Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
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University of New Mexico Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets Ahmad A. Abubaker,1, M. Palanikumar(2,∗), Abdallah Al-Husban3,4 1Faculty of Computer Studies, Arab Open University, Saudi Arabia; [email protected] 2Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai-602105, India; [email protected] 3Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan; [email protected] 4Jadara Research Center, Jadara University, Irbid 21110, Jordan. ∗Correspondence: [email protected]; In this communication, we give the theory of the 3-valued extension of neutrosophic soft set (3-valued ENSS) and define certain operations. Notably, we demonstrated an algorithm to address the decision-making problem using a soft set model. We present a similarity measure for two 3-valued ENSSs and describe its use in a pattern recognition challenge. Illustrative examples are provided to demonstrate their effectiveness in solving problems with uncertainties. Keywords: interval valued fuzzy soft set, fuzzy soft set, decision making problem, aggregation operator. ———————————————————————————————— 1. Introduction Every day, machine learning, intelligence collection, knowledge compilation, and other processes are used in the conflict resolution process. One of the biggest challenges is demonstrating the effectiveness of strategic planning. Mathematical theory enables the adoption of appropriate decision-making techniques. The DM concept might be advantageous to businesses Neutrosophic Sets and Systems, Vol. 94, 2025 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
since it assesses and ranks different points of view based on their qualities. We can then effectively select, classify, generate, and evaluate our options. MADM considers every feature and component to provide the optimal response. It used to be widely accepted that weights and attributes required to be represented as distinct numerical values. Numerous assessments and DM problems require the analysis of numerous variables and indications. Assessing and collecting data for evaluation indicators can help overcome assessment and DM problems. The complex structure of real-world systems frequently leads to MADM issues, which obscure evaluation specifics. To explain uncertainty, many theories have been proposed, such as fuzzy sets (FSs) [1], which have membership grades (MG) from 0 to 1. Each element in the intuitionistic FS (IFS) developed by Atanassov [2] the condition that 0 ℘+[1 , for ℘, [ ∈[0,1] and positive ℘and negative [. Yager [3] invented the Pythagorean FSs (PFS) concept, which is characterized by its MG and non-membership