# Analysis of Economic Relationship Using the Concept of Complex Pythagorean Fuzzy Information

Authors: Naeem Jan; Saif Ur Rehman; Abdul Nasir; Hassen Aydi; Sami Ullah Khan
Journal: Security and Communication Networks
 (2021)

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## Abstract

Fuzzy set theory and fuzzy logics are the powerful mathematical tools to model the imprecision and vagueness. In this research, the novel concept of complex Pythagorean fuzzy relation (CPFR) is introduced. Furthermore, the types of CPFRs are explained with appropriate examples such as CPF composite relation, CPF equivalence relation, CPF order relation, and CPF equivalence classes. Moreover, numerous results and interesting properties of CPFRs are discussed in detail. Furthermore, the impacts of economic parameters over each other are studied through the proposed concepts of CPFRs. In addition, the application also discusses the effects of economic parameters of one country over the other countries’ economic parameters.


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## Body

## 1. Introduction

In 1965, Zadeh [1] initiated the concepts of fuzzy sets (FSs). A FS is characterized by a mapping whose range is the unit interval [0, 1]. This mapping is called a degree of membership. Zadeh’s FS theory is used to model imprecise information and vagueness. Later in 1986, Atanassov [2] introduced the notion of intuitionistic fuzzy sets (IFSs). An IFS is a collection of objects characterized by a pair of mappings whose values range are between 0 and 1. These mappings represent the degree of membership and the degree of nonmembership of the object. Moreover, the sum of both the degrees of membership and nonmembership in an IFS must not exceed 1. This limitation persuaded Yager [3] to introduce a modified version of IFS, called a Pythagorean fuzzy set (PFS). Like IFSs, PFSs also consist of the degree of membership and the degree of nonmembership that attain values from the unit interval [0, 1] provided that the sum of squares of both the degrees does not exceed 1. Deschrijver and Kerre [4] wrote on the relationships between some extensions of FS theory. Maiers and Sherif [5] proposed the applications of FS theory. Klir [6] worked on FS interpolation of possibility theory. De et al. [7, 8] discussed some operations on IFSs and provided an application of IFSs in medical diagnosis. Szmidt and Kacprzyk [9] measured distance between IFSs. Peng et al. [10, 11] conceived some results for PFSs and provided an application.Ramot et al. [12] initiated the concept of complex fuzzy sets (CFSs). A CFS is characterized by the degree of membership that is a complex valued mapping, whose range is a unit circle in the complex plane. Since a complex number is a combination of real and imaginary numbers, the degree of membership is expressed in the polar form as ηℂℏeξℂℏ2πi, where ηℂℏ and ξℂℏ are the real numbers from the unit interval. ηℂℏ is known as the amplitude term and ξℂℏ is known as the phase term. Therefore, a CFS is capable to model problems having a periodic nature. Alkouri and Salleh [13] introduced the concept of complex intuitionistic fuzzy sets (CIFSs). These sets comprise of a pair of complex valued degrees of membership and degree of nonmembership. Both the degrees attain values from a unit circle in a complex plane provided that their sum also lied within the unit circle. In other words, the sum of amplitude terms lies in the unit interval, and the sum of both the phase terms also lies in [0, 1]. Ullah et al. [14] amended the concept of CIFSs and provided the notion of complex Pythagorean fuzzy set (CPFS). These sets also discuss the complex valued degrees of membership and nonmembership provided that both the degrees and the sum of their squares lie within the unit circle of complex plane. Yazdanbakhsh and Dick [15] carried out a systematic review of CFSs and logics. Tamir et al. [16, 17] provided the axiomatic theory of CF logics and the applications of CFS and CF logics. Rani and Garg [18] proposed the distance measures between CIFSs and applied them to decision-making processes. Ngan et al. [19] represented