Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2113 https://internationalpubls.com Aggregate Operators of Neutrosophic Vague Hypersoft Set 1Sharviya Sona S, 2Elvina Mary L 1Research Scholar, PG & Research Department of Mathematics, 2Assistant Professor, PG & Research Department of Mathematics 1, 2 Nirmala College for Women, Coimbatore, India. Email Id: sharviyasona2611@gmail.com1 , Email Id: elvinalawrence 07@gmail.com2 Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: A new set NVHSS is defined. This essay covers fundamental operators such as union, intersection, complement, subset, empty set, and same set etc., of NVHSS. Appropriate examples are provided along with the implementation and validity. Proposed operations will be crucial in future decision-making for more accuracy and precision in areas such as management issues, personal selection, and many more. Keywords: SS, Neutrosophic SS, Hyper soft set, NV , Neutrosophic Vague soft set, VHSS, Neutrosophic Vague Hypersoft set. 1. Introduction Atanassov's theory only addresses incomplete data that takes into account both membership and non- membership values; intuitionistic fuzzy set theory is unable to cope with inconsistent and imprecise data. The NS was created by Smarandache to address such inconsistent and imprecise data [3]. Molodtsov was the first person to describe the concept of SS [2] as a a brand-new numerical tool for handling problems with unclear circumstances. Maji [5] offered the concept of an NSS with the required functions. The concept of the potential NSS was created by Karaaslan. Saqlain. Samlai et al. By merging the plithogenic sets and hypersoft sets, Martin and Smarandache created the plithogenic hypersoft set in [4]. Saqlain et al. Neutrosophic vague set theory was studied by Alkhazaleh (2015). Das et al[1]. Muhammad saqlain ,Sana Moni Muhammad Naveed and Florentin smarandache[2020] established a AONHSS. Anjan Mukherjee . [7]Rana Muhammed Zulqarnain Xiao Long xin Muhammad saqlain ,Florentin smarandache[2020] presented GAONHSs .This article tries to establish a concept called as Neutrosophic Vague Hyper Soft Set (NVHSS). 