Limit properties and derivative operations in the metric space of intuitionistic fuzzy numbers

Ai, Zhenghai; Zhang, Xu; Lei, Qian

doi:10.1007/s10700-016-9239-7

Cited by 18 publications

(15 citation statements)

References 20 publications

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“…Since the q‐ ROFN is a generalization of the IFN, then on the basis of the relationship between the IFN and the q‐ ROFN, we introduce some necessary prior knowledge regarding the continuity and derivative, infinitesimal of IFFs.…”

Section: Preliminariesmentioning

confidence: 99%

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Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Zhang

2019

Int J Intell Syst

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The q‐rung orthopair fuzzy set ( q‐ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q‐rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q‐rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q‐rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q‐rung orthopair single variable fuzzy function ( q‐ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q‐ROSVFFs, and offer the relationship between the limit of q‐ROSVFF and that of q‐ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q‐ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q‐ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q‐ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q‐ROSVFFs.

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Section: Preliminariesmentioning

confidence: 99%

“…Definition Let

F false(X false)

be an IFF of

X

, if

\lim_{X \to X_{0}} F (X) = true(0, 1 true)

; then

F false(X false)

is called the IFI as

X \to X_{0}

.…”

Section: Preliminariesmentioning

confidence: 99%

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Zhang

2019

Int J Intell Syst

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“…As mentioned in Introduction, using the IFIs, we can reveal the relationships between the derivatives and the differentials, using the intuitionistic fuzzy infinitesimals and the operator |•| (if β ⊴ α, then |α ⊖ β| � α ⊖ β, and if α ⊴ β, then |α ⊖ β| � β ⊖ α.) [17].…”

Section: Preliminariesmentioning

confidence: 99%

“…More recently, the scholars have investigated a series of fundamental properties with respect to the intuitionistic fuzzy calculus (IFC) [16][17][18][19][20][21][22]; the key characteristic of the IFC is that it is based on the basic operational laws of intuitionistic fuzzy numbers (IFNs). With these operational laws, Lei and Xu [16] introduced the limits, the derivatives, and the differentials in intuitionistic fuzzy circumstance and gave their basic properties.…”

Section: Introductionmentioning

confidence: 99%

“…With these operational laws, Lei and Xu [16] introduced the limits, the derivatives, and the differentials in intuitionistic fuzzy circumstance and gave their basic properties. To further develop limit theory, the intuitionistic fuzzy infinitesimals (IFIs) [17] were introduced to reveal the relationships between the derivatives and the differentials. Later on, the completeness of intuitionistic fuzzy partially ordered set [18] was discussed.…”

Section: Introductionmentioning

confidence: 99%

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Foundation of Interval-Valued Intuitionistic Fuzzy Limit and Differential Theory and an Application to Continuous Data

Shu

Zhang

2019

Complexity

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The intuitionistic fuzzy calculus (IFC), based on the basic operational laws of intuitionistic fuzzy numbers (IFNs), has been put forward. However, the interval-valued IFC (IVIFC), based on the basic operational laws of interval-valued IFNs (IVIFNs), is only in the original stage. To further develop the theory of the IVIFC and make it be rigorous, the primary task is to systematically investigate the characteristics of the limits and differentials, which is a foundation of the IVIFC. Moreover, there is quite a lot of literature on IVIFNs; however, the scholars did not reveal the relationships between IFNs and the IVIFNs. To do that, we first investigate the limit of interval-valued intuitionistic fuzzy sequences, and then, we focus on investigating the limit, the continuity, and the differential of IVIFFs in detail and reveal their relationships. After that, due to the fact that the IFC and the IVIFC are based on the basic operational laws of IFNs and IVIFNs, respectively, we reveal the relationships between the IFNs and the IVIFNs via some homomorphic mappings. Finally, a case study about continuous data of IVIFNs is provided to illustrate the advantages of continuous data.

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Some New Ordered Semi-linear Spaces of Intuitionistic Fuzzy Processes and the Pair of Closely Related States

Phu¹,

Hung²,

Quynh³

2020

Advances in Intelligent Systems and Computing

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Limit properties and derivative operations in the metric space of intuitionistic fuzzy numbers

Cited by 18 publications

References 20 publications

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Single variable differential calculus under q ‐rung orthopair fuzzy environment: Limit, derivative, chain rules, and its application

Foundation of Interval-Valued Intuitionistic Fuzzy Limit and Differential Theory and an Application to Continuous Data

Some New Ordered Semi-linear Spaces of Intuitionistic Fuzzy Processes and the Pair of Closely Related States

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