CIFS by quaternion numbers and applied them to decision-making. Yaqoob et al. [20] applied the concept of IFSs to cellular network provider companies. Dick et al. [21] wrote on Pythagorean and CFS operations. Ma et al. [22] worked on the group decision-making framework by CPF information, and Bi et al. [23] introduced the CF arithmetic aggregation operators.The set relations have various applications in mathematics, engineering, social sciences, and many other fields. Klir and Folger [24] defined the crisp relations (CRs). Several types of relations, such as inverse relation, reflexive relation, symmetric relation, transitive relation, composite relation, and equivalence relation, are defined. Mendel [25] initiated the theory of fuzzy relations (FRs). The plus point of FRs is that these relations indicate the grade of the relationship by the degree of membership. Burillo et al. [26] instigated the notion of intuitionistic fuzzy relations (IFRs). These relations not only discuss the degree of membership of the relationship but also deliberate the degree of nonmembership. Bhattacharya and Mukherjee [27] discussed the FRs and fuzzy groups. Yeh and Bang [28] presented the FRs and fuzzy graphs and applied these notions in clustering analysis. Blin [29] proposed FRs in group decision theory. Deschrijver and Kerre [30] worked on the composition of IFRs. Bustince and Burillo [31] gave structures on IFRs. Bustince [32] constructed the IFRs with predetermined properties, and Naz et al. [33] came up with a novel approach to decision-making with PF information. Alshammari et al. [34] presented the topological structure of CPFSs, and Dinakaran [35] analyzed the online food delivery industries using PFRs and composition.This article proposes a novel concept of complex Pythagorean fuzzy relation (CPFR). The CPFRs consider the degrees of membership and nonmembership of any relation between the elements of CPFSs. Moreover, the complex nature of the relation allows to model problems with phase changes, such as periodicity. Although IFRs also cover both the degrees of membership as well as nonmembership, but they restrict the choice of numbers that could be assigned to the members of sets. For example, we cannot assign the membership degree0.7e0.5 when the degree of nonmembership is 0.4e0.6 because the sum of amplitude terms exceeds 1, i.e., 0.7+0.4=1.1. The same applies for the phase terms that their sum must not exceed 1. But, the advantage of proposed structure is that it eases these constraints by some modifications. It expands the choice of numbers to be assigned as the degrees. Furthermore, the types of CPFRs are explained with appropriate examples. Some of the types include CPF irreflexive relations, CPF antisymmetric relations, CPF equivalence relations, CPF order relations, and CPF equivalence classes. Additionally, some unique and interesting properties and results are achieved. Finally, in the support of the proposed work, an application is provided that helps in the investigation of quality of relationships among different economic indicators and the impacts that each economic parameter has on the other rest of the economic parameters. These concepts can be stretched to other frameworks of the fuzzy set theory which will give rise to many interesting structures. The range of applications of these complex natured structures might be so vast, since they can deal with problems that have multiple dimensions. In future, we would be looking to apply these notions to study the security and communication networks.The study is arranged in the following way: Section1 presents the introduction and the literature review. Section 2 reviews some of the predefined concepts that are used in current study. In Section 3, the complex Pythagorean fuzzy relations (CPFRs) and their types are discussed along with examples. Additionally, some results of the proposed relations have also been proved. The Section 4 proposes an application of CPFRs and its types for the investigation of direct and indirect effects of economic factors of one country on other countries. Finally, in Section 5, the research work is concluded.