2. Preliminaries Definition 2.1:[3] Let ꬺ be the worldwide collection and Ȩ be the collection of characteristics with respectively to ꬺ. Let Ꝑ (ꬺ) be the power set of ꬺ and Ă ⊆ Ȩ . A pair (ꟻ, Ă) is known as SS over ꬺ and its provided as ꟻ: Ă → Ꝑ(ꬺ) It is also depicted as: (ꟻ, Ă) = [ꟻ (ȩ) ∈ Ꝑ (ꬺ): ȩ (ꟻ, Ă) = {ꟻ (ȩ) ∈ Ꝑ (ꬺ): ȩ ∈ Ȩ , ꟻ (ȩ) = ∅ 𝑖𝑓 ȩ ≠ Ąȩ≠ Ă] mailto:sharviyasona2611@gmail.com1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2114 https://internationalpubls.com Definition 2.2: [6] Let ꬺ be worldwide collection and Ꝑ(ꬺ) be a power set of ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. ꟻ is a mapping from Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn to Ꝑ(ꬺ) Definition 2.3:[1] Let a NV set on the worldwide collection Ă={:x∈X} Since Ʈ (xu) is TMF , Į (xu) is IMF ,Ƒ(xu) is FMF where Ʈ (xu)=[ Ʈ+, Ʈ-] , Į(xu)=[ Ĭ+, Ĭ-] , Ƒ (xu)= [ Ƒ+, Ƒ-] Ʈ+=1-Ƒ- , Ƒ+ =1- Ʈ- 0 ≤ Ʈ (xu) + Ĭ (xu) + Ƒ(xu) ≤ 3. Definition 2.4: [7] Let ꟻ (ꬺ) be the set of NV of ꬺ and Ă ⊆ Ȩ . A pair ((ꟻ, Ă) is called a NSS over ꬺ and its mapping is provided as ꟻ: A → Ꝑ (ꬺ) Definition 2.5: [5] Let ꬺ be worldwide collection and Ꝑ(ꬺ) be a power set of ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. ꟻ is a mapping from Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn to Ꝑ(ꬺ) and ꟻ (Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn ) ={< 𝑥, Ť (xu), Ĭ (xu), Ƒ (xu) >, 𝑥 ∈ ꬺ} Where Ť(xu)=[ Ť+, Ť-] , Ĭ (xu)=[ Ĭ+,Ĭ-] , Ƒ (xu)=[ Ƒ+,Ƒ-] Ť+=1-Ƒ- , Ƒ+ =1- Ť- 0 ≤ Ť(xu) + Ĭ (xu) + Ƒ (xu) ≤ 3. Definition 2.6:[5] Consider two NVSs , ANS and BNV the union is a NVHS, CNV written as CNV=ANV U BNV, following by Ʈ CNV(xu)=[max(Ʈ𝐴𝑁𝑉 − ,Ʈ𝐵𝑁𝑉 − ),max(Ʈ𝐴𝑁𝑉 + ,Ʈ𝐵𝑁𝑉 + )] ĬCNV(xu)=[min(Ĭ𝐴𝑁𝑉 − ,Ĭ𝐵𝑁𝑉 − ),min(Ĭ𝐴𝑁𝑉 + ,Ĭ𝐵𝑁𝑉 + )] ƑCNV(xu)=[min(Ƒ𝐴𝑁𝑉 − ,Ƒ𝐵𝑁𝑉 − ),min(Ƒ𝐴𝑁𝑉 + ,Ƒ𝐵𝑁𝑉 + )] Definition 2.7:[5] Consider two NVSs , ANS and BNV the intersection is a NVHS, CNV, written as CNV=ANV ∩ BNV, following by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2115 https://internationalpubls.com ƮCNV(xu)=[min(Ʈ𝐴𝑁𝑉 − ,Ʈ𝐵𝑁𝑉 − ),min(Ʈ𝐴𝑁𝑉 + ,Ʈ𝐵𝑁𝑉 + )] ĬCNV(xu)=[max(𝐼𝐴𝑁𝑉 − ,Ĭ𝐵𝑁𝑉 − ),max(Ĭ𝐴𝑁𝑉 + ,Ĭ𝐵𝑁𝑉 + )] ƑCNV(xu)=[max(Ƒ𝐴𝑁𝑉 − ,Ƒ𝐵𝑁𝑉 − ),max(Ƒ𝐴𝑁𝑉 + ,Ƒ𝐵𝑁𝑉 + )] 3.1. Neutrosophic Vague Hypersoft Set (NVHSS) Let ꟻ(Ș1) and ꟻ(Ș2) be two worldwide collection and Ꝑ(ꬺ) be a power set ꬺ of Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. ꟻ is a mapping from Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn to Ꝑ(ꬺ) and ꟻ (Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn ) ={< 𝑥, Ť (xu), Ĭ (xu), Ƒ (xu) >, 𝑥 ∈ ꬺ} where Ť, Ĭ, Ƒ are membership values respectively such that Ť, Ĭ, Ƒ : ꬺ → [0−, 1+] and Where Ť(xu)=[ Ť+, Ť-] , Ĭ (xu)=[ Ĭ+,Ĭ-] , Ƒ (xu)=[ Ƒ+,Ƒ-] Ť+=1-Ƒ- , Ƒ+ =1- Ť- 0 ≤ Ť(xu) + Ĭ (xu) + Ƒ (xu) ≤ 3. Example 3.2: Let ꬺ be the group of decision-makers to choose the optimal car type as ꬺ = {X1, X2, X3} Consider the collection of characteristics as ¥1 = Car type, ¥2 = Seat, ¥3 = Variate. And their respective characteristics are following by ¥1 = Car type = {Convertible, Coupe, Sedon} ¥2 = Seat = {7 S, 4 S ,5 S} ¥3 = Variate = {Low end , Medium end, Top end} Let the function be ꟻ : ¥1 × ¥2 ×¥3 →Ꝑ(ꬺ) Table 1: car type ¥1(Car 𝑡𝑦𝑝𝑒) X1 X2 X3 Convertible [0.2, 0.3],[ 0.2,0.4],[0.7,0.8] [0.2, 0.4],[0.2,0.3],[0.6,0.8] [0.4,0.5],[0.1,0.5,[0.5,0.6] Coupe [0.2,0.4],[0.3,0.4],[ 0.6,0.8] [0.2,0.5],[0.1,0.2],[0.5,0.8] [0.3,0.5],[0.2,0.4],[0.5,0.7] Variate [0.4,0.6],[0.3,0.5],[0.6, 0.4] [0.4,0.5],[0.2,0.4],[0.5,0.6] [0.3,0.6],[0.1,0.2],[0.4, 0.7] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2116 https://internationalpubls.com Table 2: Seat ¥2 (Seat) X1 X2 X3 7 S [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5 0.6] [0.1,0.5],[0.2,0.3],[0.5,0.9] 5 S [0.1,0.2],[0.2,0.4],[0.8,0.9] [0.2,0.3],[0.1,0.3],[0.7,0.8] [0.6,0.8],[0.1,0.3],[0.2,0.4] 4 S [0.4,0.5],[0.2,0.4],[0.5,0.6] [0.1,0.3],[0.1,0.3],[0.7,0.9] [0.1,0.2],[0.1,0.2],[0.8,0.9] Table 3: Variate ¥3 (Variate) X1 X2 X3 Low end [0.2,0.7],[0.1,0.2],[0.3,0.8 ] [0.3,0.6],[0.3,0.4],[0.4,0.7] [0.3,0.4],[0.1,0.2],[0.6,0. 7] Medium [0.4,0.5],[0.2,0.3],[0.5,0.6 ] [0.1,0.3],[0.2,0.3],[0.7,0.9] [0.1,0.2],[0.3,0.4],[0.8,0. 9] Top end [0.2,0.5],[0.2,0.3],[0.5,0,8 ] [0.4,0.5],[0.1,0.2],[0.5,0.6] [0.1,0.5],[0.2,0.3],[0.5,0. 9] Definition 3.3 ꟻ : ¥1 × ¥2 ×¥3 →Ꝑ(ꬺ) Let’s assume ꟻ(Ș) = ꟻ(Convertible, 7 Seat, Top end) = {X1, X2}. Then NVHSS of above assumed relation is ꟻ(Ș) =ꟻ(Convertible, 7 Seat, Top end) = {< X1, (Convertible {[0.2, 0.3],[ 0.2,0.4],[0.7,0.8]}, 7 Seat {[0.2,0.5],[0.2,0.3],[0.5,0.8] }, Top end {[0.2,0.5],[0.2,0.3],[0.5,0,8]}) > < X2,(Convertible {[0.2, 0.4],[0.2, 0.3],[0.6,0.8]},7 Seat {[0.4,0.5],[0.1,0.2],[0.5 0.6]}, Top end {[0.4,0.5],[0.1,0.2],[0.5,0.6]}) >} Its tabular form is given as Table 4: NVHSS ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Definition 3.4: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2117 https://internationalpubls.com are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) is the NVHSSsubset of ꟻ(Ș2) if 𝑇(ꟻ(Ș1))) ≤ 𝑇(ꟻ(Ș1)) 𝐼(ꟻ(Ș1))) ≤ 𝐼(ꟻ(Ș1)) 𝐹(ꟻ(Ș1))) ≥ 𝐹(ꟻ(Ș1)) Example 3.5 Consider the two NVHSS, ꟻ(Ș1) and NVHSS, ꟻ(Ș2) over