grade (NMG), with the restriction that ℘+[1 to ℘2+[21. Many research have been conducted on the use of IFSs and PFSs in many fields. The constraint that the square sum of its MG and NMG degrees not exceed unity defines the extended IFSs. The concept of picture FSs is extended by FSs and IFSs [4]. Their ability to communicate information is still restricted. Because of this, the experts were still having trouble explaining the data in these sets and the associated data. Wang et al. [5] investigated the concept of complex IFS with DOMBI prioritized AOs and its application for trustworthy green supplier selection. Using the interval-valued IFS distance-based MAIRCA methodology, Mishra et al. [6] investigate a way of assessing sustainable wastewater treatment systems. Positive MG (℘), neutral MG (∝), and negative MG ([) are the three basic concepts of the picture FS, according to Cuong et al. [7]. Additionally, it offers more benefits than PFS and IFS. Since ℘, ∝, [ ∈[0,1], it has been noted that the picture FS is an enhancement of the IFS that may handle more inconsistency and 0 ℘+∝+[1. According to the picture FS description, expert opinions like ”yes,” ”abstain,” ”no,” and ”refusal” will be conveyed. It will also encourage uniformity between the assessment data and the actual decision environment and stop evaluation information from being thrown away. Molodtsov introduced the concept of soft sets (SOSs) [8]. SOSs more accurately the complexity and objectivity of DM in real world scenarios than other uncertain theories. Furthermore, a crucial area of study is the integration of SOSs with other mathematical models. Maji suggested FSOSs [9] and intuitionistic FSOSs [10]. Recently, AlHusban et al. [11–14] discussed the various extension algebraic structures via FS, IVFS and AO. These two theories are used to address a range of DM problems. For the rest of the work, I shall adhere to the structure mentioned below. Section 1 deals that the introduction. Section 2 discussed PFS and its basic concepts. The new idea of 3-valued ENSSs and its fundamental functions are explained in Section 3. The similarity measures between 3-valued ENSSss are Neutrosophic Sets and Systems, Vol. 94, 2025 33 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
discussed in Section 4. The selection tools for pattern recognition and its practical applications were covered in Section 5. The conclusion is covered in section 6. 