases in which η < 0̃, (1-4) is feasible. This is consistent with the results in real numbers, where the cube root of 8 has only one real value, 2, and the cube root of −8 has only one real value −2.

5. Discussion on the Delta n-th Root of a Fuzzy Real Number

Naturally, we can consider the delta n-th root for any n ∈ ℕ, which generalizes the delta and delta cube roots. When n is an even number, we can easily verify that we can define the delta nth root only when η ≥ 0̃. However, what we need to verify increases if n is odd. The main challenge with n increasing verifying the delta n-th root of n-th power of a fuzzy number. The number of variables increases as n increases because it produces Δ1n(ηn)(t)=supx1xn=tn(min (η(x1),,η(xn))). Although it is difficult to consider all cases, we introduce the following remarks to provide some insight.

Remark 2

(Observation about the delta 5-th root of 5-th power of fuzzy real number)

Notice that

Δ15(η5)(t)=supx1x2x3x4x5=t5(min (η(x1),η(x2),η(x3),η(x4),η(x5))),

by Definitions 2 and 5. The signs of x1x2x3 must be the same as those of x4x5. Similar to the arguments in the proof of Theorem 3, we assume that x1x2x3x4x5. The cases in Table 3.

We can verify that Δ15(η5)(t)=η(t) holds for the Cases (1) and (6) in Table 3 with similar arguments with ones in the proofs of Theorem 3 and Corollary 2. For the Cases (2) and (4) in Table 3, we suggest a way to make an example that does not satisfy Δ15(η5)(t)=η(t).

The main point is to set the value t which is strictly larger or smaller than the other variables x1, ..., x5.

First, let us consider the Cases (2) and (4) in Table 3. Because t < 0, t must be less than x1. Then, select any t that satisfies |t| > 1; for example, t = −2.

Then consider the case of |x5| = |t5| and |xi| = 1 for i = 1, 2, 3, 4. For the Case (2), let (x1, x2, x3, x4, x5) = (−1, 1, 1, 1, 32) and t = −2 for example. This enables us to create a fuzzy number that strictly increases from t to x5, as in the case of Figure 9. Finally, we verify that the equation Δ15(η5)(t)=η(t) does not hold.

Second, consider the Cases (3) and (5) in Table 3. Because t > 0, t must be greater than x5 and |t| > 1, we select t = 2. Consider the case where |x1| = |t5| and |xi| = 1 for i = 2, 3, 4, 5.

For Case (3), let (x1, x2, x3, x4, x5) = (−32,−1, 1, 1, 1) and t = 2 for an example. This enables us to create a fuzzy real number that strictly decreases from x1 to t, as shown in Figure 8. In addition, we can check whether the equation Δ15(η5)(t)=η(t) holds.

For an other odd n such as n = 7, 9, ..., we will be able to prove that the equation Δ1n(ηn)(t)=η(t) holds in the case of either η ≥ 0̃ or η ≤ 0̃. By following arguments similar to those in Remark 2, we can create examples such that Δ1n(ηn)(t)=η(t) does not hold when the fuzzy real number ηF(ℝ) has a nonzero value in each of the negative and positive domains.

6. Conclusion

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 η.

For any real number t ∈ ℝ, the root of its cube (t3)13 always becomes t. We investigate whether this property also applies to fuzzy numbers. Consequently, this holds in the case of either a nonnegative fuzzy real number or a nonpositive number. However, we identified cases and examples in which this property did not hold.

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 n-th root of a fuzzy number becomes complex as n increases, owing to the growing number of variables to consider. To determine whether the n-th power of the delta n-th root of a fuzzy number η becomes η, we presented a method for dividing the cases for consideration. By defining the fuzzy n-th root using a functional definition, we apply the n-th root concept to fuzzy numbers as if they were real numbers. Consequently, fuzzy numbers can be applied to a wider and more diverse range of fields. These generalizations open up opportunities for further applications, emphasizing the growing scope and impact of our research on operations involving fuzzy numbers.

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)

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