the same universe ꬺ = {X1, X2, X3}. The NVHSS ꟻ(Ș1) = ꟻ(Convertible𝑠, 7 Seat, Top end) = {X1, X2} is the subset of NVHSS ꟻ(Ș2) = ꟻ(Convertible𝑠,7 Seat) = {X1} if 𝑇(ꟻ(Ș1)) ≤ 𝑇(ꟻ(Ș2)) , 𝐼(ꟻ(Ș1)) ≤ 𝐼(ꟻ(Ș2) ) , 𝐹(ꟻ(Ș1)) ≥ 𝐹ꟻ(Ș2)) . It is provided in tabular form below. Table 5: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Table 6: NVHSS ꟻ(Ș2) ꟻ(Ș2)= ꟻ(Convertible, 7 Seat) X1 Convertible [0.3,0.4],[0.2,0.5],[0.6,0.7] 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] This can also be written as T(ꟻ(Ș1) ) ⊂ T(ꟻ(Ș2)) = ꟻ (Convertible, 7 Seat , Top end ) ⊂ ꟻ (Convertible, 7 Seat) = {,,} ⊂ {< X1, (Convertible{[0.3,0.4],[0.2,0.5],[0.6,0.7] },7 Seat {[0.1,0.3],[0.3,0.4],[0.7,0.9]})>} This illustrates the value of membership of Convertible for X1 in both sets is ([0.2,0.3],[0.2,0.4],[0.7, 0.8]) and ([0.3,0.4],[0.2,0.5],[0.6,0.7]) which meet the requirements of the definition of NVHSS subset . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2118 https://internationalpubls.com Definition 3.6: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) is the Neutrosophic vague equal Hypersoft of ꟻ(Ș2) if T(ꟻ(Ș1) ) = T(ꟻ(Ș2)) I(ꟻ(Ș1) ) = I(ꟻ(Ș2)) F(ꟻ(Ș1) ) = F(ꟻ(Ș2)) Example 3.7 Consider the two NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) over the same worldwide ꬺ = {X1, X2, X3 }. The NVHSS F(S1) = ꟻ(Ș1) =ꟻ (Convertible, 7 Seat, Top end ) = {X1, X2} is the equal to NVHSS ꟻ(Ș2)= ꟻ(Convertible, 7 Seat) = {X1} if 𝑇(ꟻ(Ș1)) = 𝑇(ꟻ(Ș2))) , 𝐼(ꟻ(Ș1)) = 𝐼(ꟻ(Ș2)) , 𝐹(ꟻ(Ș1)) =𝐹(ꟻ(Ș2)). Table 7: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Table 8: NVHSS ꟻ(Ș2) ꟻ(Ș2) X1 Convertible [0.2,0.3],[0.1,0.4],[0.7,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] This can also be written as (ꟻ(Ș1) =ꟻ(Ș2)= ꟻ(Convertible, 7 Seat, Top end) = (({< X1, (Convertible {[0.2,0.3],[0.2,0.4],[0.7, 0.8] },7 Seat {[0.4,0.5],[0.2,0.3],[0.5,0.6] },Top end {[0.2,0.5],[0.2,0.3],[0.5,0.8] }) >,< X2, (Convertible{[0.2,0.4],[0.2,0.3],[0.6,0.8] },7 Seat{[0.1,0.3],[0.2,0.3],[0.7,0.9] },Top end {[0.4,0.5],[0.1,0.2],[0.5,0.6]) >} ={< X1(Convertible{[0.3,0.4],[0.2,0.5],[0.6,0.7] },7 Seat {[0.1,0.3],[0.3,0.4],[0.7,0.9]}) >} This illustrates the value of membership of Convertible for X1 in both sets is([0.2,0.3],[0.2,0.4],[0.7, 0.8] ) ([0.2,0.3],[0.2,0.4],[0.7, 0.8]) which satisfy the Definition of NVEHSS. The characteristics of NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2119 