EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6050 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exponential Fuzzy Sets and Applications of AI-Powered Investment Decision-Making Using the Weighted Mean Method M. Kaviyarasu1,∗, Mohammed Alqahtani2, M. Rajeshwari3 1 Department of Mathematics, Vel Tech Rangarajan Dr Sagunthala R & D Institute of Science and Technology, Chennai, Tamilnadu-600062,India 2 Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia 3 Department of Mathematics, Presidency University, Bangalore,India Abstract. The exponential fuzzy set (EFS) is a new modification that allows for a more flexible representation of uncertainty by using an exponential function to define membership degree. In this work, we define basic operations on EFS, such as complement, union, intersection, simple difference, and limited difference functions. The equivalency formula, symmetrical difference formula, disjoint sets, disjoint sum, and disjunctive sum are further important qualities that we examine. We analyse essential laws in the exponential fuzzy framework, such as the idempotent law of union. In addition, we present a few theorems that govern the relational and algebraic structures of EFS. EFS has been compared against traditional approaches and the result- ing studies showcase its advantages in modeling uncertainty, artificial intelligence, and decision making. This paper studies the use of exponential fuzzy sets in AI driven investment decision processes using the weighted mean method of multifactor investment analysis. 2020 Mathematics Subject Classifications: 03E72, 91B06, 68T27, 91G10 Key Words and Phrases: Exponential Fuzzy Set, Operations of Exponential Fuzzy Set, Decision-Making, Weighted Mean Method 1. Introduction Zadeh’s [1] fuzzy sets have proved to be beneficial in a number of fields of mathe- matical modeling and decision making. This concept has resulted in the creation of such sets as the intuitionistic fuzzy sets [2], neutrosophic fuzzy sets [3], and even pythagorean fuzzy sets (Yager, 2013). All of which were created to solve some form of ambiguity. One of these extensions is the exponential fuzzy set, which enables the simulation of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6050 Email addresses: kavitamilm@gmail.com (M. Kaviyarasu), ,m.alqahtani@seu.edu.sa (m. Alqahtani), rajeakila@gmail.com (M. Rajeshwari) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 2 of 29 systems where there is a significant level of uncertainty and provides robust mathemati- cal support. Exponential fuzzy sets have great potential for use in optimization, control systems, as well as decision analysis. The theoretical basis for exponential fuzzy sets is the need to define uncertainty in dynamic environments with extreme rates of change. For example, the effectiveness of exponential membership functions was proved by Wu et al. [4], who employed them in a fuzzy control approach to examine the stability of nonlinear parabolic systems. Furthermore, the predicted value of exponential fuzzy numbers and their use in inventory models for degrading objects were investigated by Garai et al. [5]. These works reflect the practical use of fuzzy sets with exponential- type membership functions for information systems that are rapidly evolving and have a strong time constraint. One of the crucial advantages of exponential fuzzy sets is that unlike traditional fuzzy models, they can represent abrupt changes in uncertainty more effective. The quantitative evaluation of differences of information in uncertain situations has been enhanced by an extreme divergence measure of Tomar and Ohlan [6]. Also, Bustince et al. [7] provide an account of the history of all types of fuzzy sets and give justification for the new novel exponential fuzzy sets from a research perspective. In the mathemat- ical formulation of exponential fuzzy sets, the degree of membership is implemented in the form of exponent, which is the basic parameter of softening the set. The use of exponential functions in fuzzy set theories makes it possible to more accurately model a wide range of real life problems such as intelligent decision making systems, environ- mental, economic and sociological forecasts (Hadi-Vencheh & Mirjaberi [8]). In addition Liang & Xu [9], the growing use of fuzzy logic in practice is backed by new advances in multiple-attribute decision-making methods, e.g., hesitant Pythagorean fuzzy sets and exponential fuzzy TOPSIS. In [10] and [11] discussed group decision making problems. The aim of this work is to provide an extensive overview of exponential fuzzy sets, such as their theory and numerous applications. We survey the existing material exponential membership functions enhance fuzzy modeling techniques. Finally, we emphasize the ad- vantages of exponential fuzzy sets in dealing with dynamic and exponentially changing uncertainty and compare their efficiencies with other fuzzy extensions. From the above literature we found the research gap and exponential fuzzy concept we introduced. This paper is organized as: The basic definitions in section 2, The basic definitions and characteristics of exponential fuzzy sets are covered in Section 3, Main results of exponential fuzzy sets are shown in Section 4, application in section 5 and future research possibilities are discussed in Section 6. 