## 2. Preliminaries

This section enlightens some related prerequisites such as fuzzy set (FS), complex fuzzy set (CFS), intuitionistic fuzzy set (IFS), complex intuitionistic fuzzy set (CIFS), Pythagorean fuzzy set (PFS), and complex Pythagorean fuzzy set (CPFS). Moreover, the Cartesian product and relations in above sets are also described with some examples.Definition 1 (see [1]).
Letℬ.. be a universal set. Then, a set ℍ.. on ℬ.. is said to be an FS if it is of the form(1)ℍ..=ℏ1,mℏ1|ℏ1∈ℬ..,where mℏ1 is a fuzzy valued mapping, i.e., m:ℬ..⟶0,1 and symbolizes the membership degree of the FS ℍ...Definition 2 (see [12]).
Letℬ.. be a universal set. Then, a set ℍ.. on ℬ.. is said to be CFS if it is of the form(2)ℍ..=ℏ1,mℍ..cℏ1|ℏ1∈ℬ..,where mℍ..cℏ1 is a complex valued mapping which symbolizes the membership degree of the CFS ℍ... This mapping mℍ..cℏ1 is defined as(3)mℍ..c:ℬ..⟶ϒ|ϒ∈ℂ,ϒ≤1,where ℂ is the set of complex numbers, thus(4)ϒℏ1=α+iβ,or(5)ϒℏ1=ηℂℏ1eξℂℏ12πi,whereηℂℏ1,ξℂℏ1∈0,1.Definition 3 (see [16]).
Letℍ..=ℏj,mℍ..ℂℏj|ℏj∈ℬ.. and K..=ℏk,mK..ℂℏk|ℏk∈ℬ.., j,k∈ℕ, be two CFS on the universal set ℬ.. with the membership degrees symbolized by the complex valued mappings mℍ..ℂ and mK..ℂ, respectively. Then, the Cartesian product of ℍ.. and K.. is denoted and defined as(6)ℍ..×K..=ℏj,ℏk,mℍ..×K..ℂℏj,ℏk,|ℏj∈ℍ..,ℏk∈K..,where the mapping mℍ..×K..ℂ:ℍ..×K..⟶ϒ|ϒ∈ℂ,ϒ≤1 symbolizes the membership degree of the product ℍ..×K.. and is defined as(7)mℍ..×K..ℂℏj,ℏk≤minmℍ..ℂℏj,mK..ℂℏk.
Sincemℍ..×K..ℂ is a complex valued mapping, one writes(8)ϒℏj,ℏk=αjk+iβjk,or(9)ϒℏj,ℏk=ηℍ..×K..ℂℏj,ℏkeξℍ..×K..ℂℏj,ℏk2πi=minηℍ..ℂℏj,ηK..ℂℏkeminξℍ..ℂℏj,ξK..ℂℏk2πi,where ηℍ..×K..ℂℏj,ℏk,ξℍ..×K..ℂℏj,ℏk∈0,1.Definition 4 (see [16]).
Letℍ..=ℏj,mℍ..ℂℏj|ℏj∈ℬ.. and K..=ℏk,mK..ℂℏk|ℏk∈ℬ.. be two CFSs on ℬ... Then, a fuzzy relation (FR) ℝ is any subset of the product ℍ..×K... It is denoted and defined as(10)ℝ=ℏj,ℏk,mℝℂℏj,ℏk|ℏj,ℏk∈ℍ..×K..,where mℝℂℏj,ℏk symbolizes the membership degree of FR ℝ and(11)mℝℂℏj,ℏk=mℍ..×K..ℂℏj,ℏk.Example 1.
Consider aCFSℍ.. on ℬ...(12)ℍ..=ℏ1,0.8e0.52πi,ℏ2,0.3e0.62πi,ℏ3,0e12πi.