https://internationalpubls.com Definition 3.8: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) is Empty NVHS set if { T(ꟻ(Ș1))= 0 I(ꟻ(Ș1))= 0 F(ꟻ(Ș1))= 0 ] Example 3.9 Consider the NVHSS ꟻ(Ș1) over the universe ꬺ = {X1, X2, X3 } . The NVHSS ꟻ(Ș1)= ꟻ( Convertible, 7 Seat, Top end) = {X1, X2} is said to be null NVHSS if its NVHSS values are 0. Table 9: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0,0],[0,0],[0,0] [0,0],[0,0],[0,0] 7 Seat [0,0],[0,0],[0,0] [0,0],[0,0],[0,0] Top end [0,0],[0,0],[0,0] [0,0],[0,0],[0,0] This can also be written as ꟻ(Ș1) = ꟻ (Convertible, 7 Seat, Top end) = {< X1, (Convertible ({[0,0],[ 0, 0],[0,0]}), 7 Seat {[0,0],[ 0, 0],[0,0]}, Top end( {[0,0],[ 0, 0],[0,0]}) >, < X4, (Convertible ({[0,0],[ 0, 0],[0,0]}), 7 Seat {[0,0],[ 0, 0],[0,0]}, Top end( {[0,0],[ 0, 0],[0,0] }) >} Definition 3.10: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻc(Ș1) of ꟻ(Ș1) if ꟻ(Ș1): (⇁ Ý1 ×⇁ Ý2 ×⇁ Ý3 … ⇁ Ý𝑛) → Ꝑ(ꬺ) such that [T(ꟻ(Ș1))= T(ꟻ(Ș1)) I(ꟻ(Ș1))= I(ꟻ(Ș1)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2120 https://internationalpubls.com F(ꟻ(Ș1))= F(ꟻ(Ș1)) ] Example 3.11 Consider the NVHSS ꟻ(Ș1) over the universe ꬺ = {X1, X2, X3 }. The compliment of NVHSS ꟻ(Ș1) = ꟻ( Convertible, 7 Seat, Top end) = {X1, X2} is given as 𝑇𝐶(ꟻ(Ș1) ) = 𝐹(ꟻ(Ș1) ) , I𝐶(ꟻ(Ș1) ) = I(ꟻ(Ș1) ), F𝐶(ꟻ(Ș1) ) = T(ꟻ(Ș1) ). Table 10: NVHSS ꟻ(Ș1) F𝐶(Ș1) X1 X2 Not Convertible [0.7,0.8],[0.2,0.4],[0.2,0.3] [0.6,0.4],[0.2,0.3],[0.2,0.8] Not 7 Seat [0.5,0.6],[0.2,0.3],[0.4,0.5] [0.7,0.9],[0.2,0.3],[0.1,0.3] Not Top end [0.5,0.8],[0.2,0.3],[0.2,0.5] [0.5,0.6],[0.1,0.2],[0.4,0.5] This can also be written as ꟻ(Ș1) = ꟻ ( 𝑛𝑜𝑡 Convertible, 7 Seat, Top end) = {< X1, (𝑛𝑜𝑡 Convertible {[0.7,0.8],[0.2,0.4],[0.2,0.3] }, 𝑛𝑜𝑡 7 Seat {[0.5,0.6],[0.2,0.3],[0.4,0.5] }, 𝑛𝑜𝑡 Top end {[0.5,0.8],[0.2,0.3],[0.2,0.5] }) >,< X2(𝑛𝑜𝑡 Convertible {[0.6,0.4],[0.2,0.3],[0.2,0.8] }, 𝑛𝑜𝑡 7 Seat {[0.7,0.9],[0.2,0.3],[0.1,0.3] }, 𝑛𝑜𝑡 Top end {[0.5,0.6],[0.1,0.2],[0.4,0.5] ) >} This illustrates the value of membership of Convertible for X1 in ꟻ (S1) is {[0.7,0.8],[0.2,0.4],[0.2,0.3] }, and its compliment is([0.2,0.3],[0.2,0.4],[0.7,0.8]) which satisfy the Definition of CNVHSS. This shows that {[0.7,0.8],[0.2,0.4],[0.2,0.3] }is the compliment of ([0.2,0.3],[0.2,0.4],[0.7,0.8]) and The same applied to the remaining qualities of NVHSS ꟻ (S1). Definition 3.12: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) ∪ ꟻ(Ș2) is given as 𝑇(ꟻ(Ș1) ∪ ꟻ(Ș2)) = { 𝑇(ꟻ(Ș1)) 𝑖𝑓 𝑥 ∈ Ș1 𝑇(ꟻ(Ș2)) 𝑖𝑓 𝑥 ∈ Ș2 max (𝑇(ꟻ(Ș1)), 𝑇(ꟻ(Ș2))) 𝑖𝑓 𝑥 ∈ Ș1 ∩ Ș2 𝐼(ꟻ(Ș1) ∪ ꟻ(Ș2)) = { 𝐼(ꟻ(Ș1)) 𝑖𝑓 𝑥 ∈ Ș1 𝐼(ꟻ(Ș2)) 𝑖𝑓 𝑥 ∈ Ș2 min (𝐼(ꟻ(Ș1)), 𝐼(ꟻ(Ș2))) 𝑖𝑓 𝑥 ∈ Ș1 ∩ Ș2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2121 https://internationalpubls.com 𝐹(ꟻ(Ș1) ∪ ꟻ(Ș2)) = { 𝐹(ꟻ(Ș1)) 𝑖𝑓 𝑥 ∈ Ș1 𝐹(ꟻ(Ș2)) 𝑖𝑓 𝑥 ∈ Ș2 min (𝐹(ꟻ(Ș1)), 𝐹(ꟻ(Ș2))) 𝑖𝑓 𝑥 ∈ Ș1 ∩ Ș2 Example3.13 Consider the two NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) over the same universe ꬺ = {X1, X2, X3}. Tabular representation of NVHSS ꟻ(Ș1) = ꟻ(Convertible ,7 Seat ,Top end ) = {X1, X2} and NVHSS ꟻ(Ș2) = ꟻ(Convertible,7 Seat ) = {X1} is given below Table 11: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Table 12: NVHSS ꟻ(S2) ꟻ(Ș2) X1 Convertible [0.3,0.4],[0.2,0.5],[0.6,0.7] 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] Then the union of above NVHSS is given as Table 13: Union of NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș1) ꟻ(Ș1) ∪ ꟻ(Ș2) X1 X2 Convertible [0.3,0.4],[0.2,0.4],[0.6,0.7] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0,0],[ 0, 0] [0.4,0.5],[0.1,0.2],[0.5,0.6] This can also be written as ꟻ(Ș1) ∪ ꟻ(Ș2)= ꟻ (Convertible,7 Seat , Top end ) ∪ ꟻ (Convertible,7 Seat) = {< X1, (Convertible {[0.3,0.4],[0.2,0.4],[0.6,0.7]}, 7 Seat{[0.4,0.5],[0.2,0.3],[0.5,0.6]}, Top end{[0.2,0.5],[0,0],[ 0, 0]}) >, < X2,(Convertible {[0.2,0.4],[0.2,0.3],[0.6,0.8]}, 7 Seat{[0.1,0.3],[0.2,0.3],[0.7,0.9]}, Top end{[0.4,0.5],[0.1,0.2],[0.5,0.6]}) >} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2122 https://internationalpubls.com Definition 3.14: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) ∩ ꟻ(Ș2) 𝑇(ꟻ(𝑆1) ∩ ꟻ(𝑆2)) = { 𝑇(ꟻ(Ș1)) 𝑖𝑓 𝑥 ∈ Ș1 𝑇(ꟻ(Ș2)) 𝑖𝑓 𝑥 ∈ Ș2 min (𝑇(ꟻ(Ș1)), 𝑇(ꟻ(Ș2))) 𝑖𝑓 𝑥 ∈ Ș1 ∩ Ș2 𝐼(ꟻ(𝑆1) ∩ ꟻ(𝑆2)) = { 𝐼(ꟻ(𝑆1)) 𝑖𝑓 𝑥 ∈ Ș1 𝐼(ꟻ(Ș2)) 𝑖𝑓 𝑥 ∈ Ș2 max (𝐼(ꟻ(Ș1)), 𝐼(ꟻ(Ș2))) 𝑖𝑓 𝑥 ∈ Ș1 ∩ Ș2 𝐹(ꟻ(Ș1) ∩ ꟻ(Ș2)) = { 𝐹(ꟻ(Ș1)) 𝑖𝑓 𝑥 ∈ Ș1 𝐹(ꟻ(Ș2)) 𝑖𝑓 𝑥 ∈ Ș2 max (𝐹(ꟻ(Ș1)), 𝐹(ꟻ(Ș2))) 𝑖𝑓 𝑥 ∈ Ș1 ∩ Ș2 Example 3.15 Consider the two NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) over the same universe ꬺ = {X1, X2, X3}. Tabular representation of NVHSS ꟻ(Ș1) = ꟻ(Convertible ,7 Seat ,Top end ) = {X1, X2} and NVHSS ꟻ(Ș2) = ꟻ(Convertible,7 Seat ) = {X1} is given below Table 14: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Table 15: NVHSS ꟻ(Ș2) ꟻ(Ș2) X1 Convertible [0.3,0.4],[0.2,0.5],[0.6,0.7] 