1.1. Motivation • Traditional F-sets provide a foundation for handling uncertainty, but they have limitations in capturing rapid variations in membership values. • EFSs increase the flexibility of membership functions by introducing a non-linear transformation. • EFS increases decision-making sensitivity, especially in situations where little M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 3 of 29 changes in input have a big influence on results. • By adding exponential functions, it expands on traditional fuzzy logic and is ap- propriate for more complex uses like pattern identification and risk assessment. • EFS better captures uncertainty and hesitancy in expert opinions, leading to im- proved diagnostic accuracy. • EFS improves edge detection and noise reduction in image analysis, leading to clearer and more accurate image segmentation. 1.2. Need of EFS Exponential fuzzy sets address this issue by incorporating an exponential function into the membership structure. This allows for a more flexible and precise representation of uncertainty, especially in situations where small changes in input values lead to signif- icant variations in membership degrees. For example, in medical diagnosis, financial risk assessment, and engineering problems, uncertainty often behaves in a nonlinear manner, making exponential fuzzy sets a better choice. 1.3. Advantages of EFS • Suitable for situations where uncertainty follows an exponential pattern rather than a linear one. • Ensures gradual changes in membership values, preventing abrupt shifts. • More responsive to small variations in data, improving accuracy in decision-making. • Enhances accuracy and robustness in multi-criteria decision-making (MCDM) and expert systems. 1.4. Novelty • Describes a new membership function transformation for an EFS. • Incorporates exponential scaling to improve the representation of uncertainty in conventional fuzzy set theory. • Enhances similarity and divergence metrics to help make better decisions. 2. Preliminaries Definition 1. [1] A fuzzy set A in a universe of discourse Z is characterized by a membership function ℵA which takes the value in the unit interval [0, 1], ℵA(s) = Z → [0, 1]. The value of ℵA(s) represents the grade of membership of Z in Å and is a point in [0, 1]. M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 4 of 29 Definition 2. [1] The complement A′ is defined by ℵA′ (s) = 1 − ℵA(s).Where A is a F-set in Z. Definition 3. [12] If ℵA and ℵB are membership functions of F-sets A and B, then the union and intersection of two F-sets is ℵA∪B(s) = max {ℵA(s),ℵB(s)} , ∀s ∈ Z. ℵA∩B(s) = min {ℵA(s),ℵB(s)} ,∀s ∈ Z. Definition 4. [12] The difference of two F-sets A and B is given by A− B = A ∩ Bc. Definition 5. [12] A bounded difference of two F-sets A and B is given by ℵAoB(s) = max [o,ℵA(s),ℵB(s)] Definition 6. [12] The disjoint sum of two F-sets is given by ℵA⊗B(s) = |⊗,ℵA(s),ℵB(s)| where ℵA(s) and ℵB(s) are membership function of A and B. Definition 7. Let A and B be any two F-sets of Z then the disjunctive sum is given by: ℵA∆B(s) = (A ∩ Bc) ∪ (Ac ∩ B) = (A ∗ Bc)⊕ (Ac ∗ B) . Definition 8. Let A and B be any two F-sets of Z then the equivalence formula is (Ac ∪ B) ∩ (A ∪ Bc) = (Ac ∩ Bc) ∪ (A ∩ B) . Definition 9. Symmetrical difference formula for two fuzzy sets A and B is given by (Ac ∩ B) ∪ (A ∩ Bc) = (Ac ∪ Bc) ∩ (A ∪ B) . 3. Exponential Fuzzy Sets Definition 10. If Z is a universe discourse and s be any particular element of Z. The EFS EA defined on Z is a collection of ordered pairs, EA = {( s,ℵA(s)e −⊺ℵA(s) ) |s ∈ Z, ⊺ > 0 } , where ℵA(s)e −⊺ℵA(s) : Z → [0, 1] is called the membership function. The degree of mem- bership function 0 ≤ ℵA(s)e −⊺ℵA(s) ≤ 1. Example 1. Let Z = {1, 2, 3, 4, 5} be the universal set and fuzzy membership values of Z is ℵA(s) = {0.9, 0.7, 0.5, 0.4, 0.3}, the decay parameter ⊺ = 0.02. The exponential fuzzy membership function is given by: EA(s) = ℵA(s)e −⊺ℵA(s). The exponential fuzzy membership values EA(s) = {(1, 0.8839), (2, 0.6903), (3, 0.4950), (4, 0.3968), (5, 0.2982)}. M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 5 of 29 Figure 1: Exponential Fuzzy Set Definition 11. Let EA and EB be two EFSs on Z with their grade values given by: ℵEA(s) = ℵA(s)e −⊺ℵA(s) ℵEB(s) = ℵB(s)e −⊺ℵB(s) The exponential fuzzy intersection of EA and EB is defined as: ℵEA∩EB(s) = min {ℵEA(s),ℵEB(s)} = min { ℵA(s)e −⊺ℵA(s),ℵB(s)e −⊺ℵB(s) } (3.1) Similarly, the exponential fuzzy union of EA and EB is defined as: ℵEA∪EB(s) = max {ℵEA(s),ℵEB(s)} = max { ℵA(s)e −⊺ℵA(s),ℵB(s)e −⊺ℵB(s) } (3.2) Example 2. Let Z = {s1, s2, s3, s4, s5} be a finite universe, and the membership functions of two exponential F-sets EA and EB are ℵEA(s) = ℵA(s)e −⊺ℵA(s) M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 6 of 29 ℵEB(s) = ℵB(s)e −⊺ℵB(s) Now, we compute the corresponding EA ∩ EB and EA ∪ EB membership values: s ℵA(s) EA(s) ℵB(s) EB(s) EA ∩ EB EA ∪ EB s1 0.8 0.7873 0.7 0.6903 0.6903 0.7873 s2 0.6 0.5928 0.5 0.4950 0.4950 0.5928 s3 0.4 0.3968 0.3 0.2982 0.2982 0.3968 s4 0.2 0.1992 0.2 0.1992 0.1992 0.1992 s5 0.1 0.0998 0.1 0.0998 0.0998 0.0998 Definition 12. Consider two EFSs EA, EB, and ℵA(s)e −⊺ℵA(s),ℵB(s)e −⊺ℵB(s) denotes the membership functions EA and EB. The simple difference EA −EB of these two EFSs EA and EB is given by EA − EB = EA ∩ EBc = ℵEA(s) ∗ ℵ c EB(s) = ℵA(s)e −⊺ℵA(s) ∗ ℵB(s) ce−⊺ℵc B(s) Example 3. Let EA = { 0.8e−⊺0.8 s1 + 0.5e−⊺0.5 s2 + 0.6e−⊺0.6 s3 } and EB = { 0.4e−⊺0.4 s1 + 0.3e−⊺0.3 s2 + 0.3e−⊺0.3 s3 } be two two EFSs. The simple difference is EA − EB = EA ∩ EBc = { 0.8e−⊺0.8 s1 + 0.5e−⊺0.5 s2 + 0.6e−⊺0.6 s3 } ∗ { 0.6e−⊺0.6 s1 + 0.7e−⊺0.7 s2 + 0.7e−⊺0.7 s3 } = { 0.6e−⊺0.6 s1 + 0.5e−⊺0.5 s2 + 0.6e−⊺0.6 s3 } . Definition 13. Let ℵA(s)e −⊺ℵA(s) and ℵB(s)e −⊺ℵB(s) be the membership functions of EFS EA and EB. The bounded difference is EA ◦ EB(s) = max [ ◦,ℵA(s)e −⊺ℵA(s),ℵB(s)e −⊺ℵB(s) ] . Example 4. Let EA = ( 0.9e−⊺0.9 s1 + 0.8e−⊺0.8 s2 + 0.6e−⊺0.6 s3 ) and EB = ( 0.1e−⊺0.1 s1 + 0.4e−⊺0.4 s2 + 0.5e−⊺0.5 s3 ) be two two EFSs. The bounded difference of these two EFSs is: EA ◦ EB(s) = ( 0.8e−⊺0.8 s1 + 0.4e−⊺0.4 s2 + 0.1e−⊺0.1 s3 ) Definition 14. A disjoint sum of EA and EB is as follows EA ⊗ EB(s) = ∣∣∣∣∣ℵA(s)e −⊺ℵA(s) − ℵB(s)e −⊺ℵB(s) ∣∣∣∣∣ M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 7 of 29 Example 5. Let EA = ( 0.2e−⊺0.2 s1 + 0.3e−⊺0.3 s2 + 0.5e−⊺0.5 s3 ) and EB = ( 0.15e−⊺0.15 s1 + 0.35e−⊺0.35 s2 + 0.65e−⊺0.65 s3 ) be two two EFSs. Using the max function for calculating the phase term, the disjoint sum of theses two two EFSs is: EA ⊗ EB(s) = ( 0.05e−⊺0.05 s1 + 0.05e−⊺0.05 s2 + 0.05e−⊺0.05 s3 ) Definition 15. Let ℵA(s)e −⊺ℵA(s) and ℵB(s)e −⊺ℵB(s) denotes the membership functions EA and EB. The disjunctive sum is EA∆EB(s) = (EA ∩ EBc) ∪ (EAc ∩ EB) . The membership function of EA∆EB(s) is: ℵEA∆EB(s) = [ℵEA∩EBc(s)⊕ ℵEAc∩EB(s)] = [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] . Example 6. Suppose EA = ( 0.6e−⊺0.6 s1 + 0.7e−⊺0.7 s2 + 0.5e−⊺0.5 s3 ) and EB = ( 0.3e−⊺0.3 s1 + 0.4e−⊺0.4 s2 + 0.7e−⊺0.7 s3 ) . Then the disjunctive sum of these two EFSs is define as ℵEA∆EB(s) = [ℵEA∩EBc(s)⊕ ℵEAc∩EB(s)] = [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] . ℵEA∆EB(s) = ( 0.6e−⊺0.6 s1 + 0.6e−⊺0.6 s2 + 0.3e−⊺0.3 s3 ) ⊕ ( 0.3e−⊺0.3 s1 + 0.3e−⊺0.3 s2 + 0.5e−⊺0.5 s3 ) . ℵEA∆EB(s) = ( 0.6e−⊺0.6 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ) . Definition 16. For any two EFSs EA and EB, the equivalence formula is; (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB). The membership function of two EFSs EA and EB are given below [ℵEAc∪EB(s) ∩ ℵEA∪EBc(s)] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] [ℵEAc∩EBc(s) ∪ ℵEA∩EB(s)] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] Example 7. Suppose EA = ( 0.6e−⊺0.6 s1 + 0.8e−⊺0.8 s2 + 0.7e−⊺0.7 s3 ) and EB = ( 0.8e−⊺0.8 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ) . The equivalence formula is; (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB). (EAc ∪ EB) ∩ (EA ∪ EBc) = ( 0.8e−⊺0.8 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ) ∗ ( 0.6e−⊺0.6 s1 + 0.8e−⊺0.8 s2 + 0.7e−⊺0.7 s3 ) = ( 0.6e−⊺0.6 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ) (3.3) M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 8 of 29 (EAc ∩ EBc) ∪ (EA ∩ EB) = ( 0.2e−⊺0.2 s1 + 0.2e−⊺0.2 s2 + 0.3e−⊺0.3 s3 ) ⊕ ( 0.6e−⊺0.6 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ) = ( 0.6e−⊺0.6 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ) (3.4) From equation 3.3 and 3.4, we have (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Definition 17. The symmetrical difference formula for two EFSs EA and EB are denoted by (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∩ EB) . The symmetrical difference formula two EFSs EA and EB are given below [ℵEAc∩EB(s) ∪ ℵEA∩EBc(s)] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] [ℵEAc∪EBc(s) ∩ ℵEA∪EB(s)] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] Example 8. Suppose EA = ( 0.5e−⊺0.5 s1 + 0.7e−⊺0.7 s2 + 0.8e−⊺0.8 s3 ) and EB = ( 0.3e−⊺0.3 s1 + 0.5e−⊺0.5 s2 + 0.2e−⊺0.2 s3 ) . The symmetrical difference formula is (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∩ EB) . (EAc ∩ EB) ∪ (EA ∩ EBc) = ( 0.3e−⊺0.3 s1 + 0.5e−⊺0.5 s2 + 0.2e−⊺0.2 s3 ) ⊕ ( 0.5e−⊺0.5 s1 + 0.5e−⊺0.5 s2 + 0.8e−⊺0.8 s3 ) = ( 0.5e−⊺0.5 s1 + 0.5e−⊺0.5 s2 + 0.8e−⊺0.8 s3 ) (3.5) (EAc ∪ EBc) ∩ (EA ∩ EB) = ( 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.8e−⊺0.8 s3 ) ⊕ ( 0.5e−⊺0.5 s1 + 0.7e−⊺0.7 s2 + 0.8e−⊺0.8 s3 ) = ( 0.5e−⊺0.5 s1 + 0.5e−⊺0.5 s2 + 0.8e−⊺0.8 s3 ) (3.6) From equation 3.5 and 3.6, we have (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∩ EB) . Definition 18. Let EA and EB and EFSs be three EFSs then, the distributive law are EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) EA ∩ (EB ∪ EC) = (EA ∩ EB) ∪ (EA ∩ EC) . Theses two law are said to be distributive law of union over intersection and intersec- tion over union. If EA = ℵEA(s) = ℵA(s)e −⊺ℵA(s), EB = ℵEB(s) = ℵB(s)e −⊺ℵB(s) and EC = ℵEC(s) = ℵC(s)e −⊺ℵC(s), the distributive law of union intersection become: [ℵEA(s)⊕ (ℵEB(s) ∗ ℵEC(s))] = [ ℵA(s)e −⊺ℵA(s) ⊕ [ ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) ]] [ℵEA(s)⊕ ℵEB(s)]∗[ℵEA(s)⊕ ℵEC(s)] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] . M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 9 of 29 Example 9. EA = ( 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ) , EB = ( 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ) and EC = ( 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 1e−⊺1 s3 ) be the three exponential fuzzysets. EA ∪ (EB ∩ EC) = [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ⊕ [[ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ] ∗ [ 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 1e−⊺1 s3 ]] = [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ⊕ [ 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 0.4e−⊺0.4 s3 ] = [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] (3.7) (EA ∪ EB) ∩ (EA ∪ EC) = [[ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ⊕ [ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ]] ∗ [[ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ⊕ [ 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 1e−⊺1 s3 ]] = [[ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ∗ [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 1e−⊺1 s3 ]] = [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] (3.8) From equation 3.7 and 3.8 we have EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC). Next, we see the distributive condition of intersection over union is EA ∩ (EB ∪ EC) = [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ∗ [[ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ] ⊕ [ 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 1e−⊺1 s3 ]] = [ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ∗ [ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 1e−⊺1 s3 ] = [ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.5e−⊺0.5 s3 ] (3.9) (EA ∩ EB) ∪ (EA ∩ EC) = [[ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ∗ [ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ]] ⊕ [[ 0.9e−⊺0.9 s1 + 0.6e−⊺0.6 s2 + 0.5e−⊺0.5 s3 ] ∗ [ 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 1e−⊺1 s3 ]] = [[ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.4e−⊺0.4 s3 ] ⊕ [ 0.6e−⊺0.6 s1 + 0.3e−⊺0.3 s2 + 0.5e−⊺0.5 s3 ]] M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 10 of 29 = [ 0.7e−⊺0.7 s1 + 0.5e−⊺0.5 s2 + 0.5e−⊺0.5 s3 ] (3.10) From equation 3.9 and 3.10 we have EA ∩ (EB ∪ EC) = (EA ∩ EB) ∪ (EA ∩ EC). Definition 19. The union of idempotent law in a EFSs EA is EA ∪ EA = EA and the idempotent law of intersection is EA ∩ EA = EA. If a membership value of EA is ℵEA(s) = ℵA(s)e −⊺ℵA(s) the idempotent law of union becomes ℵEA(s) = ℵEA∪EA(s). This prove this, we have ℵEA∪EA(s) = [ℵEA(s)⊕ ℵEA(s)] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = ℵA(s)e −⊺ℵA(s) = ℵEA(s). Similarly ℵEA∪EA(s) = ℵEA(s). Example 10. Let EA = ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) , be a EFS, the idempotent law of union is, ℵEA∪EA(s) = ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) ⊕ ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) = ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) = ℵEA(s). The idempotent law of intersection is , ℵEA∩EA(s) = ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) ∗ ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) = ( 0.8e−⊺08 s1 + 0.9e−⊺0.9 s2 + 0.7e−⊺0.7 s3 ) = ℵEA(s). The union and intersection law of idempotent laws are hold. Definition 20. EFS satisfied the involution law using standard complement function. The involution law for a EFS EA is (EAc)c = EA. If a membership value of EA is ℵEA(s) = ℵA(s)e −⊺ℵA(s) the involution law is ℵA(s)e −⊺ℵA(s) = (ℵc A(s)) ce−⊺(ℵc A(s))c . M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 11 of 29 Example 11. Let EA = ( 0.7e−⊺0.7 s1 + 0.6e−⊺0.6 s2 + 0.4e−⊺0.4 s3 ) , be a EFS. By using standard complement function, the involution law valid. EAc = 0.3e−⊺0.3 s1 + 0.7e−⊺0.7 s2 + 0.6e−⊺0.6 s3 EAcc = 0.7e−⊺0.7 s1 + 0.6e−⊺0.6 s2 + 0.4e−⊺0.4 s3 = EA. 4. Main Results of Exponential Fuzzy Sets Theorem 1. Let EA and EB be EFSs over the classical set Z, the symmetrical difference condition is satisfied for union, intersection and their complement functions of phase term. Proof. EA and EB be two EFSs. To demonstrate the formula for symmetrical differ- ence (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) , To determine the phase term, the max function. Case 1 . ℵEA(s) ≤ ℵEB(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ] (4.1) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ] (4.2) From equation 4.1 and 4.2 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Case 2 . ℵEA(s) ≤ ℵEB(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵB(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.3) M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 12 of 29 (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.4) From equation 4.3 and 4.4 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Case 3 . ℵEA(s) ≤ ℵEB(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ⊕ ℵc A(s)e −⊺ℵc A(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.5) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.6) From equation 4.5 and 4.6 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Case 4 . ℵEA(s) ≤ ℵEB(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.7) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.8) From equation 4.7 and 4.8 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 13 of 29 Case 5 . ℵEB(s) ≤ ℵEA(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.9) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ∗ ℵA(s)e −⊺ℵA(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.10) From equation 4.9 and 4.10 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Case 6 . ℵEB(s) ≤ ℵEA(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵB(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.11) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ∗ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.12) From