The Cartesian productℍ..×ℍ.. is(13)ℍ..×ℍ..=ℏ1,ℏ10.8e0.52πi,ℏ1,ℏ20.3e0.52πi,ℏ1,ℏ30e0.52πi,ℏ2,ℏ10.3e0.52πi,ℏ2,ℏ20.3e0.62πi,ℏ2,ℏ30e0.62πi,ℏ3,ℏ10e0.52πi,ℏ3,ℏ20e0.62πi,ℏ3,ℏ30e12πi,.
TheCFRℝ is(14)ℝ=ℏ1,ℏ10.8e0.52πi,ℏ1,ℏ20.3e0.52πi,ℏ2,ℏ30e0.62πi,ℏ3,ℏ10e0.52πi.Definition 5 (see [2]).
Letℬ.. be a universal set. Then, a set ℍ.. on ℬ.. is said to be an IFS if it is of the form(15)ℍ..=ℏ1,mℏ1,nℏ1|ℏ1∈ℬ..,provided that mℏ1+nℏ1∈0,1, where mℏ1 and nℏ1 are the fuzzy valued mappings which symbolize the membership and nonmembership degrees of the IFS ℍ.., respectively.Definition 6 (see [13]).
Letℬ.. be a universal set. Then, a set ℍ.. on ℬ.. is said to be a CIFS if it is of the form(16)ℍ..=ℏ1,mℍ..ℂℏ1,nℍ..ℂℏ1|ℏ1∈ℬ..,where mℍ..ℂ and nℍ..ℂ are the complex valued mappings which represent the membership degree and nonmembership degree of CIFS ℍ.., respectively. These mappings are defined as(17)mℍ..ℂ:ℬ..⟶ϒ1|ϒ1∈ℂ,ϒ1≤1,nℍ..ℂ:ℬ..⟶ϒ2|ϒ2∈ℂ,ϒ2≤1,where ℂ is the set of complex numbers. Thus,(18)ϒ1ℏ1=α1+iβ1,ϒ2ℏ1=α2+iβ2,provided that ϒ1ℏ12+ϒ2ℏ12∈0,1 or(19)ϒ1ℏ1=η1ℍ..ℂℏ1eξ1ℍ..ℂℏ12πi,ϒ2ℏ1=η2ℍ..ℂℏ1eξ2ℍ..ℂℏ12πi,where η1ℍ..ℂℏ1,η2ℍ..ℂℏ1,ξ1ℍ..ℂℏ1,ξ2ℍ..ℂℏ1∈0,1, provided that η1ℍ..ℂℏ12+η2ℍ..ℂℏ12∈0,1 and ξ1ℍ..ℂℏ12+ξ2ℍ..ℂℏ12∈0,1.Definition 7.
([3]). Let ℬ.. be a universal set. Then, a set ℍ.. on ℬ.. is said to be a Pythagorean fuzzy set (PFS) if it is of the form(20)ℍ..=ℏ1,mℏ1,nℏ1|ℏ1∈ℬ..,provided that m2ℏ1+n2ℏ1∈0,1, where mℏ1 and nℏ1 are the fuzzy valued mappings which symbolize the membership and nonmembership degrees of the PFS ℍ.., respectively.Definition 8 (see [14]).
Letℬ.. be a universal set. Then, a set ℍ.. on ℬ.. is said to be a CPFS if it is of the form(21)ℍ..=ℏ1,mℍ..ℂℏ1,nℍ..ℂℏ1|ℏ1∈ℬ..,where mℍ..ℂℏ1 and nℍ..ℂℏ1 are the complex valued mappings which symbolize the membership degree and nonmembership degree of CPFS ℍ.., respectively. These mappings are defined as(22)mℍ..ℂ:ℬ..⟶ϒ1|ϒ1∈ℂ,ϒ1≤1,nℍ..ℂ:ℬ..⟶ϒ2|ϒ2∈ℂ,ϒ2≤1,where ℂ is the set of complex numbers. Thus,(23)ϒ1ℏ1=α1+iβ1,ϒ2ℏ1=α2+iβ2,provided that ϒ1ℏ12+ϒ2ℏ12∈0,1 or(24)ϒ1ℏ1=η1ℍ..ℂℏ1eξ1ℍ..ℂℏ12πi,ϒ2ℏ1=η2ℍ..ℂℏ1eξ2ℍ..ℂℏ12πi,where η1ℍ..ℂℏ1,η2ℍ..ℂℏ1,ξ1ℍ..ℂℏ1,ξ2ℍ..ℂℏ1∈0,1, provided that η1ℍ..ℂℏ12+η2ℍ..ℂℏ12∈0,1 and ξ1ℍ..ℂℏ12+ξ2ℍ..ℂℏ12∈0,1.


