Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 386 https://internationalpubls.com Application of Beal’s Fuzzy Sets in Pattern Recognitions under Similarity Measures Dr.K.Balasubramaniyan1 C.Dilly Rani 2 1 Assistant Professor, Department of Mathematics, (Deputed to Annamalai University, Annamalai nagar, Chidambaram) Arignar Anna Government Arts College Vadachennimalai-636121, Attur, Tamilnadu, India. Email id:kgbalumaths@gmail. 2 Assistant Professor Department of Mathematics, J.J.College of Engineering and Technology (Sowdambikaa Groups of Institutions), Tiruchirappalli-600 009, Tamilnadu, India. Email id: kani21bala@gmail.com Article History: Received: 10-11-2024 Revised: 25-12-2024 Accepted: 12-01-2025 Abstract: In this article, we established to basic collection of operations and construct the abstract properties that can be transmitted to the different models they are in joined with then to rank Beal’s fuzzy set. we obtain the functions of score and accuracy, and formulate aggregate operators to be used with Beal’s fuzzy sets. Alternatively, we develop the successful techniques “aggregative operators” to handle multi-criteria decision making problems in the Beal’s fuzzy set environment. The proposed techniques has been illustrated and analysed through suitable example. Keywords: Fuzzy set, Beal’s fuzzy set, Score function, Accuracy function, Aggregate operators, Decision making. AMS Subject Classification (2010): 20N25, 03F72, 20N99 1. Introduction The problems in the real world are too complicated since it includes ambiguity, uncertainty, or insufficient knowledge. So, decision-makers treat these problems using the methodology of fuzzification which is a vital method to address humanistic systems existing in real-world problems. L.A.Zadeh [19] in 1965, created fuzzy set (FSs) as a generalization of classical collection which are characterized by membership functions from the universe of discourse to the closed interval [0,1] . FS theory is applicable in various areas such as control theory, artificial intelligence, pattern recognition, database systems, and medical diagnosis. The fusion of technology and generalized forms of classical sets is very useful to solve many real-world complex problems that involve vague and uncertain information. A applicable set is defined by its members function from the universe of discourse to the two-point set {0, 1}. Classical set theory is insufficient to take the complex problems involving vague and fuzzy information. To handle the vagueness and uncertainty fuzzy sets (IFS). IFSs are widely used in many fields of mathematics, computer science, management and medical sciences. Szmidt [15] and Kacprzyk [16] and Wang and