2. Preliminaries Let Xbe a universal set for the entirety of this section. The fundamental concepts of the neutrosophic set (NSS), which are well-known in the literature, are reviewed and introduced in this section. Definition 2.1. The neutrosophic interval valued FS (NIVFS) Z={x, >Z(∂),=Z(∂),`Z(∂)|x∈X}, where >Z(∂)=[>l Z(∂),>u Z(∂)] and =Z(∂) = [=l Z(∂),=u Z(∂)] and `Z(∂) = [`l Z(∂),`u Z(∂)] called the value of truth, interminacy and false membership of Z, respectively. The function >Z:X→D[0,1], =Z:X→D[0,1], `Z:X→D[0,1] and 0 (>Z(∂)) + (=Z(∂)) + (`Z(∂)) 3. Here Z=D[>l Z,>u Z],[=l Z,=u Z],[`l Z,`u Z]Eis mentioned a NIVF number (NIVFN). Definition 2.2. Suppose that Z=h>Z,=Z,`Ziand 0=h>0,=0,`0iare any two NIVFNs over (X, E). Then (1) Zc=h`Z,=Z,>Zi (2) Zt0=Dmax(>Z,>0),min(=Z,=0),min(`Z,`0)E (3) Zu0=Dmin(>Z,>0),min(=Z,=0),max(`Z,`0)E (4) Z0iff >Z >0and =Z =0and `Z`0 (5) Z=0iff >Z=>0and =Z==0and `Z=`0. Definition 2.3. Let Ebe a set of parameter. Given ZvEand F:Z→PF(X), where PF(X) is the collection of all fuzzy subsets of X, the pair (F, Z) is referred to as a Pythagorean FSOS (PFSOS) on X. Definition 2.4. A neutrosophic refined set (NRS) Z={hx, (>1 Z(∂),>2 Z(∂), ..., , >P Z(∂)), (=1 Z(∂),=2 Z(∂), ..., , =P Z(∂)),(`1 Z(∂),`2 Z(∂), ..., , `P Z(∂))i:∂∈E}, where, >1 Z(∂),>2 Z(∂), ..., , >P Z(∂) : E→[0,1],=1 Z(∂),=2 Z(∂), ..., , =P Z(∂) : E→ [0,1],`1 Z(∂),`2 Z(∂), ..., , `P Z(∂) : E→[0,1] such that 0 >i Z(∂) + =i Z(∂) + `i Z(∂)3(i= 1,2, ..., P) and >1 Z(∂) >2 Z(∂)... >i Z(∂), for any ∂∈E. Here >1 Z(∂),>2 Z(∂),>P Z(∂), =1 Z(∂),=2 Z(∂),=P Z(∂) and `1 Z(∂),`2 Z(∂),`P Z(∂) represents the TMG, IMG, and FMG sequences of the element x. Additionally, Pis referred to as the dimension of NRS Z. TMG sequences grow, but other sequences (IMG, FMG) do not increase or decrease. However, there is no rise or decrease in TMG, IMG, or FMG sequences throughout this paper. NRS(E) denotes the collection of all neutrosophic refined sets in E. Neutrosophic Sets and Systems, Vol. 94, 2025 34 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
3. Novel approach towards 3-valued ENS The notion of 3-valued extended neutrosophic soft sets will be presented. Definition 3.1. Let E={κ1, κ2, ..., κm}be a set of parameters and Xbe the universal set. A soft universe is the pair (X, E). Let Tand Fbe unchanged, and let the indeterminacy Ibe refined as either the same or (Unknown, contradiction), which are subsets of [0,1]. Next, we obtain the 3-valued neutrosophic soft set scenario as follows: We provide the following numerical example to demonstrate the Definition 3.1: Example 3.2. A collection of parameters is E={κ1, κ2, κ3}. Assume M:E→M(X) is provided by M1(∂) = κ1 (h0.35,0.45i,h0.05,0.10i,h0.45,0.60i) κ2 (h0.50,0.55i,h0.05,0.15i,h0.40,0.45i) κ3 (h0.30,0.35i,h0.20,0.25i,h0.40,0.50i) ;M2(∂) = κ1 (h0.30,0.35i,h0.10,0.15i,h0.50,0.60i) κ2 (h0.35,0.45i,h0.20,0.30i,h0.40,0.50i) κ3 (h0.30,0.40i,h0.10,0.20i,h0.45,0.60i) ; M3(∂) = κ1 (h0.05,0.10i,h0.30,0.35i,h0.50,0.60i) κ2 (h0.15,0.20i,h0.40,0.45i,h0.35,0.40i) κ3 (h0.05,0.10i,h0.15,0.25i,h0.45,0.50i) ; Definition 3.3. Assume that two 3-valued ENSs on (X, E) are Mand N. As indicated by MvNif and only if Mi(∂)vNi(∂) if >i(∂) >i(∂),=i(∂) =i(∂),`i(∂)`i(∂), ∀∂∈X. We propose the following numerical example to demonstrate the previously mentioned definition: Example 3.4. Consider the 3-valued ENS Min Example 3.2. Let Nbe another 3-valued ENS is defined as follows: N1(∂) = κ1 (h0.53,0.63i,h0.18,0.38i,h0.27,0.32i) κ2 (h0.63,0.78i,h0.18,0.38i,h0.07,0.12i) κ3 (h0.48,0.68i,h0.38,0.58i,h0.17,0.22i) ;N2(∂) = κ1 (h0.48,0.58i,h0.28,0.48i,h0.12,0.17i) κ2 (h0.53,0.68i,h0.43,0.68i,h0.02,0.07i) κ3 (h0.48,0.78i,h0.38,0.48i,h0.17,0.22i) ; N3(∂) = κ1 (h0.33,0.48i,h0.48,0.63i,h0.07,0.12i) κ2 (h0.43,0.58i,h0.68,0.68i,h0.07,0.12i) κ3 (h0.33,0.58i,h0.43,0.53i,h0.12,0.17i) ; Definition 3.5. Suppose Mand Nare two 3-valued ENSs on (X, E). These two 3-valued ENSs are identical (denoted by M=N) if and only if MivNiand MiwNi. Neutrosophic Sets and Systems, Vol. 94, 2025 35 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
Definition 3.6. Let Mand Ndenote two 3-valued ENSs on (X, E). The union and intersection of Mand Nover (X, E) are indicated by MtNand MuN, respectively, J:E→M(X), I:E→M(X) such that J(∂) = M(∂)tN(∂), I(∂) = M(∂)uN(∂), for all ∂∈X. Example 3.7. Let Mand Nbe the two 3-valued ENSs on (X, E) that are defined as M1(∂) = κ1 (h0.30,0.50i,h0.20,0.30i,h0.30,0.50i) κ2 (h0.20,0.40i,h0.50,0.60i,h0.30,0.40i) κ3 (h0.30,0.60i,h0.20,0.40i,h0.20,0.30i) ;M2(∂) = κ1 (h0.20,0.50i,h0,0.10i,h0.50,0.60i) κ2 (h0.30,0.40i,h0.10,0.30i,h0.50,0.60i) κ3 (h0.40,0.60i,h0.10,0.20i,h0.30,0.50i) ; and N1(∂) = κ1 (h0.30,0.40i,h0.20,0.40i,h0.40,0.60i) κ2 (h0.20,0.30i,h0,0.20i,h0.50,0.70i) κ3 (h0.40,0.50i,h0.30,0.40i,h0.20,0.30i) ;N2(∂) = κ1 (h0.10,0.40i,h0.20,0.30i,h0.50,0.60i) κ2 (h0.20,0.30i,h0.30,0.40i,h0.40,0.50i) κ3 (h0.20,0.40i,h0.30,0.50i,h0.40,0.50i) ; M1(∂)tN1(∂) = κ1 (h0.30,0.50i,h0.20,0.30i,h0.30,0.50i) κ2 (h0.20,0.40i,h0,0.20i,h0.30,0.40i) κ3 (h0.40,0.60i,h0.20,0.40i,h0.20,0.30i) ;M2(∂)tN2(∂) = κ1 (h0.20,0.50i,h0,0.10i,h0.50,0.60i) κ2 (h0.30,0.40i,h0.10,0.30i,h0.40,0.50i) κ3 (h0.40,0.60i,h0.10,0.20i,h0.30,0.50i) ; M1(∂)uN1(∂) = κ1 (h0.30,0.40i,h0.20,0.30i,h0.40,0.60i) κ2 (h0.20,0.30i,h0,0.20i,h0.50,0.70i) κ3 (h0.30,0.50i,h0.20,0.40i,h0.20,0.30i) ;M2(∂)uN2(∂) = κ1 (h0.10,0.40i,h0,0.10i,h0.50,0.60i) κ2 (h0.20,0.30i,h0.10,0.30i,h0.50,0.60i) κ3 (h0.20,0.40i,h0.10,0.20i,h0.40,0.50i) ; 4. Determine similarity measure Methods: Let Zand Ybe the two 3-valued ENSs. The similarity measure between Zand Yis defined as Sim(Z, Y ) = Φ(Z, Y ). Since Φ(Z, Y ) = 1 n0n n0 M k=p n M i=p min (Tl 1(Z(∂k), Y (∂k)) , Tl 2(Z(∂k), Y (∂k)) , Sl(Z(∂k), Y (∂k)) ), max (Tu 1(Z(∂k), Y (∂k)) , Tu 2(Z(∂k), Y (∂k)) , Su(Z(∂k), Y (∂k)) ) (1) and T1(Z(∂k), Y (∂k)) = Ln0 k=pLn i=p>il Z(∂k)· >il Y(∂k) Ln0 k=pLn i=pp−qp−>i2l Z(∂k)·p−>i2l Y(∂k),Ln0 k=pLn i=p>iu Z(∂k)· >iu Y(∂k) Ln0 k=pLn i=pp−qp−>i2u Z(∂k)·p−>i2u Y(∂k)! (2) T2(Z(∂k), Y (∂k)) = Ln0 k=pLn i=p=i2l Z(∂k)· =i2l Y(∂k) Ln0 k=pLn i=pp−qp−=i4l Z(∂k)·p−=i4l Y(∂k),Ln0 k=pLn i=p=i2u Z(∂k)· =i2u Y(∂k) Ln0 k=pLn i=pp−qp−=i4u Z(∂k)·p−=i4u Y(∂k)! (3) Neutrosophic Sets and Systems, Vol. 94, 2025 36 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
S(Z(∂k), Y (∂k)) = p−v u u tLn0 k=pLn i=p(`i2l Z(∂k)−`i2u Y(∂k)) Ln0 k=pLn i=pp+(`i2u Z(∂k)·`i2u Y(∂k)),Ln0 k=pLn i=p(`i2u Z(∂k)−`i2l Y(∂k)) Ln0 k=pLn i=pp+(`i2l Z(∂k)·`i2l Y(∂k))!(4) Theorem 4.1. Let Z, Y and Wdenote any three 3-valued ENSs over (X, E). Show that Z⊆YvW=⇒Sim(Z, W )Sim(Y, W ). Proof. For i= 1,2, ..., n and k= 1,2, ..., n0 ZvY=⇒ >il Z(∂k),>iu Z(∂k)>il Y(∂k),>iu Y(∂k) =il Z(∂k),=iu Z(∂k)=il Y(∂k),=iu Y(∂k) `il Z(∂k),`iu Z(∂k)`il Y(∂k),`iu Y(∂k) ZvW=⇒ >il Z(∂k),>iu Z(∂k)>il W(∂k),>iu W(∂k) =il Z(∂k),=iu Z(∂k)=il W(∂k),=iu W(∂k) `il Z(∂k),`iu Z(∂k)`il W(∂k),`iu W(∂k) YvW=⇒ >il Y(∂k),>iu Y(∂k)>il W(∂k),>iu W(∂k) =il Y(∂k),=iu Y(∂k)=il W(∂k),=iu W(∂k) `il Y(∂k),`iu Y(∂k)`il W(∂k),`iu W(∂k) (5) Clearly, >il Z(∂k)· >il W(∂k),>iu Z(∂k)· >iu W(∂k)>il Y(∂k)· >il W(∂k),>iu Y(∂k)· >iu W(∂k) n0 M k=p n M i=p>il Z(∂k)· >il W(∂k), n0 M k=p n M i=p>iu Z(∂k)· >iu W(∂k)n0 M k=p n M i=p>il Y(∂k)· >il W(∂k), n0 M k=p n M i=p>iu Y(∂k)· >iu W(∂k) (6) Clearly, >i2l Z(∂k),>i2u Z(∂k)>i2l Y(∂k),>i2u Y(∂k)>i2l W(∂k),>i2u W(∂k) and − >i2u Z(∂k),−>i2l Z(∂k)− >i2u Y(∂k),−>i2l Y(∂k)− >i2u W(∂k),−>i2l W(∂k) and p−>i2l Z(∂k),p−>i2u Z(∂k)p−>i2l Y(∂k),p−>i2u Y(∂k)p−>i2l W(∂k),1− >i2u W(∂k) and p−>i2l Z(∂k)·p−>i2l W(∂k),p−>i2u Z(∂k)·p−>i2u W(∂k) p−>i2l Y(∂k)·p−>i2l W(∂k),p−>i2u Y(∂k)·p−>i2u W(∂k) Neutrosophic Sets and Systems, Vol. 94, 2025 37 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
and qp−>i2l Z(∂k)·p−>i2l W(∂k),qp−>i2u Z(∂k)·p−>i2u W(∂k) qp−>i2l Y(∂k)·p−>i2l W(∂k),qp−>i2u Y(∂k)·p−>i2u W(∂k) and p−qp−>i2l Z(∂k)·p−>i2l W(∂k)qp−>i2u Z(∂k)·p−>i2u W(∂k) p−qp−>i2l Y(∂k)·p−>i2l W(∂k)qp−>i2u Y(∂k)·p−>i2u W(∂k) p−qp−>i2u Z(∂k)·p−>i2u W(∂k)p−qp−>i2l Z(∂k)·p−>i2l W(∂k) p−qp−>i2u Y(∂k)·p−>i2u W(∂k)p−qp−>i2l Y(∂k)·p−>i2l W(∂k) and Ln0 k=pLn i=pp−qp−>i2u Z(∂k)·p−>i2u W(∂k) ,Ln0 k=pLn i=pp−qp−>i2l Z(∂k)·p−>i2l W(∂k) ! Ln0 k=pLn i=pp−qp−>i2u Y(∂k)·p−>i2u W(∂k) ,Ln0 k=pLn i=pp−qp−>i2l Y(∂k)·p−>i2l W(∂k) ! (7) Equations (6) is divided by (7), Ln0 k=pLn i=p>il Z(∂k)· >il W(∂k) Ln0 k=pLn i=pp−rp−>i2u Z(∂k)·p−>i2u W(∂k) ,Ln0 k=pLn i=p>iu Z(∂k)· >iu W(∂k) Ln0 k=pLn i=pp−rp−>i2l Z(∂k)·p−>i2l W(∂k) Ln0 k=pLn i=p>il Y(∂k)· >il W(∂k) Ln0 k=pLn i=pp−rp−>i2u Y(∂k)·p−>i2u W(∂k) ,Ln0 k=pLn i=p>iu Y(∂k)· >iu W(∂k) Ln0 k=pLn i=pp−rp−>i2l Y(∂k)·p−>i2l W(∂k) Hence Neutrosophic Sets and Systems, Vol. 94, 2025 38 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