## 3. Main Results

Definition 9.
The Cartesian product of the CPFSsℍ..=ℏj,mℍ..ℂℏj,nℍ..ℂℏj|ℏj∈ℬ.. and K..=ℏk,mK..ℂℏk,nK..ℂℏk|ℏk∈ℬ..,j,k∈ℕ is defined by(25)ℍ..×K..=ℏj,ℏk,mℍ..×K..ℂℏj,ℏk,nℍ..×K..ℂℏj,ℏk|ℏi∈ℍ..,ℏk∈K...
The mappingsmℍ..×K..ℂ:ℬ..⟶ϒm|ϒm∈ℂ,ϒm≤1 and nℍ..×K..ℂ:ℬ..⟶ϒn|ϒn∈ℂ,ϒn≤1 represent the degrees of membership and nonmembership ℍ..×K.. and is defined as mℍ..×K..ℂℏj,ℏk=minmℍ..ℂℏj,mK..ℂℏk and nℍ..×K..ℂℏj,ℏk=maxnℍ..ℂℏj,nK..ℂℏk. Moreover, the complex numbers ϒn and ϒm for ℍ..×K.. can be defined as(26)ϒmℏj,ℏk=ηℍ..×K..mℏj,ℏkeξℍ..×K..mℏj,ℏk2πi=minηℍ..mℏj,ηK..mℏkeminξℍ..mℏj,ξK..mℏk2πi,ϒnℏj,ℏk=ηℍ..×K..nℏj,ℏkeξℍ..×K..nℏj,ℏk2πi=maxηℍ..nℏj,ηK..nℏkemaxξℍ..nℏj,ξK..nℏk2πi,with the conditions(27)0≤ηℍ..×K..mℏj,ℏk2+ηℍ..×K..nℏj,ℏk2≤1,0≤ξℍ..×K..mℏj,ℏk2+ξℍ..×K..nℏj,ℏk2≤1.Definition 10.
A complex Pythagorean fuzzy relation (CPFR) denoted byℝ.. is a nonempty subset of ℍ..×K.., where ℍ.. and K.. are the CPFSs.Example 2.
Take the CPFS,(28)ℍ..=ℏ1,0.6e0.52πi,0.7e0.62πi,ℏ2,0.7e0.62πi,0.5e0.72πi.
Now, findℍ..×ℍ... We have(29)ℍ..×ℍ..=ℏ1,ℏ1,0.6e0.52πi,0.7e0.62πi,ℏ1,ℏ2,0.5e0.52πi,0.5e0.62πi,ℏ2,ℏ1,0.5e0.52πi,0.6e0.52πi,ℏ2,ℏ2,0.7e0.62πi,0.5e0.72πi.
The CPR is(30)ℝ..=ℏ1,ℏ2,0.5e0.52πi,0.5e0.62πi,ℏ2,ℏ2,0.7e0.62πi,0.5e0.72πi.Definition 11.
The inverse of the CPFRℝ..−1 of the CPFR ℝ..=ℏj,ℏk,mℝ..ℂℏj,ℏk,nℝ..ℂℏj,ℏk|ℏj,ℏk∈ℝ.. is ℝ..−1=ℏk,ℏj,mℝ..ℂℏk,ℏj,nℝ..ℂℏk,ℏj|ℏk,ℏj∈ℝ...Example 3.
Let the CPFRℝ.. from (29),(31)ℝ..1=ℏ1,ℏ2,0.5e0.52πi,0.5e0.62πi,ℏ2,ℏ1,0.5e0.52πi,0.6e0.52πi,where ℝ..1⊂ℝ... Now, the inverse of ℝ..1 is(32)ℝ..1−1=ℏ2,ℏ1,0.5e0.52πi,0.5e0.62πi,ℏ1,ℏ2,0.5e0.52πi,0.6e0.52πi.Definition 12.
The CPFRℝ..2 is called a CPF reflexive relation if(33)∀ℏ1,mℍ..ℂℏ1,nℍ..ℂℏ1∈ℍ..⇒ℏ1,ℏ1,mℍ..ℂℏ1,ℏ1,nℍ..ℂℏ1,ℏ1∈ℝ..2,while the CPFR ℝ..3 is an irreflexive relation if ℏ1,ℏ1,mℍ..ℂℏ1,ℏ1,nℍ..ℂℏ1,ℏ1∉ℝ..3.Example 4.
From Example2, in (29), the following relations ℝ..2 and ℝ..3 are CPF reflexive relation and irreflexive relation, respectively:(34)ℝ..2=ℏ1,ℏ1,0.6e0.52πi,0.7e0.62πi,ℏ2,ℏ2,0.7e0.62πi,0.5e0.72πi,ℝ..3=ℏ1,ℏ2,0.5e0.52πi,0.5e0.62πi,ℏ2,ℏ1,0.5e0.52πi,0.6e0.52πi.Definition 13.
A CPFRℝ.. is called a CPF symmetric relation if(35)∀ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ...
A CPFRℝ..1 is called a CPF asymmetric relation if(36)∀ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..1⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∉ℝ..1.
A CPFRℝ..2 is called a CPF antisymmetric relation if(37)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..2,ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..2⇒ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2=ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1.Example 5.
The following CPF relationsℝ..1, ℝ..2, and ℝ..3 are CPF symmetric relation, CPF asymmetric relation, and CPF antisymmetric relation, respectively:(38)ℝ..1=ℏ1,ℏ2,0.5e0.72πi,0.6e0.52πi,ℏ2,ℏ1,0.5e0.72πi,0.6e0.52πi,ℝ..2=ℏ1,ℏ2,0.6e0.52πi,0.7e0.62πi,ℏ1,ℏ3,0.7e0.62πi,0.6e0.52πi,ℝ..3=ℏ1,ℏ1,0.4e0.52πi,0.9e0.72πi,ℏ3,ℏ3,0.8e0.62πi,0.4e0.72πi.Theorem 1.
A CPFRℝ.. is a symmetric relation iff ℝ..=ℝ..−1.Proof.
Letℝ..=ℝ..−1, then we have(39)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..−1.
Sinceℝ..=ℝ..−1, one writes(40)ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ...
Hence proved thatℝ.. is a CPF symmetric relation.