Xin [17] developed various distances and similarity measures between IFSs and studied applications of distance and of Atanassov [2] paper, several generalizations of IFSs have appeared in the literature. In 2020, a new notion called n- Pythagorean fuzzy sets (n-PFS) was created by Bryniarska [3] as a super class of FFSs and studied Yager’s aggregation operations for n-PFSs. The distance and similarity measure on n-PFSs and their applications in MCDM problems were studied by Liu, Chen, and Peng [10] and Peng and Liu [11] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 387 https://internationalpubls.com Ibrahim and his coworkers [6,7] initiated the study of (3,2)-Fuzzy sets was created by Alshami [1] and presented their applications to MCDM methods. Recently, Jun and his co-workers [8] created the class of Beal’s fuzzy sets (BFSs) as a super class of n-PFSs. In this paper, we establish to basic set of operations and investigate the abstract properties that can be transmitted to the different models they are in connection with then to rank Beal’s fuzzy sets, we obtain the mapping of score and accuracy, and formulate aggregate operators to be applied with Beal’s fuzzy sets. Ultimately, we develop the successful techniques “aggregative operators” to handle multi-criteria decision making problems in the Beal’s fuzzy sets environment. The proposed technique has been explained and analysed through suitable example. 1.1. Extension structure of fuzzy sets 2. Preliminaries Throughout this article X be a universe of discourse N referred to the collection all natural numbers and 𝑚,𝑛 ∈ 𝑁. Definition 2.1 [Fuzzy set] Let X be a nonempty set. Then a fuzzy set on x is defined by 𝐽𝐴: 𝑥 → [0,1]. 𝐽𝐴 is called the membership function. 𝐽𝐴(𝑥) is called the membership grade of X in 𝐽𝐴. we also write 𝐽𝐴 = {(𝑥, 𝐽𝐴(𝑥))/𝑥 ∈ 𝑋}. Example 2.2 Let 𝑥 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝐽𝐴: 𝑥 → [0,1] defined by 𝐽𝐴(𝑎) = 0, 𝐽𝐴(𝑏) = 0.6, 𝐽𝐴(𝑐) = 0.3, 𝐽𝐴(𝑑) = 1. Definition 2.3 [Intuitionistic fuzzy set] : An IFS is an extension of a fuzzy set introduced by K.Atanassov in 1983. An intuitionistic fuzzy set A in X is defined as 𝐴 = {〈𝑥, 𝐽𝐴(𝑥), 𝐾𝐴(𝑥)〉/ 𝑥 ∈ 𝑋}, where 𝐽𝐴: 𝑋 → [0,1] and 𝐾𝐴: 𝑋 → [0,1] are respectively degree of membership and degree of non- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 388 https://internationalpubls.com membership for every 𝑥 ∈ 𝑋 with 0 ≤ 𝐽𝐴(𝑥) + 𝐾𝐴(𝑥) ≤ 1 and 𝜋𝐴(𝑥) = 1 − (𝐽𝐴(𝑥) + 𝐾𝐴(𝑥)) is the degree of indeterminacy of 𝑥 ∈ 𝑋. Sometimes IFS is obviously called bifuzzy set. Definition 2.4 A structure 𝐴 = {〈𝑥, 𝐽𝐴(𝑥), 𝐾𝐴(𝑥)〉/ 𝑥 ∈ 𝑋}, where 𝐽𝐴: 𝑋 → [0,1] and 𝐾𝐴: 𝑋 → [0,1] are respectively degree of membership and degree of non-membership for every 𝑥 ∈ 𝑋 to A is called (i) Intuitionistic fuzzy set in X if 0 ≤ 𝐽𝐴(𝑥) + 𝐾𝐴(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. (ii) Pythagorean fuzzy set in X if 0 ≤ 𝐽𝐴 2(𝑥) + 𝐾𝐴 2(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. (iii) (2, 1) - fuzzy set in X if 0 ≤ 𝐽𝐴 2(𝑥) + 𝐾𝐴(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. (iv) (3, 2) - fuzzy set in X of 0 ≤ 𝐽𝐴 3(𝑥) + 𝐾𝐴 2(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. (v) Fermatean fuzzy set in X if 0 ≤ 𝐽𝐴 3(𝑥) + 𝐾𝐴 3(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. (vi) (4, 2) – fuzzy set in X if 0 ≤ 𝐽𝐴 4(𝑥) + 𝐾𝐴 2(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. (vii) N – fuzzy set when 𝑚,𝑛 ∈ 𝑁 in X if 0 ≤ 𝐽𝐴 𝑚(𝑥) + 𝐾𝐴 𝑛(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋 and 𝑚 ≥ 4 and 𝑛 ≥ 4. (ix) q-runk fuzzy set when 𝑚 = 𝑛 = 𝑞 i.e., 0 ≤ 𝐽𝐴 𝑞(𝑥) + 𝐾𝐴 𝑞(𝑥) ≤ 1 for all 𝑥 ∈ 𝑋. In what follows, we compare Beal’s fuzzy set with the previous generalization of intuitionstic fuzzy sets. Proposition 2.5 (i) Every bifuzzy set is a Beal’s fuzzy set. (ii) If 𝑚 ≥ 2 and 𝑛 ≥ 2, then a Pythagorean fuzzy set is a Beal’s fuzzy set. (iii) If 𝑚 ≥ 3 and 𝑛 ≥ 3, then a Fermatean fuzzy set is a Beal’s fuzzy set. (iv) If 𝑚 ≥ 𝑞 and 𝑛 ≥ 𝑞, then a q-runk ortho pair fuzzy set is a Beal’s fuzzy set. (v) If 𝑚 ≥ 2 and 𝑛 ≥ 3, then a (2, 3) – fuzzy set is a Beal’s fuzzy set. Proof: The proof is straight forward. Remark: For all 𝑎, 𝑏 ∈ [0,1], we have 𝑎 + 𝑏 ≤ 1 ⟹ 𝑎2 + 𝑏2 ≤ 1 ⟹ 𝑎3 + 𝑏3 ≤ 1 ⟹ 𝑎4 + 𝑏4 ≤ 1⟹ 𝑎𝑛 + 𝑏𝑛 ≤ 1 ⟹ 𝑎𝑚 + 𝑏𝑛 ≤ 1, 𝑚 ≥ 4 𝑎𝑛𝑑 𝑛 ≥ 4 from Definition 2.1. Definition 2.6 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑋) 𝑎𝑛𝑑 𝑥 ∈ 𝑋. Then the expression ∏ (𝑥) = (1− 𝐽𝐴 𝑚(𝑥) − 𝐾𝐴 𝑛(𝑥)) 2 𝑚+𝑛 𝐴 is said to be the degree of indeterminacy of x to A. Remark: Clearly, ∏ (𝑥) (𝑚+𝑛) 2 𝐴 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. Definition 2.7 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑋). Then the complement of A, denoted by 𝐴𝑐, is defined as follows: 𝐴𝑐 = (𝐽𝐴 𝑐 , 𝐾𝐴 𝑐) = (𝐾𝐴 𝑛 𝑚 , 𝐽𝐴 𝑛 𝑚). Theorem 2.8 Let 𝐴 = (𝐽𝐴, 𝐾𝐴), 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋). Then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 389 https://internationalpubls.com (𝑖) 𝐴𝑐 ∈ 𝐵𝑚 𝑛 (𝑋) (𝑖𝑖)(𝐴𝑐)𝑐 = 𝐴 (𝑖𝑖𝑖)(𝐴1 ∪ 𝐴2) 𝑐 = 𝐴1 𝑐 ∩ 𝐴2 𝑐 (𝑖𝑣)(𝐴1 ∩ 𝐴2) 𝑐 = 𝐴1 𝑐 ∪ 𝐴2 𝑐. Proof: (i) Since, 𝐴𝑐 = (𝐽𝐴 𝑐 , 𝐾𝐴 𝑐) = (𝐾𝐴 𝑚 𝑛 , 𝐽𝐴 𝑚 𝑛), we have 0 ≤ 𝐽𝐴𝑐 𝑚 + 𝐾𝐴𝑐 𝑛 = ((𝐾𝐴) 𝑛 𝑚) 𝑚 + ((𝐽𝐴) 𝑚 𝑛) 𝑛 = 𝐾𝐴 𝑛 + 𝐽𝐴 𝑚 = 𝐽𝐴 𝑚 + 𝐾𝐴 𝑛 ≤ 1 Therefore, 𝐴𝑐 ∈ 𝐵𝑚 𝑛 (𝑋). (ii) Easy and left to the reader. (iii) We have (𝐴1 ∪ 𝐴2) 𝑐 = (max {𝐽𝐴1, 𝐽𝐴2) ,min {𝐾𝐴1, 𝐾𝐴2}) 𝑐 = (min { 𝐾𝐴1 𝑛 𝑚 , 𝐾𝐴2 𝑛 𝑚 } ,max{ 𝐽𝐴1 𝑚 𝑛 , 𝐽𝐴2 ( 𝑚 𝑛 )}) = (𝐾𝐴1 𝑛 𝑚 , 𝐽𝐴1 𝑚 𝑛 ) ∩ (𝐾𝐴2 𝑛 𝑚 , 𝐽𝐴2 ( 𝑚 𝑛 )) = 𝐴1 𝑐 ∩ 𝐴2 𝑐. (i) Follows by a similar process to (iii). Definition 2.9 Let 𝐴 = (𝐽𝐴 , 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑋). Then we need of terms and the possibility measure on A are called as follows: (𝑖) □A = (𝐽𝐴 , (1− 𝐽𝐴 𝑚) 1 𝑛 ). (𝑖𝑖)◊A = ((1− 𝐾𝐴 𝑛) 1 𝑚 , 𝐾𝐴). Example 2.10 Let 𝑋 = {𝑥} 𝑎𝑛𝑑 𝐴 = {〈0.75,0.8〉} ∈ 𝐵6 5(𝑥). Then ∏ (𝑥) = 0.6321432,𝐴 □A ={{〈0.75,0.8〉} , ◊A = 〈𝑥, 0.75932, 0.8〉}, 𝐴𝑐 = {〈𝑥, 0.8,0.723〉}. Theorem 2.11 𝐼𝑓 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑋), Then (𝑖) □A ∈ 𝐵𝑚 𝑛 (𝑋), (𝑖𝑖)◊A ∈ 𝐵𝑚 𝑛 (𝑋), Proof: (i) Follows on noting that: 𝐽□A 𝑚 + 𝐾□A 𝑚 = 𝐽𝐴 𝑚 + ((1− 𝐽𝐴 𝑚) 1 𝑛) 𝑛 = 𝐽𝐴 𝑚 + (1 − 𝐽𝐴 𝑚) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 390 https://internationalpubls.com = 1. (ii)Follows on noting that 𝐽◊A 𝑚 + 𝐾◊A 𝑛 = ((1− 𝐾𝐴 𝑛) 1 𝑚) 𝑚 + 𝐾𝐴 𝑛 = (1− 𝐾𝐴 𝑛) + 𝐾𝐴 𝑛 = 1. Definition 2.12 Let 𝐴 = (𝐽𝐴, 𝐾𝐴), 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋) and 𝑞 ∈ 𝑁. Then the operations 𝐴1⊕𝐴2, 𝐴1 ⨂𝐴2 , 𝑞𝐴, and 𝐴𝑞are defined as follows (i) 𝐴1⊕𝐴2 = (𝐽𝐴1 𝑚 + 𝐽𝐴2 𝑚 − 𝐽𝐴1 𝑚𝐽𝐴2 𝑚 , 𝐾𝐴1 𝑛𝐾𝐴2 𝑛), (ii) 𝐴1 ⨂𝐴2 = (𝐽𝐴1 𝑚𝐽𝐴2 𝑚 , 𝐾𝐴1 𝑛 + 𝐾𝐴2 𝑛 − 𝐾𝐴1 𝑛𝐾𝐴2 𝑛) (iii) 𝑞𝐴 = (1− (1 − 𝐽𝐴 𝑚)𝑞, 𝐾𝐴 𝑛), (iv) 𝐴𝑞 = (𝐽𝐴 𝑚𝑞, 1− (1− 𝐾𝐴 𝑛)𝑞). Theorem 2.13 Let 𝐴 = (𝐽𝐴, 𝐾𝐴), 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋) and 𝑞 ∈ 𝑁. Then (i) 𝐴1⊕𝐴2 ∈ 𝐵𝑚 𝑛 (𝑋), (ii) 𝐴1 ⨂𝐴2 ∈ 𝐵𝑚 𝑛 (𝑋), (iii) 𝐴1 ∪ 𝐴2 ∈ 𝐵𝑚 𝑛 (𝑋), (iv) 𝐴1 ∩ 𝐴2 ∈ 𝐵𝑚 𝑛 (𝑋), (v) 𝑞𝐴 ∈ 𝐵𝑚 𝑛 (𝑋), (vi) 𝐴𝑞 ∈ 𝐵𝑚 𝑛 (𝑋). Proof: (i) Since, 𝐴1⊕𝐴2 = (𝐽𝐴1 𝑚 + 𝐽𝐴2 𝑚 − 𝐽𝐴1 𝑚𝐽𝐴2 𝑚 , 𝐾𝐴1 𝑛𝐾𝐴2 𝑛), we have, 𝐽𝐴1⊕𝐴2 𝑚 + 𝐾𝐴1⊕𝐴2 𝑛 = (𝐽𝐴1 𝑚 + 𝐽𝐴2 𝑚 − 𝐽𝐴1 𝑚𝐽𝐴2 𝑚) 𝑚 + (𝐾𝐴1 𝑛𝐾𝐴2 𝑛) 𝑛 = 𝐽𝐴1 𝑚(1− 𝐽𝐴2 𝑚) + 𝐽𝐴2 𝑚 + (𝐾𝐴1 𝑛𝐾𝐴2 𝑛 ) 𝑛 ≥ 0 and 𝐽𝐴1⊕𝐴2 𝑚 + 𝐾𝐴1⊕𝐴2 𝑛 = (𝐽𝐴1 𝑚 + 𝐽𝐴2 𝑚 − 𝐽𝐴1 𝑚𝐽𝐴2 𝑚) 𝑚 + (𝐾𝐴1 𝑛𝐾𝐴2 𝑛) 𝑛 ≤ ((1 − 𝐾𝐴1 𝑛 ) + (1 − 𝐾𝐴2 𝑛 ) − (1− 𝐾𝐴1 𝑛 )(1− 𝐾𝐴2 𝑛 )) 𝑚 + (𝐾𝐴1 𝑛𝐾𝐴2 𝑛) 𝑛 = (1− 𝐾𝐴1 𝑛𝐾𝐴1 𝑛 ) 𝑚 + (𝐾𝐴1 𝑛𝐾𝐴2 𝑛 ) 𝑛 ≤ 1 because 0 ≤ 𝐾𝐴1 𝑛𝐾𝐴2 𝑛 ≤ 1 and 𝑚,𝑛 ≥ 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 391 https://internationalpubls.com Hence, 𝐴1⊕𝐴2 ∈ 𝐵𝑚 𝑛 (𝑋), (ii) Similar to (i). (iii)Suppose max {𝐾𝐴1 , 𝐾𝐴2} ≤ 𝐾𝐴1 , we have 0 ≤ 𝐽𝐴1∪𝐴2 𝑚 + 𝐾𝐴1∪𝐴2 = (max{𝐽𝐴1 , 𝐽𝐴2}) 𝑚 + ((min{𝐾𝐴1 , 𝐾𝐴2}) 𝑛 ≤ 𝐽𝐴1 𝑚 + 𝐾𝐴1 𝑛 ≤ 1 Suppose now max {𝐽𝐴1, 𝐽𝐴2} = 𝐽𝐴2 . Since min{𝐾𝐴1 , 𝐾𝐴2} ≤ 𝐾𝐴2 , we have 0 ≤ 𝐽𝐴1∪𝐴2 𝑚 + 𝐾𝑛𝐴1∪𝐴2, = (max{𝐽𝐴1 , 𝐽𝐴2}) 𝑚 + ((min{𝐾𝐴1 , 𝐾𝐴2}) 𝑛 ≤ 𝐽𝐴1 𝑚 + 𝐾𝐴1 𝑛 ≤ 1 Thus, the proof of (iii) is completed. (iv) Similar to (iii) (v) Since 𝐴 ∈ 𝐵𝑚 𝑛 (𝑋), we have 0 ≤ 𝐽𝐴 𝑚 ≤ 1 and 0 ≤ 𝐾𝐴 𝑛 ≤ 1. 