Ln0 k=pLn i=p(>il Z(∂k)·>il W(∂k))! Ln0 k=pLn i=pp−q((p−>i2l Z(∂k))·(p−>i2l W(∂k))) !, Ln0 k=pLn i=p(>iu Z(∂k)·>iu W(∂k))! Ln0 k=pLn i=pp−q((p−>i2u Z(∂k))·(p−>i2u W(∂k))) ! Ln0 k=pLn i=p(>il Y(∂k)·>il W(∂k))! Ln0 k=pLn i=pp−q((p−>i2l Y(∂k))·(p−>i2l W(∂k))) !, Ln0 k=pLn i=p(>iu Y(∂k)·>iu W(∂k))! Ln0 k=pLn i=pp−q((p−>i2u Y(∂k))·(p−>i2u W(∂k))) ! ; (8) Therefore T1(Z(∂k), W(∂k)) T1(Y(∂k), W (∂k)) Clearly, =i2l Z(∂k)· =i2l W(∂k),=i2u Z(∂k)· =i2u W(∂k)=i2l Y(∂k)· =i2l W(∂k),=i2u Y(∂k)· =i2u W(∂k). Hence n0 M k=p n M i=p=i2l Z(∂k)· =i2l W(∂k), n0 M k=p n M i=p=i2u Z(∂k)· =i2u W(∂k) n0 M k=p n M i=p=i2l Y(∂k)· =i2l W(∂k), n0 M k=p n M i=p=i2u Y(∂k)· =i2u W(∂k)(9) Clearly, =i4l Z(∂k),=i4u Z(∂k)=i4l Y(∂k),=i4u Y(∂k)=i4l W(∂k),=i4u W(∂k) implies that − =i4u Z(∂k),−=i4l Z(∂k)− =i4u Y(∂k),−=i4l Y(∂k)− =i4u W(∂k),−=i4l W(∂k) p−=i4l Z(∂k),p−=i4u Z(∂k)p−=i4l Y(∂k),p−=i4u Y(∂k)p−=i4l W(∂k),1− =i4u W(∂k) and p−=i4l Z(∂k)·p−=i4l W(∂k),p−=i4u Z(∂k)·p−=i4u W(∂k) p−=i4l Y(∂k)·p−=i4l W(∂k),p−=i4u Y(∂k)·p−=i4u W(∂k) Neutrosophic Sets and Systems, Vol. 94, 2025 39 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
qp−=i4l Z(∂k)·p−=i4l W(∂k),qp−=i4u Z(∂k)·p−=i4u W(∂k) qp−=i4l Y(∂k)·p−=i4l W(∂k),qp−=i4u Y(∂k)·p−=i4u W(∂k) p−qp−=i4l Z(∂k)·p−=i4l W(∂k),qp−=i4u Z(∂k)·p−=i4u W(∂k) p−qp−=i4l Y(∂k)·p−=i4l W(∂k),qp−=i4u Y(∂k)·p−=i4u W(∂k) p−qp−=i4u Z(∂k)·p−=i4u W(∂k),p−qp−=i4l Z(∂k)·p−=i4l W(∂k) p−qp−=i4u Y(∂k)·p−=i4u W(∂k),p−qp−=i4l Y(∂k)·p−=i4l W(∂k) Ln0 k=pLn i=pp−rp−=i4u Z(∂k)·p−=i4u W(∂k) ,Ln0 k=pLn i=pp−rp−=i4l Z(∂k)·p−=i4l W(∂k) ! Ln0 k=pLn i=pp−rp−=i4u Y(∂k)·p−=i4u W(∂k) ,Ln0 k=pLn i=pp−rp−=i4l Y(∂k)·p−=i4l W(∂k) ! ; (10) Equation (9) is divided by (10), Ln0 k=pLn i=p=i2l Z(∂k)· =i2l W(∂k),Ln0 k=pLn i=p=i2u Z(∂k)· =i2u W(∂k) Ln0 k=pLn i=pp−rp−=i4u Z(∂k)·p−=i4u W(∂k) ,Ln0 k=pLn i=pp−rp−=i4l Z(∂k)·p−=i4l W(∂k) Ln0 k=pLn i=p=i2l Y(∂k)· =i2l W(∂k),Ln0 k=pLn i=p=i2u Y(∂k)· =i2u W(∂k) Ln0 k=pLn i=pp−rp−=i4u Y(∂k)·p−=i4u W(∂k) ,Ln0 k=pLn i=pp−rp−=i4l Y(∂k)·p−=i4l W(∂k) Hence Ln0 k=pLn i=p(=i2l Z(∂k)·=i2l W(∂k))! Ln0 k=pLn i=pp−q((p−=i4l Z(∂k))·(p−=i4l W(∂k))) !, Ln0 k=pLn i=p(=i2u Z(∂k)·=i2u W(∂k))! Ln0 k=pLn i=pp−q((p−=i4u Z(∂k))·(p−=i4u W(∂k))) ! Ln0 k=pLn i=p(=i2l Y(∂k)·=i2l W(∂k))! Ln0 k=pLn i=pp−q((p−=i4l Y(∂k))·(p−=i4l W(∂k))) !, Ln0 k=pLn i=p(=i2u Y(∂k)·=i2u W(∂k))! Ln0 k=pLn i=pp−q((p−=i4u Y(∂k))·(p−=i4u W(∂k))) ! ; Therefore T2(Z(∂k), W(∂k)) T2(Y(∂k), W (∂k)) (11) Clearly, `i2l Z(∂k),`i2u Z(∂k)`i2l Y(∂k),`i2u Y(∂k)`i2l W(∂k),`i2u W(∂k) and Neutrosophic Sets and Systems, Vol. 94, 2025 40 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