Conversely, assume thatℝ.. is a CPF symmetric relation. Then,(41)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ...
But, if(42)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..−1,this implies that ℝ..=ℝ..−1.Theorem 2.
Letℝ..1 and ℝ..2 are CPF symmetric relations, then their intersection, i.e, ℝ..1∩ℝ..2 is also a CPF symmetric relation.Proof.
Assume thatℝ..1 and ℝ..2 are two CPFR symmetric relations on CPFS ℍ... Then, from the definition of CPFR, ℝ..1⊆ℍ..×ℍ.. and ℝ..2⊆ℍ..×ℍ..⇒ℝ..1∩ℝ..2⊆ℍ..×ℍ..⇒ℝ..1∩ℝ..2 is CPR on ℍ... Now, assume that(43)ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..1∩ℝ..2⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..1andℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..2.
Since,ℝ..1 and ℝ..2 are CPF symmetric relations. Therefore,(44)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..1,ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..2⇒ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..1∩ℝ..2.Definition 14.
A CPFRℝ.. is called a CPF transitive relation if(45)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..,ℏ2,ℏ3,mℝ..ℂℏ2,ℏ3,nℝ..ℂℏ2,ℏ3∈ℝ..,this implies that(46)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ...Example 6.
A CPFRℝ.. is a CPF transitive relation if(47)ℝ..=ℏ1,ℏ2,0.6e0.52πi,0.4e0.72πi,ℏ2,ℏ3,0.5e0.82πi,0.6e0.32πi,ℏ1,ℏ3,0.7e0.42πi,0.5e0.62πi.Definition 15.
A CPF composite relationℝ..1∘ℝ..2 combines the CPFRs ℝ..1 and ℝ..2 as for(48)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..1,ℏ2,ℏ3,mℝ..ℂℏ2,ℏ3,nℝ..ℂℏ2,ℏ3∈ℝ..2,this implies that(49)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ..1∘ℝ..2.Theorem 3.
A CPFRℝ.. is a transitive relation iff ℝ..∘ℝ..⊆ℝ...Proof.
Assume thatℝ.. is a CPF transitive relation, then for(50)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..,ℏ2,ℏ3,mℝ..ℂℏ2,ℏ3,nℝ..ℂℏ2,ℏ3∈ℝ..,one has(51)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ...
But,(52)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ..∘ℝ...
Hence,ℝ..∘ℝ..⊆ℝ..
Conversely assume thatℝ..∘ℝ..⊆ℝ.., then for(53)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..,ℏ2,ℏ3,mℝ..ℂℏ2,ℏ3,nℝ..ℂℏ2,ℏ3∈ℝ..,one has(54)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ..∘ℝ...
But,ℝ..∘ℝ..⊆ℝ..; this implies that(55)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ...
Hence, it is proved thatℝ.. is a CPF transitive relation.Definition 16.
A CPFRℝ..1 is called a CPF equivalence relation, if ℝ..1 satisfies the properties of a CPF reflexive relation, a CPF symmetric relation, and a CPF transitive relation. While, a CPFR ℝ..2 is called a CPF-order relation if ℝ..2 satisfies the properties of a CPF reflexive relation, a CPF antisymmetric relation, and a CPF transitive relation.Theorem 4.
A CPF equivalence relation implies thatℝ..∘ℝ..=ℝ...Proof.
We know that a CPF equivalence relationℝ.. is also a CPF transitive relation, so in view of Theorem 3,(56)ℝ..×ℝ..⊆ℝ...
We assume that(57)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ...
Asℝ.. is an equivalence relation and verifies the axioms of a CPF symmetric relation and CPF is a transitive relation,(58)ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ...
In view of (57) and (58), we have that ℏ1,ℏ1,mℝ..ℂℏ1,ℏ1,nℝ..ℂℏ1,ℏ1∈ℝ... By using the concept of a CPF composite relation, we have(59)ℏ1,ℏ1,mℝ..ℂℏ1,ℏ1,nℝ..ℂℏ1,ℏ1∈ℝ..∘ℝ...
Now, from (57) and (59), we get that(60)ℝ..⊆ℝ..×ℝ...