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] Then the intersection of above NVHSS is given as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2123 https://internationalpubls.com Table 16: NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) ꟻ(Ș1) ∩ ꟻ(Ș2) X1 Convertible [0.2,0.3],[0.2,0.5],[0.7,0.8] 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] Top end [ 0 , 0 ],[ 0.2, 0.3],[0.5,0.8] This can also be written as ꟻ(Ș1) ∩ ꟻ(Ș2) = ꟻ(Convertible,7 Seat , Top end ) ∩ ꟻ (Convertible,7 Seat ) = {< X1, (Convertible{[0.2,0.3],[0.2,0.5],[0.7,0.8]}, 7 Seat{[0.1,0.3],[0.3,0.4],[0.7,0.9]}, Top end{[0,0],[0.2,0.3],[0.5,0.8]}) >} Definition 3.16: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) ∧ ꟻ(Ș2) = ꟻ ( Ș1 ×Ș2) T( Ș1 ×Ș2) = 𝑚in (T(ꟻ(Ș1) ), 𝑇(ꟻ(Ș2))) I( Ș1 ×Ș2) = 𝑚ax (I(ꟻ(Ș1) ), 𝑇(ꟻ(Ș2))) F( Ș1 ×Ș2) = 𝑚ax (F(ꟻ(Ș1) ), 𝑇(ꟻ(Ș2))) Example:3.17 Consider the two NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) over the same universe ꬺ = {X1, X2, X3}. Tabular representation of NVHSS ꟻ(Ș1) = ꟻ(Convertible ,7 Seat ,Top end ) = {X1, X2} and NVHSS ꟻ(Ș2) = ꟻ(Convertible,7 Seat ) = {X1} is given below Table 17: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2124 https://internationalpubls.com Table 18: NVHSS ꟻ(Ș2) ꟻ(Ș2) X1 Convertible [0.3,0.4],[0.2,0.5],[0.6,0.7] 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] Then the AND Operation of above NVHSS is given as Table 19: AND of NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) ꟻ(Ș1) ∧ ꟻ(Ș2) X1 X2 Convert × Convert [0.2,0.3],[0.2,0.5],[0.7,0.8] [0,0],[0.2,0.3],[0.6,0.8] Convert × 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] [0,0],[0.2,0.3],[0.6,0.8] 7 Seat × Convert [0.3,0.4],[0.2,0.5],[0.6,0.7] [0,0],[0.2,0.3],[0.7,0.9] 7 Seat × 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] [0,0],[0.2,0.3],[0.7,0.9] Top end × Convert [0.2,0.4],[0.2,0.5],[0.6,0.8] [0,0],[0.1,0.2],[0.5,0.6] Top end × 7 Seat [0.1,0.3],[0.3,0.4],[0.7,0.9] [0,0],[0.1,0.2],[0.5,0.6] Definition 3.18: Let ꟻ(Ș1) and ꟻ(Ș2) be two Neutrosophic Vague Hypersoft set over ꬺ. Consider as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and for 𝑛 ≥ 1, There are a few unique characteristics such as ꝁ1, ꝁ2, ꝁ3,….. ꝁn and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn are sets with the following constraints for corresponding values and characteristics, correspondingly Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn with Ꞵk∩Ꞵl = ∅, The relationship between k ≠ l and k, l𝜖{1,2,3 … 𝑛} and Ꞵ1, Ꞵ2,Ꞵ3,….... Ꞵn = S. then ꟻ(Ș1) ˅ ꟻ(Ș2) = ꟻ ( Ș1 ×Ș2) is given as T( Ș1 ×Ș2) = 𝑚𝑎𝑥 (T(ꟻ(Ș1) ), 𝑇(ꟻ(Ș2))) I( Ș1 ×Ș2) = 𝑚𝑖𝑛 (I(ꟻ(Ș1) ), 𝑇(ꟻ(Ș2))) F( Ș1 ×Ș2) = 𝑚𝑖𝑛 (F(ꟻ(Ș1) ), 𝑇(ꟻ(Ș2))) Example 3.18 Consider the two NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) over the same universe ꬺ = {X1, X2, X3}. Tabular representation of NVHSS ꟻ(Ș1) = ꟻ(Convertible ,7 Seat ,Top end ) = {X1, X2} and NVHSS ꟻ(Ș2) = ꟻ(Convertible,7 Seat ) = {X1} is given below Table 20: NVHSS ꟻ(Ș1) ꟻ(Ș1) X1 X2 Convertible [0.2,0.3],[0.2,0.4],[0.7, 0.8] [0.2,0.4],[0.2,0.3],[0.6,0.8] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2125 https://internationalpubls.com 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0.2,0.3],[0.7,0.9] Top end [0.2,0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0.1,0.2],[0.5,0.6] Table 21: NVHSS ꟻ(Ș2) ꟻ(Ș2) X1 Samsung [0.3,0.4],[0.2,0.5],[0.6,0.7] 6 GB [0.1,0.3],[0.3,0.4],[0.7,0.9] Then the OR Operation of above NVHSS is given as Table 22: OR of NVHSS ꟻ(Ș1) and NVHSS ꟻ(Ș2) ꟻ(Ș1) ˅ ꟻ(Ș2) X1 X2 Convert× Convert [0.3,0.4],[0.2,0.4],[0.6,0.7] [0.2,0.4],[0,0],[0,0] Convert × 7 Seat [0.2,0.3],[0.2,0.4],[0.7,0.8] [0.2,0.4],[0,0],[0,0] 7 Seat × Convert [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0,0],[0,0] 7 Seat × 7 Seat [0.4,0.5],[0.2,0.3],[0.5,0.6] [0.1,0.3],[0,0],[0,0] Top end × Convert [0.3,0.5],[0.2,0.3],[0.5,0.7] [0.4,0.5],[0,0],[0,0] Top end × 7 Seat [0.2, 0.5],[0.2,0.3],[0.5,0.8] [0.4,0.5],[0,0],[0,0] References [1] Alkhazaleh, S. (2015). Neutrosophic vague set theory. Critical Review, 10, 29-39. [2] D. Molodtsov, Soft Set Theory First Results, Computers & Mathematics with Applications, 37(1999) 19–31. [3] F. Smarandache, Neutrosophic set – a generalization of intuitionistic fuzzy sets, International Journal of Pure and Applied Mathematics, 24(3)(2005) 287–297. [4] Muhammad saqlain ,Sana Moni Muhammad Naveed Jafar Muhammad Saeed and Florentin smarandache[2020] established Aggregate operators of Neutrosophic Hypersoftset.Vol 32(2020) [5] P.K. Maji, A.R. Roy, R. Biswas, An application of soft sets in a decision making problem, Computers and mathematics with applications. 44 (2002) 1077-1083. [6] Rana Muhammed Zulqarnain Xiao Long xin Muhammad saqlain,Florentin smarandache[2020] presented Generalized Aggregate Operators on Neutrosophic Hypersoftset.Vol.36 (2020)