equation 4.11 and 4.12 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Case 7 . ℵEB(s) ≤ ℵEA(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc B(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.13) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ∗ ℵA(s)e −⊺ℵA(s) ] M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 14 of 29 = [ ℵc B(s)e −⊺ℵc B(s) ] (4.14) From equation 4.13 and 4.14 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Case 8 . ℵEB(s) ≤ ℵEA(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∩ EB) ∪ (EA ∩ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵB(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.15) (EAc ∪ EBc) ∩ (EA ∪ EB) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ∗ ℵA(s)e −⊺ℵA(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.16) From equation 4.15 and 4.16 (EAc ∩ EB) ∪ (EA ∩ EBc) = (EAc ∪ EBc) ∩ (EA ∪ EB) . Therefore, the formula for symmetrical difference is valid for all cases. Theorem 2. The union, intersection and complement function of exponential EFSs EA and EB is an equivalence relation. Proof. EA and EB be two EFSs. To demonstrate the equivalence relation. (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 1 . ℵEA(s) ≤ ℵEB(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵB(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.17) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.18) From equation 4.17 and 4.18 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 15 of 29 Case 2 . ℵEA(s) ≤ ℵEB(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.19) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.20) From equation 4.19 and 4.20 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 3 . ℵEA(s) ≤ ℵEB(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵR(s) ∗ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.21) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ⊕ ℵA(s)e −⊺ℵA(s) ] = [ ℵA(s)e −⊺ℵA(s) ] (4.22) From equation 4.21 and 4.22 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 4 . ℵEA(s) ≤ ℵEB(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EB(s) ≤ ℵc EA(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ] (4.23) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc B(s)e −⊺ℵc B(s) ⊕ ℵA(s)e −⊺ℵA(s) ] M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 16 of 29 = [ ℵc B(s)e −⊺ℵc B(s) ] (4.24) From equation 4.23 and 4.24 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 5 . ℵEB(s) ≤ ℵEA(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵB(s) ∗ ℵA(s)e −⊺ℵA(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.25) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.26) From equation 4.25 and 4.26 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 6 . ℵEB(s) ≤ ℵEA(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = ℵc A(s)e −⊺ℵc A(s) (4.27) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ] (4.28) From equation 4.27 and 4.28 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 7 . ℵEB(s) ≤ ℵEA(s),ℵc EA(s) ≤ ℵEB(s),ℵc EB(s) ≤ ℵEA(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵB(s)e −⊺ℵB(s) ∗ ℵA(s)e −⊺ℵA(s) ] M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 17 of 29 = ℵB(s)e −⊺ℵB(s) (4.29) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵB(s)e −⊺ℵB(s) ] (4.30) From equation 4.29 and 4.30 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . Case 8 . ℵEB(s) ≤ ℵEB(s),ℵEB(s) ≤ ℵc EA(s),ℵEA(s) ≤ ℵc EB(s) and ℵc EA(s) ≤ ℵc EB(s). (EAc ∪ EB) ∩ (EA ∪ EBc) = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc B(s)e −⊺ℵc B(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] = ℵc A(s)e −⊺ℵc A(s) (4.31) (EAc ∩ EBc) ∪ (EA ∩ EB) = [ ℵc A(s)e −⊺ℵc A(s) ∗ ℵc B(s)e −⊺ℵc B(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ⊕ ℵB(s)e −⊺ℵB(s) ] = [ ℵc A(s)e −⊺ℵc A(s) ] (4.32) From equation 4.31 and 4.32 (EAc ∪ EB) ∩ (EA ∪ EBc) = (EAc ∩ EBc) ∪ (EA ∩ EB) . From the all cases EA and EB is equivalence relation. Theorem 3. Any finite collection of EFSs is always an EFSs for union and intersec- tion. Proof. case i . Let EA1, EA2, EA3, .....EAm be EFSs and its membership functions is ℵEA1 ,ℵEA2 ,ℵEA3 , ......,ℵEAm . ℵ′ EA(s) = max [ ℵEA1 ,ℵEA2 ,ℵEA3 , ......,ℵEAm ] . Now, EA1 ∪ EA2 ∪ ...... ∪ EAm = [ ℵA1(s)e −⊺ℵA1 (s) ⊕ ℵA2(s)e −⊺ℵA2 (s) ⊕ .....⊕ ℵAm(s)e −⊺ℵAm (s) ] = ℵ′ A1 (s)e −⊺ℵ′ A1 (s) = E ′ A1 . M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 18 of 29 case ii . EA1, EA2, EA3, .....EAm be anym EFSs and ℵA1(s)e −⊺ℵA1 (s),ℵA2(s)e −⊺ℵA2 (s), .....ℵAm(s)e −⊺ℵAm (s) denotes the membership functions of these EFSs. ℵ′ EA(s) = min [ ℵEA1 ,ℵEA2 ,ℵEA3 , ......,ℵEAm ] . Now, EA1 ∩ EA2 ∩ ...... ∩ EAm = [ ℵA1(s)e −⊺ℵA1 (s) ∗ ℵA2(s)e −⊺ℵA2 (s) ∗ ..... ∗ ℵAm(s)e −⊺ℵAm (s) ] = ℵ′ A1 (s)e −⊺ℵ′ A1 (s) = E ′ A1 . Which is also a EFS. Theorem 4. For any two EFS EA and EB, the union and intersection function with the same function for determining the phase term satisfy: M∑ j=1,sj∈Z |ℵEA∩EB(si)| ≤ M∑ j=1,sj∈Z |ℵEA∪EB(si)| . Proof. The expression function of union and intersection are define by ℵEA∪EB(s) = max [ℵEA(s),ℵEB(s)] and ℵEA∩EB(s) = min [ℵEA(s),ℵEB(s)] . As |ℵEA∩EB(u)| ≤ |ℵEA∪EB(u)| |ℵEA∩EB(v)| ≤ |ℵEA∪EB(v)| ... ... ... |ℵEA∩EB(sm)| ≤ |ℵEA∪EB(sm)| . Sum of all above inequalities we obtained M∑ i=1,si∈Z |ℵEA∩EB(si)| ≤ M∑ i=1,si∈Z |ℵEA∪EB(si)| . Theorem 5. For