Hence, from (56) and (60), we get that ℝ..∘ℝ..=ℝ...Theorem 5.
The inverse CPFRℝ..−1 of a CPF order relation is also a CPF-order relation.Proof.
The inverse CPFRℝ..−1 of a CPF-order relation is also a CPF-order relation, if ℝ..−1 verifies the following three properties of a CPF-order relation:(P1) ∀ℏ1∈ℍ..,ℏ1,ℏ1,mℍ..ℂℏ1,ℏ1,nℍ..ℂℏ1,ℏ1∈ℝ... Since ℝ.. is a CPyF reflexive relation, ⇒ℏ1,ℏ1,mℍ..ℂℏ1,ℏ1,nℍ..ℂℏ1,ℏ1∈ℝ..−1.Hence,ℝ..−1 is a CPF reflexive relation(P2) Let ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..andℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ.., ⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..−1andℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..−1.Sinceℝ.. is a CPF antisymmetric relation, one gets(61)ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2=ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1.Hence,ℝ..−1 is a CPF antisymmetric relation(P3) Assume that ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..andℏ2,ℏ3,mℝ..ℂℏ2,ℏ3,nℝ..ℂℏ2,ℏ3∈ℝ..,⇒ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..−1andℏ3,ℏ2,mℝ..ℂℏ3,ℏ2,nℝ..ℂℏ3,ℏ2∈ℝ..−1
Sinceℝ.. is a CPF transitive relation,(62)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ..⇒ℏ3,ℏ1,mℝ..ℂℏ3,ℏ1,nℝ..ℂℏ3,ℏ1∈ℝ..−1.
Hence,ℝ..−1 is a CPF transitive relation.
Thus, from (P1), (P2), and (P3), ℝ..−1 is also a CPF-order relation.Definition 17.
For a CPF equivalence relationℝ.., ℏ1,mℍ..ℂℏ1,nℍ..ℂℏ1, the equivalence class of ℏ1 modulo ℝ.., is ℝ..ℏ1=ℏ2,mℍ..ℂℏ2,nℍ..ℂℏ2|ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ...Example 7.
Let the CPF equivalence relation(63)ℝ..=ℏ1,ℏ1,0.6e0.52πi,0.5e0.72πi,ℏ1,ℏ2,0.5e0.62πi,0.5e0.72πi,ℏ2,ℏ1,0.5e0.42πi,0.6e0.62πi,ℏ2,ℏ2,0.7e0.62πi,0.4e0.52πi.
Now, the equivalence classes of a CPF relation are(64)ℝ..ℏ1=ℝ..ℏ2=ℏ1,0.6e0.52πi,0.5e0.72πi,ℏ2,0.7e0.62πi,0.4e0.52πi.Theorem 6.
For a CPF equivalence relationℝ..,ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ.. iff ℝ..ℏ1=ℝ..ℏ2.Proof.
Assume thatℝ..ℏ1=ℝ..ℏ2, then for ℏ3∈ℬ..,(65)ℏ3,mℍ..ℂℏ3,nℍ..ℂℏ3∈ℝ..ℏ1⇒ℏ3,ℏ1,mℝ..ℂℏ3,ℏ1,nℝ..ℂℏ3,ℏ1∈ℝ...
Sinceℝ.. is symmetric, one writes(66)ℏ1,ℏ3,mℝ..ℂℏ1,ℏ3,nℝ..ℂℏ1,ℏ3∈ℝ...
Hence,ℝ.. is a CPF symmetric relation. Similarly,(67)ℏ3,mℍ..ℂℏ3,nℍ..ℂℏ3∈ℝ..ℏ2⇒ℏ3,ℏ2,mℝ..ℂℏ3,ℏ2,nℝ..ℂℏ3,ℏ2∈ℝ..,since ℝ.. is transitive,(68)⇒ℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ...
Hence,ℝ.. is a CPF transitive relation.
Conversely assume thatℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..andℏ3,mℍ..ℂℏ3,nℍ..ℂℏ3∈ℝ..ℏ1, then(69)ℏ3,ℏ1,mℝ..ℂℏ3,ℏ1,nℝ..ℂℏ3,ℏ1∈ℝ...
Thus,(70)ℏ3,ℏ1,mℝ..ℂℏ3,ℏ1,nℝ..ℂℏ3,ℏ1∈ℝ..andℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..⇒ℏ3,ℏ2,mℝ..ℂℏ3,ℏ2,nℝ..ℂℏ3,ℏ2∈ℝ..,as ℝ.. is a CPF transitive relation.(71)ℏ3,mℍ..ℂℏ3,nℍ..ℂℏ3∈ℝ..ℏ2.
Hence, we get that(72)ℝ..ℏ1⊆ℝ..ℏ2.
Similarly, we assume thatℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ..andℏ3,mℍ..ℂℏ3,nℍ..ℂℏ3∈ℝ..ℏ2, then(73)ℏ3,ℏ2,mℝ..ℂℏ3,ℏ2,nℝ..ℂℏ3,ℏ2∈ℝ...
Sinceℏ1,ℏ2,mℝ..ℂℏ1,ℏ2,nℝ..ℂℏ1,ℏ2∈ℝ.., one writes ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ.., as ℝ.. is a CPF symmetric relation. Now, we have(74)ℏ3,ℏ2,mℝ..ℂℏ3,ℏ2,nℝ..ℂℏ3,ℏ2∈ℝ..,ℏ2,ℏ1,mℝ..ℂℏ2,ℏ1,nℝ..ℂℏ2,ℏ1∈ℝ..⇒ℏ3,ℏ1,mℝ..ℂℏ3,ℏ1,nℝ..ℂℏ3,ℏ1∈ℝ..,as ⇒ℏ3,ℏ1,mℝ..ℂℏ3,ℏ1,nℝ..ℂℏ3,ℏ1∈ℝ.. is a CPF transitive relation. This implies that(75)ℏ3,mℍ..ℂℏ3,nℍ..ℂℏ3∈ℝ..ℏ1.
Hence,(76)ℝ..ℏ2⊆ℝ..ℏ1.
Now, from (72) and (76), we get that ℝ..ℏ2=ℝ..ℏ1.