Table 12. 3-valued ENS for the financial sector Ti T(∂)κ4κ5 T1(∂)h0.3,0.4i,h0.65,0.7i,h0.2,0.4i h0.25,0.35i,h0.5,0.55i,h0.2,0.3i T2(∂)h0.5,0.55i,h0.45,0.5i,h0.25,0.45i h0.4,0.45i,h0.3,0.7i,h0.3,0.5i T3(∂)h0.5,0.55i,h0.15,0.4i,h0.2,0.3i h0.4,0.5i,h0.55,0.6i,h0.3,0.35i The experts present the 3-valued ENS and its values in Tables 3 to 12, based on their assessment of the alternatives against the criteria under discussion. In this example, we should compute the similarity measure of the 3-valued ENSs in Table 3 to Table 12 with the one in Table 1 using method. The similarity measure for pattern recognition Pito Tiis calculated as shown in the table below. Tables 13 and 14 show different values. Table 13. Different values T1 1(x1)T1 2(x1)S1(x1)T2 1(x2)T2 2(x2) (L, P)h0.800316,0.84237i h0.758793,0.762018i h0.509328,0.765444i h0.671653,0.740835i h0.70213,0.791527i (L, Q)h0.719578,0.794162i h0.792615,0.873883i h0.84641,0.579972i h0.916231,0.932051i h0.844602,0.866913i (L, R)h0.895836,0.91482i h0.867642,0.895062i h0.648341,0.923459i h0.888981,0.935176i h0.557979,0.741576i (L, S)h0.781427,0.819973i h0.679001,0.781781i h0.477713,0.693705i h0.854992,0.884033i h0.550141,0.763987i (L, T)h0.939365,0.948507i h0.728212,0.806633i h0.512322,0.737975i h0.869847,0.895939i h0.785653,0.815063i Table 14. Different values S2(x2)T3 1(x3)T3 2(x3)S3(x3)Similarity (L, P)h0.535115,0.893521i h0.652244,0.706035i h0.470374,0.533671i h0.481875,0.652386i0.142049 (L, Q)h0.457751,0.574692i h0.782963,0.865623i h0.700021,0.790201i h0.463118,0.589981i0.143720 (L, R)h0.600575,0.865618i h0.856691,0.877117i h0.665936,0.684441i h0.595246,0.773675i0.132584 (L, S)h0.46486,0.572492i h0.802296,0.841273i h0.709213,0.794079i h0.676507,0.950697i0.14246 (L, T)h0.478567,0.666398i h0.828071,0.890342i h0.73,0.82751i h0.640519,0.901223i0.145903 The pattern recognition similarity measure follows the order T > Q > S > P > R, as shown in the preceding results. As a result, we conclude that the patient Trepresents the best alternative. 6. Conclusion The primary purpose of this study is to provide a three-valued ENS and investigate some of its features. The similarity measure of two 3-valued ENS is addressed, and an example of rel-life is shown. In the future, we shall use the generalized hesitant cubic fuzzy soft sets and generalized hesitant intervalued fuzzy soft sets theories. Acknowledgments: The authors extend their appreciation to the Arab Open University for supporting this work. Neutrosophic Sets and Systems, Vol. 94, 2025 47 Ahmad A. Abubaker, M. Palanikumar, Abdallah Al-Husban, Soft set models applied to pattern recognition via 3-valued extension of neutrosophic soft sets
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