any EFSs EA, EB and EC, the intersection union functions with the same function for determining the phase term stratify the distributive law. Proof. First, we prove the distributive law for any EFSs EA, EB and EC , six cases arise here. We prove distributive law of union over intersection. M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 19 of 29 Case 1 ℵEA(s) ≤ ℵEB(s) ≤ ℵEC(s) EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] EA ∪ (EB ∩ EC) = ℵB(s)e −⊺ℵB(s). (4.33) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = [ ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) ] (EA ∪ EB) ∩ (EA ∪ EC) = ℵB(s)e −⊺ℵB(s) (4.34) From equation (4.33) and (4.34), we have EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) Case 2 ℵEB(s) ≤ ℵEC(s) ≤ ℵEA(s) EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] EA ∪ (EB ∩ EC) = ℵA(s)e −⊺ℵA(s). (4.35) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] (EA ∪ EB) ∩ (EA ∪ EC) = ℵA(s)e −⊺ℵA(s) (4.36) From equation (4.35) and (4.36) EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) Case 3 ℵEA(s) ≤ ℵEC(s) ≤ ℵEB(s) EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] EA ∪ (EB ∩ EC) = ℵC(s)e −⊺ℵC(s). (4.37) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = [ ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) ] (EA ∪ EB) ∩ (EA ∪ EC) = ℵC(s)e −⊺ℵC(s) (4.38) From equation (4.37) and (4.38) EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) . M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 20 of 29 Case 4 ℵEC(s) ≤ ℵEB(s) ≤ ℵEA(s) EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] EA ∪ (EB ∩ EC) = ℵA(s)e −⊺ℵA(s). (4.39) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = [ ℵA(s)e −⊺ℵA(s) ∗ ℵA(s)e −⊺ℵA(s) ] (EA ∪ EB) ∩ (EA ∪ EC) = ℵA(s)e −⊺ℵA(s) (4.40) From equation (4.39) and (4.40), EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) Case 5 ℵEB(s) ≤ ℵEA(s) ≤ ℵEC(s) EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] EA ∪ (EB ∩ EC) = ℵA(s)e −⊺ℵA(s). (4.41) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = [ ℵA(s)e −⊺ℵA(s) ∗ ℵC(s)e −⊺ℵC(s) ] (EA ∪ EB) ∩ (EA ∪ EC) = ℵA(s)e −⊺ℵA(s) (4.42) From equation (4.41) and (4.42) EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) Case 6 ℵEC(s) ≤ ℵEA(s) ≤ ℵEB(s) EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] EA ∪ (EB ∩ EC) = ℵA(s)e −⊺ℵA(s). (4.43) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = [ ℵB(s)e −⊺ℵB(s) ∗ ℵR(s)e −⊺ℵR(s) ] (EA ∪ EB) ∩ (EA ∪ EC) = ℵA(s)e −⊺ℵA(s) (4.44) M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 21 of 29 From equation (4.43) and (4.44), we have EA ∪ (EB ∩ EC) = (EA ∪ EB) ∩ (EA ∪ EC) The Law is valid for all above cases. Similar way the distributive law of intersection over union is prove. EA ∩ (EB ∪ EC) = (EA ∩ EB) ∪ (EA ∩ EC) Theorem 6. For any EFS EA, the union, intersection, complement function with the same function for determining the phase term satisfy the following: i . EA ∪ EAc = EA or EA ∪ EAc = EAc . ii . EA ∩ EAc = EA or EA ∩ EAc = EAc. Proof. To prove (i) and (ii), two cases arise here. i . EA ∪ EAc = EA or EA ∪ EAc = EAc . Case 1 . ℵc A(s) ≤ ℵA(s) EA ∪ EAc = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc A(s)e −⊺ℵc A(s) ] = ℵA(s)e −⊺ℵA(s) = EA. Case 2 . ℵA(s) ≤ ℵc A(s) EA ∪ EAc = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵc A(s)e −⊺ℵc A(s) ] = ℵc A(s)e −⊺ℵc A(s) = EAc. ii .EA ∩ EAc = EA or EA ∩ EAc = EAc. Case 1 . ℵc A(s) ≤ ℵA(s) EA ∩ EAc = [ ℵA(s)e −⊺ℵA(s) ∗ ℵc A(s)e −⊺ℵc A(s) ] = ℵc A(s)e −⊺ℵc A(s) = EAc. M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 22 of 29 Case 2 . ℵA(s) ≤ ℵc A(s) EA ∩ EAc = [ ℵA(s)e −⊺ℵA(s) ∗ ℵc A(s)e −⊺ℵc A(s) ] = ℵA(s)e −⊺ℵA(s) = EA. Theorem 7. For any EFSs EA and EB over a crisp set Z, union and intersection function with the max function for determining the phase term does not satisfy the absorption law. Proof. The absorption laws for crisp set are EFSs EA and EB, the absorption laws do not hold. If ℵA(s) ≤ ℵB(s). EA ∩ (EA ∪ EB) = [ ℵA(s)e −⊺ℵA(s) ∗ ( ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) )] = ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) = ℵA(s)e −⊺ℵB(s) ̸= EA. EA ∪ (EA ∩ EB) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) )] = ℵA(s)e −⊺ℵA(s) ⊕ ℵA(s)e −⊺ℵA(s) = ℵA(s)e −⊺ℵB(s) ̸= EA. Also if ℵB(s) ≤ ℵA(s). EA ∪ (EA ∩ EB) = [ ℵA(s)e −⊺ℵA(s) ∗ ( ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) )] = ℵA(s)e −⊺ℵA(s) ∗ ℵA(s)e −⊺ℵA(s) = ℵA(s)e −⊺ℵB(s) ̸= EA. EA ∪ (EA ∩ EB) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) )] = ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) = ℵA(s)e −⊺ℵB(s) ̸= EA. Hence, the absorption law does not hold for any EFSs. Theorem 8. For any EFSs EA, EB and EC , the complement, intersection, union func- tion for determining the phase term does not satisfy the distributive laws. Proof. The distributive law of union over intersection is EA∪(EB ∩ EC) = (EA ∪ EB)∩ (EA ∪ EC) . If ℵEA(s) ≤ ℵEB(s) ≤ ℵEC(s). EA ∪ (EB ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ( ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) )] M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 23 of 29 = ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵC(s) = ℵB(s)e −⊺ℵC(s) (4.45) (EA ∪ EB) ∩ (EA ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ⊕ ℵB(s)e −⊺ℵB(s) ] ∗ [ ℵA(s)e −⊺ℵA(s) ⊕ ℵC(s)e −⊺ℵC(s) ] = ℵB(s)e −⊺ℵB(s) ∗ ℵC(s)e −⊺ℵC(s) = ℵB(s)e −⊺ℵA(s). (4.46) From