𝑞𝐴 = (1− (1 − 𝐽𝐴 𝑚)𝑞 , 𝐾𝐴 𝑛𝑞), we have ≤ 𝐽𝑞𝐴 𝑚 + 𝐾𝑞𝐴 𝑛 = (1− (1 − 𝐽𝐴 𝑚)𝑞)𝑚 + (𝐾𝐴 𝑛𝑞) 𝑛 ≤ (1− (1− 𝐽𝐴 𝑚)𝑞)𝑚 + (𝐾𝐴 𝑛𝑞) 𝑛 ≤ (1− (𝐾𝐴 𝑛𝑞) 𝑚 ) + (𝐾𝐴 𝑛𝑞) 𝑛 ≤ 1 Theorem 2.14 Let 𝐴 = (𝐽𝐴, 𝐾𝐴), 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋) and 𝑞, 𝑞1, 𝑞2 ∈ 𝑁. Then (𝑖) 𝐴1⊕𝐴2 = 𝐴2⊕𝐴1 (𝑖𝑖) 𝐴1 ⨂𝐴2 = 𝐴2 ⨂𝐴1 (𝑖𝑖𝑖) 𝑞(𝐴1⊕𝐴2) = 𝑞𝐴1⊕𝑞𝐴2 (𝑖𝑣)(𝑞1 + 𝑞2)𝐴 = 𝑞1𝐴 ⊕ 𝑞2𝐴, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 392 https://internationalpubls.com (𝑣)(𝐴1⊗𝐴2) 𝑞 = 𝐴1 𝑞 ⊗ 𝐴2 𝑞 (𝑣𝑖) 𝐴𝑞1 ⊗𝐴𝑞2 = 𝐴𝑞1+𝑞2. Proof: The proof of above can easily understand. Theorem 2.15 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑋) and 𝑥 ∈ 𝑋. If ∏ (𝑥) = 0,𝐴 then ∏ (𝑥) = 0𝐴𝑞 for all 𝑞 ∈ 𝑁. Proof: Since ∏ (𝑥) = (1− 𝐽𝐴 𝑚(𝑥) − 𝐾𝐴 𝑛(𝑥)) 2 𝑚+𝑛 𝐴 we have ∏ (𝑥) = 0 ⟹ (1− 𝐽𝐴 𝑚(𝑥) − 𝐾𝐴 𝑛(𝑥)) 2 𝑚+𝑛 𝐴 ⟹ 𝐽𝐴(𝑥) + 𝐾𝐴 𝑛(𝑥) = 1 ⟹ 𝐽𝐴(𝑥) = 1 − 𝐾𝐴 𝑛(𝑥). By using this result, we have 𝐴𝑞 = (𝐽𝐴 𝑚𝑞, 1− (1− 𝐾𝐴 𝑛)𝑞) = (𝐽𝐴 𝑚𝑞, 1− (1− 𝐽𝐴 𝑚)𝑞) = (𝐽𝐴 𝑚𝑞, 1− 𝐽𝐴 𝑚𝑞) Hence, ∏ (𝑥) = (1 − (𝐽𝐴 𝑚𝑞 (𝑥)) 𝑚 − (1 − 𝐽𝐴 𝑚𝑞 (𝑥)) 𝑛 ) 2 𝑚+𝑛 𝐴𝑞 ⟹ ∏ (𝑥) = (1− (𝐽𝐴 𝑚𝑞 (𝑥)) 𝑚 − (1 − 𝐽𝐴 𝑚𝑞 (𝑥)) 𝑛 ) 2 𝑚+𝑛 𝐴𝑞 ⟹ ∏ (𝑥) = 0𝐴𝑞 Theorem 2.16 Let 𝐴 = (𝐽𝐴, 𝐾𝐴 ) ∈ 𝐵𝑚 𝑛 (𝑥) and 𝑥 ∈ 𝑋 𝑎𝑛𝑑 𝑞, 𝑞1, 𝑞2 ∈ 𝑁. Then (𝑖)𝑞1 ≥ 𝑞2 ⟹ 𝐴𝑞1 ⊂ 𝐴𝑞2 (𝑖𝑖) 𝑞1 ≥ 𝑞2 ⟹ 𝑞2𝐴 ⊂ 𝑞1𝐴 1− 𝐽𝐴2 𝑚 ≤ 1 − 𝐽𝐴1 𝑚 ⟹ (1 − 𝐽𝐴2 𝑚) 𝑞 ≤ (1− 𝐽𝐴1 𝑚) 𝑞 ⟹ (1− (1 − 𝐽𝐴1 𝑚) 𝑞 ) ≤ (1− (1− 𝐽𝐴2 𝑚) 𝑞 ) ⟹ 𝐽𝑞𝐴1 ≤ 𝐽𝑞𝐴2 and 𝐾𝐴1 ≥ 𝐾𝐴2 ⟹𝐾𝐴1 𝑛 ≥ 𝐾𝐴2 𝑛 ⟹ 𝐾𝐴1 𝑛𝑞 ≥ 𝐾𝐴2 𝑛𝑞 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 393 https://internationalpubls.com ⟹ 𝐾𝑞𝐴1 ≥ 𝐾𝑞𝐴2 Hence, 𝑞𝐴1 ⊂ 𝑞𝐴2. (i)Similar to that of (i) (ii)Follow since: 𝐴1 ∪ 𝐴2 = (max{ 𝐽𝐴1, 𝐽𝐴2} ,min{𝐾𝐴1, 𝐾𝐴2}) and (𝐴1 ∪ 𝐴2) 𝑞 = (𝐽𝐴1∪𝐴2, 𝐾𝐴1∪𝐴2 ) = ((max {𝐽𝐴1, 𝐽𝐴2}) 𝑚𝑞 , 1− (1 − (min{𝐾𝐴1, 𝐾𝐴2}) 𝑛 ) 𝑞 ) = (max{ {𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞}, 1− (1−min{𝐾𝐴1 𝑛, 𝐾𝐴2 𝑛}) 𝑞) = (max{ {𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞}, 1− max{1− 𝐾𝐴1 𝑛, 1− 𝐾𝐴2 𝑛}) 𝑞) = (max{ {𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞}, 1− max{1− 𝐾𝐴1 𝑛, 1− 𝐾𝐴2 𝑛}) 𝑞) = (max{ {𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞}, 1 − (max{1 − 𝐾𝐴1 𝑛) 𝑞 , (1 − 𝐾𝐴2 𝑛) 𝑞 }) . Proof: (i) Since 𝐴𝑞1 = (𝐽𝐴 𝑚𝑞1 , 1− (1− 𝐾𝐴 𝑛)𝑞1) 𝐴𝑞2 = (𝐽𝐴 𝑚𝑞2 , 1− (1− 𝐾𝐴 𝑛)𝑞2) we have, 𝑞1 ≥ 𝑞2 ⟹ 𝐽𝐴 𝑞2 ≥ 𝐽𝐴 𝑞1 𝑎𝑛𝑑 (1 − 𝐾𝐴 𝑛)𝑞1 ≤ (1− 𝐾𝐴 𝑛)𝑞2 ⟹ 𝐽𝐴 𝑚𝑞2 ≥ 𝐽𝑚𝐴 𝑞1 𝑎𝑛𝑑 1 − (1− 𝐾𝐴 𝑛)𝑞2 ≤ (1− 𝐾𝐴 𝑛)𝑞1 ⟹ 𝐽𝐴 𝑞2 ≥ 𝐽𝐴 𝑞1 𝑎𝑛𝑑 𝐾𝐴 𝑞2 ≤ 𝐾𝐴 𝑞1. Hence 𝐴𝑞1 ⊂ 𝐴𝑞2. (ii) Similar to that of (i). Theorem 2.17 Let 𝐴1 = (𝐽𝐴1 , 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋) 𝑎𝑛𝑑 𝑞 ∈ 𝑁. Then (𝑖) 𝐴1 ⊂ 𝐴2 ⟹ 𝑞𝐴1 ⊂ 𝑞𝐴2 (𝑖𝑖) 𝐴1 ⊂ 𝐴2 ⟹ 𝐴1 𝑞 ⊂ 𝐴2 𝑞 (𝑖𝑖𝑖)(𝐴1 ∪ 𝐴2) 𝑞 ⟹ 𝐴1 𝑞 ∪ 𝐴2 𝑞 (𝑖𝑣) 𝑞(𝐴1 ∪ 𝐴2) ⟹ 𝑞𝐴1 ∪ 𝑞𝐴2 (𝑣)(𝐴1 ∩ 𝐴2) ⟹ 𝐴1 𝑞 ∩ 𝐴2 𝑞 (𝑣𝑖)𝑞(𝐴1 ∩ 𝐴2) ⟹ 𝑞𝐴1 ∩ 𝑞𝐴2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 394 https://internationalpubls.com Proof: Since 𝐴1 ⊂ 𝐴2, we have 𝐽𝐴1 ≤ 𝐽𝐴2 → 𝐽𝐴1 𝑚 ≤ 𝐽𝐴2 𝑚 = max{ 𝐽𝐴2 𝑚𝑞 , 𝐽𝐴2 𝑚𝑞} ,min{ (1− (1 − 𝐾𝐴1 𝑛 ) 𝑞 )} = 𝐴1 𝑞 ∪ 𝐴2 𝑞 (iv) Follows since 𝐴1 ∩ 𝐴2 = (min{𝐽𝐴1, 𝐽𝐴2} ,max{𝐾𝐴1, 𝐾𝐴2}) 𝑎𝑛𝑑 (𝐴1 ∩ 𝐴2) 𝑞 = (𝐽𝐴1∩𝐴2 , 𝐾𝐴1∩𝐴2) = ((min{𝐽𝐴1, 𝐽𝐴2}) 𝑚𝑞 , 1− (1− {max{𝐾𝐴1, 𝐾𝐴2}) 𝑛 ) 𝑞 } = ((min{𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞}) , 1 − (1−max{𝐾𝐴1 𝑛 , 𝐾𝐴2 𝑛 }) 𝑘 ) = (min{𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞 }, 1 − (min(1− 𝐾𝐴1 𝑛 , 1− 𝐾𝐴2 𝑛 ) 𝑞 )) = (min{𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞 } ,max{1− (1 − 𝐾𝐴1 𝑛)𝑞, (1− 𝐾𝐴2 𝑛 ) 𝑞 }) = (𝑚𝑖𝑛{𝐽𝐴1 𝑚𝑞, 𝐽𝐴2 𝑚𝑞 },max{1− (1 − 𝐾𝐴1 𝑛 ) 𝑞 (1− 𝐾𝐴2 𝑛 ) 𝑞 }) = 𝐴1 𝑘 ∩ 𝐴2 𝑘 (v) The same that of result (iii) (vi) Some way that 𝑓(𝑣) Definition 2.18 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑥) and 𝛼 ∈ [0,1]. Then the operator 𝐺𝛼(𝐴) is expressed as follows: 𝐺𝛼(𝐴) = ((𝐽𝐴 𝑚 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 , 𝐽𝐴 𝑛 + ((1− 𝛼)∏ 𝑚+𝑛 2 ) 1 𝑛 ) Theorem 2.19 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑥) and 𝛼, 𝛽 ∈ [0,1]. Then (𝑖)𝛼 ≤ 𝛽 ⟹ 𝐺𝛼(𝐴) ⊂ 𝐺𝛽1(𝐴) (𝑖𝑖)𝐺0(𝐴) = □ (𝑖𝑖𝑖)𝐺1(𝐴) = ◊A Proof: (i) Immediate result to obtained (ii) Since, 𝐺0(𝐴) = ((𝐽𝐴 𝑚 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 (𝐾𝐴 𝑛 + (1 − 𝛼)∏ 𝑚+𝑛 2 )( 1 𝑛 )) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 395 https://internationalpubls.com = (𝐽𝐴, (𝐾𝐴 𝑛 + ∏ 𝑚+𝑛 2 ) 1 𝑛 ) = (𝐽𝐴, (𝐾𝐴 + 1− 𝐽𝐴 𝑚 − 𝐾𝐴 𝑛) 1 𝑛) = (𝐽𝐴, (1− 𝐽𝐴 𝑚) 1 𝑛) = □A The proof is completed. It follows on nothing that, 𝐺1(𝐴) = ( (𝐽𝐴 𝑚 + (1)∏ 𝑚+𝑛 2 ) 1 𝑚 , 𝐾𝐴 + (1− 1)∏ 𝑚+𝑛 2 ) 1 𝑛 ) = (( 𝐽𝐴 𝑚 + ∏ 𝑚+𝑛 2 ) 1 𝑚 , (𝐾𝐴 𝑛) 1 𝑛 ) = ((𝐽𝐴 𝑚 + (1 − 𝐽𝐴 𝑚 − 𝐾𝐴 𝑛) 1 𝑚 , 𝐾𝐴)) = ((1− 𝐾𝐴 𝑛) 1 𝑚 , 𝐾𝐴) = ◊A Definition 2.20 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑥) and 𝛼, 𝛽 ∈ [0,1] where 𝛼 + 𝛽 ≤ 1. we define the operator 𝐻𝛼,𝛽(𝐴) as 𝐻𝛼,𝛽(𝐴) = ((𝐽𝐴 𝑚 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 , (𝐾𝐴 𝑛 + 𝛽∏ 𝑚+𝑛 2 ) 1 𝑛 ) Theorem 2.21 For any 𝐴 = (𝐽𝐴, 𝐾𝐴) ∈ 𝐵𝑚 𝑛 (𝑥) and 𝛼, 𝛽 ∈ [0,1] where 𝛼 + 𝛽 ≤ 1. we have (𝑖)𝐻𝛼,𝛽(𝐴) ∈ 𝐵𝑚 𝑛 (𝑥) (𝑖𝑖) 0 ≤ 𝑡 ≤ 𝛼 ⟹ 𝐻𝛼,𝛽(𝐴) ⊂ 𝐻𝛼,𝛽(𝐴) (𝑖𝑖𝑖) 0 ≤ 𝑡 ≤ 𝛽 ⟹ 𝐻𝛼,𝛽(𝐴) ⊂ 𝐻𝛼,𝑡(𝐴) (𝑖𝑣)𝐺𝛼(𝐴) = 𝐻𝛼 ,1−𝛼 (𝐴) (𝑣)□A = 𝐻0,1 (𝐴) (𝑣𝑖) ◊A = H1,0 (A) (𝑣𝑖𝑖)𝐻𝛼,3 𝑐 (𝑀𝑐) = 𝐻3,𝛼(𝐴) Proof: (i) Follows since, 𝐽1+𝛼,3 𝑚 (𝐴) + 𝐾1+𝛼,3(𝐴) = ((𝐽𝐴 𝑚 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 ) 𝑚 + ((𝐾𝐴 𝑛 + 𝛽∏ 𝑚+𝑛 2 ) 1 𝑛 ) 𝑛 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 396 https://internationalpubls.com = 𝐺𝛼(𝐴) (v) Follows since 𝐻𝛼,1−𝛼(𝑥) = 𝐺𝛼(𝐴) ⟹ 𝐻0,1(𝐴) = 𝐺0(𝐴). ⟹𝐻0,1(𝐴) = □A (vi) Follows on nothing that, 𝐻𝛼,1−2(𝐴) = 𝐺𝛼(𝐴) ⟹ 𝐻1,0 (𝐴) = 𝐺1 (𝐴) ⟹𝐻1,0(𝐴) = ◊ A (vii)Since 𝑀𝑐 = (𝐾𝐴 𝑛 𝑚 , 𝐽𝑛 𝑚 𝑛 ) we have 𝐻𝛼,𝛽(𝑀 𝑐 ) = ((𝐾𝐴 𝑛 𝑚) 𝑚 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 , ((𝐽𝐴 𝑚 𝑛) 𝑛 + 𝛽∏ 𝑚+𝑛 2 ) 𝟏 𝒏 = ((𝐾𝐴 𝑛 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 (𝐽𝐴 𝑚 + 𝛽∏ 𝑚+𝑛 2 ) 1 𝑛 ) 𝑑𝐵𝑚𝑛 𝐻 (𝐴1, 𝐴2) = 1 2 ∑{|𝐽𝐴1 (𝑥𝑖) − 𝐽𝐴2 (𝑥𝑖)| 𝑛 𝑖=1 + |𝐾𝐴1 (𝑥𝑖) − 𝐾𝐴2 (𝑥𝑖)| + |∏ (𝑥𝑖) − ∏ (𝑥𝑖)|} 𝐴2𝐴1 The normalize Hamming distance is explained as 𝑑𝐵𝑚𝑛 𝐻 (𝐴1, 𝐴2) = 1 2𝑟 ∑ {|𝐽𝐴1(𝑥𝑖) − 𝐽𝐴2(𝑥𝑖)| 𝑛 𝑖=1 + |𝐾𝐴1(𝑥𝑖) − 𝐾𝐴2(𝑥𝑖)| + |∏ (𝑥𝑖) − ∏ (𝑥𝑖𝐴2𝐴1 )| The Euclidean distance is defined as 𝑑𝐵𝑚𝑛 𝐻 (𝐴1, 𝐴2) = √ 1 2 ∑ {(𝐽𝐴1(𝑥𝑖) − 