## 4. Application

In the section, an application of the proposed concepts is presented, which discusses the relationships among different economic indicators. Moreover, it also specifies the level of positive and negative impacts of one parameter on the others with respect to time phase.Figure1 demonstrates the algorithm and the procedure used in the application. The flow chart describes the stepwise progression for the investigation of quality of relationships between any two economic indicators. Initially, the economic parameters or indicators to be studied are collected. After that, the Cartesian product is found to achieve all the possible relations, which proves to be helpful in learning the relationships and the level of impacts of one parameter over the others. Next, different types and properties of CPFRs can be utilized to find out the unknown and indirect relationships. Last, the experiment is concluded by reading the complex Pythagorean fuzzy information, such that the first element in the pair of relation affects the second element. The amplitude term represents the level of effectiveness, and the phase term refers to the time phase. The degree of membership characterizes the positive effects, while the degree of nonmembership characterizes the negative effects of one parameter over the others.Figure 1
Flowchart of the method used in the application.Consider the setℍ=Import,export,GDP,industry,agriculture,employment of a country’s economic indicators. The goal is to find out the impacts of each indicator over the other with respect to time. Note that, in this experiment, the unit of time is taken to be a year. Hence, a CPFS ℍ.. is induced from the set ℍ as(77)ℍ..=Import,0.3e0.52πi,0.6e0.52πi,export,0.8e0.52πi,0.2e0.752πi,industry,0.7e0.52πi,0.4e0.252πi,agriculture,0.7e0.252πi,0.5e0.752πi,employment,0.9e0.252πi,0.3e0.752πi.The investigation of relationships among the economic indicators tempts to find out the Cartesian product ofℍ..×ℍ... Hence,(78)ℍ..×ℍ..=Import,import,0.3e0.52πi,0.6e0.52πi,import,export,0.3e0.52πi,0.6e0.752πi,import,industry,0.3e0.52πi,0.6e0.52πi,import,agriculture,0.3e0.252πi,0.6e0.752πi,import,employment,0.3e0.252πi,0.6e0.752πi,export,import,0.3e0.52πi,0.6e0.752πi,export,export,0.8e0.52πi,0.2e0.752πi,export,industry,0.7e0.52πi,0.4e0.752πi,export,agriculture,0.7e0.252πi,0.5e0.752πi,export,employment,0.8e0.252πi,0.3e0.752πiindustry,import,0.3e0.52πi,0.6e0.52πi,industry,export,0.7e0.52πi,0.4e0.752πiindustry,industry,0.7e0.52πi,0.4e0.252πi,industry,agriculture,0.7e0.252πi,0.5e0.752πiindustry,employment,0.7e0.252πi,0.4e0.252πi,agriculture,import,0.3e0.252πi,0.6e0.752πiagriculture,export,0.7e0.252πi,0.5e0.752πi,agriculture,industry,0.7e0.252πi,0.5e0.752πiagriculture,agriculture,0.7e0.252πi,0.5e0.752πi,agriculture,employment,0.3e0.52πi,0.6e0.52πiemployment,import,0.3e0.252πi,0.6e0.752πi,employment,export,0.8e0.252πi,0.3e0.752πiemployment,industry,0.7e0.252πi,0.4e0.252πi,employment,agriculture,0.7e0.252πi,0.3e0.752πiemployment,employment,0.9e0.252πi,0.3e0.752πi.Each element of the Cartesian product given by equation (78) describes the magnitude of impacts of one factor over the other. For instance, industry,export,0.7e0.52πi,0.4e0.752πi conveys that the degree of good impacts of the industry on the export of the country is 0.7 with respect to a year. Since, the imaginary part represents the time phase, so 0.52=1 year. In addition, the degree of bad impacts is 0.4 with respect to one and a half year. Likewise, agriculture,industry,0.7e0.252πi,0.5e0.752πi expresses that the degree of good and positive impacts of the agriculture over the industry of a country is 0.7 with respect to half a year, and the degree of bad and negative impacts is 0.5 with respect to one and a half year.Suppose that the impacts of export on industry and the impacts of industry on employment are known, but the impacts of export over employment are not known. These unknown impacts can be easily found by using the complex Pythagorean fuzzy transitive relations because the Cartesian product is an equivalence relation.Moreover, multiple sets of economic indicators of different countries can be considered, and the indirect impacts of one country’s economy over the other countries’ can be found using the CPF composite relations.Consider three different countries: Pakistan (P), China (C), and Iran (I). The aim is to relate their economies through some economic parameters. Henceforth following are the three ℂPy FSs,(79)P=Trade,0.7e0.52πi,0.5e0.52πi,agriculture,0.8e0.752πi,0.3e0.52πi,C=Industry,0.8e0.52πi,0.5e0.52πi,agriculture,0.5e0.752πi,0.6e0.252πi,I=Trade,0.6e0.252πi,0.5e0.52πi,employment,0.7e0.52πi,0.5e0.752πi.The CPFRℝ..1 between the set P and C is found to be(80)ℝ..1=P×C=Trade,industry,0.7e0.52πi,0.5e0.52πi,trade,agriculture,0.5e0.52πi,0.6e0.52πi,agriculture,industry,0.8e0.52πi,0.5e0.52πi,agriculture,agriculture,0.5e0.752πi,0.6e0.52πi.The CPFRℝ..2 between the sets C and I is found to be(81)ℝ..2=C×I=Industry,trade,0.6e0.252πi,0.5e0.52πi,industry,employment,0.7e0.52πi,0.5e0.752πi,agriculture,trade,0.5e0.252πi,0.6e0.52πi,agriculture,employment,0.5e0.52πi,0.6e0.752πi.Now, by using the CPF composition relationℝ.., the indirect relation between the sets P and I are found as given in Table 1. Trade (I) and employment (I) are the parameters of Iran, while trade (P) and agriculture (P) are the parameters of Pakistan.Table 1
The CPyF composition relationℝ...
ℝ..=ℝ..1∘ℝ..2Trade (I)Employment (I)Trade (P)0.6e0.252πi,0.5e0.52πi0.7e0.52πi,0.5e0.752πiAgriculture (P)0.6e0.252πi,0.3e0.52πi0.7e0.52πi,0.3e0.752πi


## 5. Conclusion

In this study, the novel concept of complex Pythagorean fuzzy relation (CPFR) is defined. Additionally, the types of CPFRs are described with suitable examples. Some of the types are a CPF irreflexive relation, a CPF inverse relation, a CPF asymmetric relation, a CPF composite relation, a CPF equivalence relation, a CPF order relation, and a CPF equivalence class. Furthermore, some useful and interesting properties and results of CPFRs are elaborated in detail. Moreover, an application of the proposed method is presented that discusses the impacts of economic parameters of a country over each other. Furthermore, the application also explains the effects of economic parameters of one country over the economic parameters of the other countries. The introduced concepts can be extended to other generalizations of fuzzy sets which will produce very interesting structures. These novel structures might have a rich variety of applications in sciences, technology, and economics.



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*Source: 4513992-2021-06-09.xml*