equation 4.45 and 4.46, we have EA ∪ (EB ∩ EC) ̸= (EA ∪ EB) ∩ (EA ∪ EC) Now distributive law of intersection over union EA ∩ (EB ∪ EC) = [ ℵA(s)e −⊺ℵA(s) ∗ ( ℵB(s)e −⊺ℵB(s) ⊕ ℵC(s)e −⊺ℵC(s) )] = ℵA(s)e −⊺ℵA(s) ∗ ℵC(s)e −⊺ℵC(s) = ℵA(s)e −⊺ℵC(s) (4.47) (EA ∩ EB) ∪ (EA ∩ EC) = [ ℵA(s)e −⊺ℵA(s) ∗ ℵB(s)e −⊺ℵB(s) ] ⊕ [ ℵA(s)e −⊺ℵA(s) ∗ ℵC(s)e −⊺ℵC(s) ] = ℵA(s)e −⊺ℵA(s) ⊕ ℵA(s)e −⊺ℵA(s) = ℵA(s)e −⊺ℵA(s) (4.48) From equation 4.47 and 4.48, we have EA ∩ (EB ∪ EC) ̸= (EA ∩ EB) ∪ (EA ∩ EC) . Hence, the absorption law does not hold for any EFSs. 5. Application: AI-Powered Investment Decision-Making Using the Weighted Mean Method Investments are essential for people because they help grow wealth, provide financial security, and ensure a stable future. By investing, individuals can increase their money over time through interest, dividends, or asset appreciation, rather than relying solely on savings. Investments also protect against inflation, which reduces the value of money, ensuring that purchasing power remains strong. They also act as a source of financial relief in cases of emergencies and enable individuals to fulfill long-term objectives like purchasing a house, covering education expenses, or saving for retirement. Investments are also capable of creating passive income from dividends, rental properties, or bonds, giving financial security without hard labor. Investment diversification among various assets minimizes risk even more, provid- ing financial security and a balanced future. Finally, achieving long-term wealth and financial freedom requires investment. Nowadays, investing money poses a variety of M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 24 of 29 difficulties for individuals, making it difficult to increase wealth and safeguard one?s fi- nancial future. A significant obstacle is a lack of financial literacy, since most individuals do not know where or how to start or how investments work. Limited capital is another problem; some individuals believe that only rich people can invest since they lack suf- ficient funds. Making the right investment decision can be complex due to uncertainty, market fluctuations, and multiple investment options. Our exponential fuzzy investment decision system is designed to assist investors in choosing the most suitable investment scheme based on a exponential fuzzy set approach. This application evaluates investment opportunities by considering multiple factors, such as risk level, expected return, investment horizon, and financial goals. Unlike fuzzy logic decision-making models, exponential fuzzy logic enables a more flexible and human-like reasoning process, allowing for better handling of imprecise and uncertain data this application helps you make data-driven investment decisions with confidence. In this section, we use the weighted mean method for AI-powered investment decision- making, where financial experts (with different importance weights) assess stocks, and their opinions are aggregated using EFSs. Let S = {s1, s2, s3, s4, s5, s6, s7, s8, s9, s10} be the set of Investments where s1 (Bonds), s2 (Public provident fund), s3 (Stocks), s4 (Real estate), s5 (Treasurys), s6 (Cryp- tocurrencies), s7 (Mutual funds), s8 (Fixed deposits), s9 (Gold), s10 (National Pension Scheme) . The decision-makers are: • E1: Junior Analyst (Weight = w1 = 0.2) • E2: Senior Analyst (Weight = w2 = 0.3) • E3: AI Model (Weight = w3 = 0.5) where ∑3 j=1wj = 1 and ⊺ = 2. Experts assign fuzzy membership values ℵAj (s) to each stock: Investments ℵA1(s) ℵA2(s) ℵA3(s) s1 0.3 0.4 0.5 s2 0.6 0.7 0.8 s3 0.8 0.9 1.0 s4 0.5 0.6 0.7 s5 0.9 1.0 0.9 s6 0.7 0.4 0.8 s7 0.5 1.0 0.3 s8 1.0 0.3 0.2 s9 0.1 0.5 0.9 s10 0.7 1.0 0.4 Table 1: Fuzzy Membership Values for Each Stock The exponential fuzzy membership formula is ℵEAj (s) = ℵAj (s)e −2ℵAj (s) (5.1) M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 25 of 29 Investment ℵEA1 (s) ℵEA2 (s) ℵEA3 (s) s1 0.1646 0.1797 0.1839 s2 0.1807 0.1726 0.1615 s3 0.1615 0.1487 0.1353 s4 0.1839 0.1807 0.1726 s5 0.1487 0.1353 0.1487 s6 0.1726 0.1797 0.1615 s7 0.1839 0.1353 0.1646 s8 0.1353 0.1646 0.1340 s9 0.0818 0.1839 0.1487 s10 0.1726 0.1353 0.1737 Table 2: exponential fuzzy membership values for each stock Weighted Mean Aggregation Using the formula: ℵEG(s) = 3∑ j=1 wjℵEAj (s) (5.2) We compute the aggregated fuzzy values: Investment ℵE(s) s1 0.1787 s2 0.1686 s3 0.1445 s4 0.1772 s5 0.1446 s6 0.1691 s7 0.1596 s8 0.1434 s9 0.1458 s10 0.1619 Table 3: Aggregated Membership Values of EFS The stock with the highest aggregated value in Table 3: max s∈Z ℵE(s) = ℵE(s1) = 0.1784 (5.3) Thus, the best investment decision is Bonds. 5.1. Sensitivity Analysis Sensitivity Analysis Sensitivity analysis assists in assessing the effect of changes in input parameters on the investment decision. By varying expert-specified weights or M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 26 of 29 membership values, we can find the stability of the exponential fuzzy investment decision system. Sensitivity analysis investigates the effect of membership value changes on the investment ranking. 5.1.1. Initial Aggregation Analysis With the fuzzy membership values for each stock from Table 2, the fuzzy aggregated value for investment decision is: max s∈Z ℵE(s) = ℵE(s1) = 0.1784 Therefore, the optimal investment choice is Bonds (s1). 5.1.2. Scenario 1: Membership Value Changes Scenario 1: Membership Value Changes To measure stability, we adjust the mem- bership values by raising and lowering each entry by 5% and recompute the aggre- gated values. Case 1.1: Raise Membership Values by 5% If membership values are raised by 5%, new fuzzy values are recomputed, and the new maximum aggregated value is: max s∈Z ℵE(s) = ℵE(s1 ′ ) = 0.1832 The best investment choice does not change Bonds (s1). Case 1.2:Lower Membership Values by 5% If all membership values are reduced by 5%, the new maximum aggregated value is: max s∈Z ℵE(s) = ℵE(s1 ′′ ) = 0.1741 Once again, Bonds (s1) is the best choice. Scenario 2: Varying Expert Weights Final ranking is determined by weights allo- cated to investment criteria. Let’s assume two varied weight scenarios Case 2.1: Same Weights for All Factors Allocating same weights to all three fuzzy membership values: w1 = w2 = w3 = 0.33 Recalculating aggregated values, we observe s1 continues to have the maximum value, verifying the consistency of the decision. Case 2.2:Increased Weight on Risk Factor If the expert gives greater weight (0.5) to the risk factor but leaves the others unchanged at 0.25, the ranking is slightly different. But Bonds ( s1) is still among the best investment options. M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 27 of 29 Figure 2: Investment: Stocks 5.2. Comparison Analysis of Exponential Fuzzy Sets EFS with Tradi- tional Fuzzy Models One of the key advantages of EFS over traditional fuzzy models is their ability to handle uncertainty more effectively. Traditional fuzzy models assign a direct membership degree to an investment option, which may not fully capture the gradual decline in confidence as uncertainty increases. In contrast, EFS introduces an exponential decay function, which provides a more refined approach to uncertainty modeling. This is particularly beneficial in investment decision-making, where risk levels vary significantly, and a more nuanced representation of uncertainty leads to better investment choices. Flexibility in decision-making is another critical factor where EFS outperforms tra- ditional fuzzy models. Traditional fuzzy logic uses a linear membership assignment, making it less adaptable in differentiating between investments with similar member- ship values. EFS, however, applies an exponential transformation, ensuring a smoother transition between choices. This allows for a more sensitive response to small variations in investment attributes, improving the accuracy of financial assessments. Numerical stability and sensitivity analysis further highlight the advantages of EFS. Traditional fuzzy models often experience abrupt shifts in decision outcomes when input data changes. In contrast, EFS incorporates a sensitivity mechanism that stabilizes rankings even with small fluctuations in the membership values. As demonstrated in the sensitivity analysis, adjusting membership values by 5% does not significantly alter the ranking of the best investment choice in the EFS model, confirming its robustness in real-world applications. A graphical comparison further strengthens the argument for EFS. In a traditional fuzzy model, membership values are assigned directly, leading to a more rigid classifi- cation of investment options. The EFS model, however, applies an exponential trans- M. Kaviyarasu, M. Rajeshwari, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (2) (2025), 6050 28 of 29 formation that results in a sharper differentiation of choices. This means that EFS can more effectively distinguish between investments with slight variations in risk and return, leading to more precise decision-making. Finally, in practical applications, EFS proves to be a more effective tool for invest- ment decision-making. Traditional fuzzy models rely on fixed weight allocations and may struggle in highly uncertain investment environments. EFS, by incorporating an exponential factor, ensures a more dynamic and realistic risk assessment. This feature makes it particularly useful for financial forecasting, portfolio optimization, and strate- gic investment planning. By improving uncertainty management and decision flexibility, EFS provides investors with a more reliable approach to selecting optimal investment schemes. 6. Conclusion The EFS offers a strong tool for managing uncertainty in mathematical modeling and decision-making. With the use of an exponential membership function, EFS success- fully describes systems with high uncertainty rates of change. This paper has discussed the basic properties and operations of EFS, as well as its uses in AI-based investment decisions. Our exponential fuzzy investment decision system showcases the strengths of EFS in making optimal investment decisions. Compared to conventional fuzzy models, EFS supports more accurate and dynamic decision-making, especially in unstable fi- nancial markets. The integration of the weighted mean method within the EFS system maximizes investment analysis by combining expert opinions efficiently. 6.1. Future Research Directions The EFS concept we can extend in Intuitionistic fuzzy set, Neutrosophic fuzzy set all areas like graph theory, BCI Algebra and different type of algebras. Then decision making problems. References [1] L. A. 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