𝐽𝐴2(𝑥2)) 2 + (𝐾𝐴1(𝑥𝑖) − 𝐾𝐴2(𝑥𝑖)) 2 + (∏ (𝑥𝑖) − ∏ (𝑥𝑖 ))2𝐴2𝐴1 𝑟 𝑖=0 } The normalized Euclidean distance is defined as 𝑑𝐵𝑚𝑛 𝐻 (𝐴1, 𝐴2) = √ 1 2𝑟 ∑ {(𝐽𝐴1(𝑥𝑖) − 𝐽𝐴2(𝑥𝑖)) 2 + (𝐾𝐴1(𝑥𝑖) − 𝐾𝐴2(𝑥𝑖)) 2 + (∏ (𝑥𝑖) − ∏ (𝑥𝑖 )) 2 𝐴2𝐴1 𝑟 𝑖=0 } and so, (𝐻𝛼,3(𝑀 𝑐))𝑐 = (((𝐽𝐴 𝑚 + 𝛽∏ 𝑚+𝑛 2 ) 1 𝑛 ) 𝑚 𝑛 + ((𝐾𝐴 𝑛 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑚 ) 𝑚 𝑛 ) = ((𝐽𝑎 𝑚 + 𝛽∏ 𝑚+𝑛 2 ) 1 𝑚 , (𝐾𝐴 𝑛 + 𝛼∏ 𝑚+𝑛 2 ) 1 𝑛 ) = 𝐻𝛽,𝛼 (𝐴). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 397 https://internationalpubls.com Hence the proof. 3. Various types of distance function over Beal’s fuzzy set Definition 3.1 Let 𝑋 = { 𝑥1, 𝑥2, 𝑥3… , 𝑥𝑟} be a whole of discourse and 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2), 𝐴3 = (𝐽𝐴3, 𝐾𝐴3) ∈ 𝐵𝑚 𝑛 (𝑋), the distance function 𝑑: 𝐵𝑚 𝑛 (𝑋) × 𝐵𝑚 𝑛 (𝑋) → [0,1] is defined as (𝑖)0 ≤ 𝑑(𝐴1, 𝐴2) ≤ 1 (Boundness) (𝑖𝑖)𝑑(𝐴1, 𝐴2) = 0 ⟹ 𝐴1 = 𝐴2 (Separable) (𝑖𝑖𝑖)𝑑(𝐴1, 𝐴2) = 𝑑(𝐴2, 𝐴1) (Symmetric) (𝑖𝑣)𝑑(𝐴1, 𝐴3) + 𝑑(𝐴2, 𝐴3) ≥ 𝑑(𝐴1, 𝐴2) (Triangle Inequality) Definition 3.2 Let 𝑋 = (𝑥1, 𝑥2} be a whole of discourse and 𝐴1, 𝐴2, 𝐴3 ∈ 𝐵7 5(𝑋), where 𝐴1 = {〈𝑥1, 0.5,0.6〉, 〈𝑥2, 0.4, 0.5〉}, 𝐴2 = {〈𝑥1, 0.8,0.3〉, 〈𝑥2, 0.9, 0.5〉}, 𝐴3 = {〈𝑥1, 0.6,0.8〉, 〈𝑥2, 0.8, 0.5〉}, Find the following: (1) Hamming Distance (2) Normalized Hamming Distance (3) Euclidean Distance (4) Normalized Euclidean Distance 𝑑𝐵7 5(𝐴1, 𝐴2) = 0.6 𝑑𝐵7 5 (𝐴3, 𝐴1) = 0.5 𝑑𝐸𝐵7 5 (𝐴1, 𝐴2) = 0.3 𝑑𝐵7 5 (𝐴3, 𝐴1) = 0.2 𝑑𝑛𝐵𝑚𝑛 5 (𝐴1, 𝐴2) = 0.6 𝑑𝐵7 5 (𝐴3, 𝐴1) = 0.5 𝑑𝐻𝐵7 5 (𝐴2, 𝐴3) = 0.5 𝑑𝑛𝐵7 5 (𝐴3, 𝐴1) = 0.3 𝑑𝐻𝐵7 5 (𝐴2, 𝐴3) = 0.2 𝑑𝐸𝐵7 5 = 0.5 𝑑𝑛𝐵7 5 = 0.4 Example 3.3 Let 𝑋 = {𝑥1, 𝑥2, 𝑥3, 𝑥4} and 𝐴1, 𝐴2, 𝐴3 ∈ 𝐵8 6(𝑋) where 𝐴1 = {〈𝑥1, 0.2,0.7〉, 〈𝑥2, 0.9, 0.7〉}, 𝐴2 = {〈𝑥1, 0.9,0.3〉, 〈𝑥2, 0.5, 0.5〉}, 𝐴3 = {〈𝑥1, 0.6,0.7〉, 〈𝑥2, 0.8, 0.5〉}, 𝑑𝐵8 5(𝐴1, 𝐴2) = 0.9 𝑑𝐻𝐵8 5(𝐴1, 𝐴2) = 0.3 𝑑𝑛 𝐵8 5(𝐴1, 𝐴2) = 0.6 𝑑𝐸𝐵8 5(𝐴1, 𝐴2) = 0.8 Definition 3.4 Let 𝑋 = { 𝑥1, 𝑥2, 𝑥3… , 𝑥𝑟} be a whole of discourse and 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2), 𝐴3 = (𝐽𝐴3, 𝐾𝐴3) ∈ 𝐵𝑚 𝑛 (𝑋). The similarity measure 𝑆: 𝐵𝑚 𝑛 (𝑋) × 𝐵𝑚 𝑛 (𝑋) → [0,1] is expressed as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 398 https://internationalpubls.com (𝑖)0 ≤ 𝑆(𝐴1, 𝐴2) ≤ 1 (Boundedness) (𝑖𝑖)𝑆(𝐴1, 𝐴2) = 1 ⟺ 𝐴1 = 𝐴2 (Separability) (𝑖𝑖𝑖)𝑆(𝐴1, 𝐴2) = 𝑆(𝐴2, 𝐴1) (Symmetric) (𝑖𝑣)𝑆(𝐴1, 𝐴3) + 𝑆(𝐴2, 𝐴1) ≥ 𝑆(𝐴1, 𝐴2) (Triangle Inequality) Using distance functions and similarity measures, the below of result proved. Theorem 3.5 Let𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋). If 𝑑(𝐴1, 𝐴2) is a distance measure between Beal’s fuzzy sets 𝐴1 and 𝐴2, then 𝑆(𝐴1, 𝐴2) = 1− 𝑑(𝐴1, 𝐴2) is a similarity measure of 𝐴1 and 𝐴2. Theorem 3.6 Let 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋). Suppose 𝐴1 ⊂ 𝐴2 ⊂ 𝐴3, then (𝑖)𝑑(𝐴1, 𝐴2) ≥ 𝑑(𝐴1, 𝐴2)𝑎𝑛𝑑 (𝑖𝑖)𝑑(𝐴1, 𝐴3) ≥ 𝑑(𝐴2, 𝐴3) (𝑖𝑖𝑖)𝑆(𝐴1, 𝐴3) ≤ 𝑆(𝐴1, 𝐴2)𝑎𝑛𝑑 (𝑖𝑣)𝑆(𝑀1, 𝑀3) ≤ 𝑆(𝑀2, 𝑀3). Theorem 3.7 Let 𝐴1 = (𝐽𝐴1, 𝐾𝐴1), 𝐴2 = (𝐽𝐴2, 𝐾𝐴2) ∈ 𝐵𝑚 𝑛 (𝑋). Then (𝑖)𝑑(𝐴1, 𝐴2) = 𝑑(𝐴1 𝑐, 𝐴2 2) (𝑖𝑖)𝑆(𝐴1, 𝐴2) = 𝑆(𝐴1 𝑐, 𝐴2 𝑐). Note. The two similarity measures are defined are defined as follows 𝑆1 = 1− 𝑑𝐵𝑚𝑛 𝑛 𝐻(𝐴1, 𝐴2), 𝑆2 = 1− 𝑑𝐵𝑚𝑛 𝑛 𝐸(𝐴1, 𝐴2) Note: 𝐴1, 𝐴2 ∈ 𝐵𝑚 𝑛 (𝑋), 𝑋 = {𝑥1, 𝑥2}, 𝑆1 ∈ (𝐴1, 𝐴2), 𝑆2 ∈ (𝐴1, 𝐴2) , 𝐴1, 𝐴2, 𝐴3 ∈ 𝐵𝑚 𝑛 (𝑋) 𝑆1(𝐴1, 𝐴2) = 0.6, 𝑆2(𝐴1, 𝐴2) = 0.8, 𝑆1(𝐴1, 𝐴3) = 0.6, 𝑆2(𝐴1, 𝐴3) = 0.8. In order to rank Beal’s fuzzy sets, we give the score mapping and accuracy, mapping of the Beal’s fuzzy set. Definition 3.8 The score mapping function of a Beal’s fuzzy set 𝐴 = (𝐽𝐴, 𝐾𝐴) can be given as 𝑆(𝐴) = 𝐽𝐴 𝑛 − 𝐾𝐴 𝑚. The accuracy function of a Beal’s fuzzy set 𝐴 = (𝐽𝐴, 𝐾𝐴) can be given as 𝛼 (A) = 𝐽𝐴 𝑛 + 𝐾𝐴 𝑚. Example 3.9 Consider the Beal’s fuzzy set 𝐴 = (0.62, 0.54) then the score function is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 399 https://internationalpubls.com 𝑆(𝐴) = { −0.14 𝑖𝑓 𝑛 = 10, 𝑚 = 3 −0.00 𝑖𝑓 𝑛 = 8, 𝑚 = 6 −0.01 𝑖𝑓 𝑛 = 7, 𝑚 = 5 0.06 𝑖𝑓 𝑛 = 5, 𝑚 = 6 and the accuracy function is 𝛼(𝐴) = { 0.16 𝑖𝑓 𝑛 = 10, 𝑚 = 3 0.05 𝑖𝑓 𝑛 = 8, 𝑚 = 6 0.08 𝑖𝑓 𝑛 = 7, 𝑚 = 5 0.11 𝑖𝑓 𝑛 = 5, 𝑚 = 6 Theorem 3.10 Let 𝐴 = (𝐽𝐴, 𝐾𝐴) be a Beal’s fuzzy set. Then the suggested score fuzzy mapping 𝑆(𝐴) ∈ [−1,1] . Proof: Since for any Beal’s fuzzy set A. we have 𝐽𝐴 𝑛 + 𝐾𝐴 𝑛 ≤ 1 Hence, 𝐽𝐴 𝑛 − 𝐾𝐴 𝑛 ≤ 𝐽𝐴 𝑛 ≤ 1 𝑎𝑛𝑑 𝐽𝐴 𝑛 − 𝐾𝐴 𝑛 ≥ −𝐾𝐴 𝑚 ≥ −1 Thus −1 ≤ 𝐽𝐴 𝑛 − 𝐾𝐴 𝑛 ≤ 1, namely 𝑆(𝐴) ∈ [−1,1]. In such way that, if 𝐴 = (0,1) then 𝑆(𝐴) = −1 and if 𝐴 = (0,1) then 𝑆(𝐴) = 1. Remark: For any Beal’s fuzzy set 𝐴1 = (𝐽𝐴1, 𝐾𝐴1) and 𝐴2 = (𝐽𝐴2, 𝐾𝐴2), the comparison technique is supposed as (𝑖)𝐼𝑓 𝑆(𝐴1) < 𝑆(𝐴2) 𝑡ℎ𝑒𝑛 𝐴1 < 𝐴2 (𝑖𝑖)𝐼𝑓 𝑆(𝐴1) > 𝑆(𝐴2) 𝑡ℎ𝑒𝑛 𝐴1 > 𝐴2 (𝑖𝑖𝑖)𝐼𝑓 𝑆(𝐴1) = 𝑆(𝐴2) 𝑡ℎ𝑒𝑛 𝐴1 = 𝐴2 (𝑖𝑣) 𝛼 ̅(𝐴1) < 𝛼 ̅(𝐴2) 𝑡ℎ𝑒𝑛 𝐴1 < 𝐴2 (𝑣)𝐼𝑓 𝛼 ̅(𝐴1) > 𝛼 ̅(𝐴2) 𝑡ℎ𝑒𝑛 𝐴1 > 𝐴2 (𝑣𝑖)𝐼𝑓 𝛼 ̅(𝐴1) = 𝛼 ̅(𝐴2) 𝑡ℎ𝑒𝑛 𝐴1 = 𝐴2. Beal’s fuzzy weighted power average In this section, we study the various operation on Beal’s fuzzy sets, and some intellectual structure are indicated in detail. Definition 4.1. Let 𝐴𝑖 = (𝐽𝐴𝑖 , 𝐾𝐴𝑖 ) (𝑖 = 1,2, … . 𝑘) be a value of Beal’s fuzzy sets and ∈= (∈1, ∈2, … ∈𝑘) 𝑇 be a weighted vector of 𝐴𝑖 with ∈𝑖> 0, ∑ ∈𝑖= 1.𝑘 𝑖=1 Then a Beal’s fuzzy weighted power mean operator is a function 𝐵𝐹𝑊𝑃𝐴: 𝐴𝑘 → 𝐴 where Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 400 https://internationalpubls.com 𝐵𝐹𝑊𝑃𝐴(𝐴1, 𝐴2, …𝐴𝑘) = ((∑ ∈𝑖 𝐽𝐴𝑖 𝑛𝑘 𝑖=1 ) 1 𝑛 , (∑ ∈𝑖 𝐾𝐴𝑖 𝑚𝑘 𝑖=1 ) 1 𝑚 ). Example 4.2. Consider 𝐴1 = (0.8, 0.4), 𝐴2 = (0.3, 0.9), 𝐴3 = (0.6, 0.7), 𝐴4 = (0.8, 0.7) 𝑎𝑛𝑑 𝐴5 = (0.9, 0.5) as five Beal’s fuzzy sets and let ∈= (0.18, 0.27, 0.25, 0.16, 0.14)𝑇 be a weighted vector of 𝐴𝑖(𝑖 = 1,2,3,4,5). Then BFWPA (𝐴1,𝐴2, … . 𝐴5) = ((0.8𝑛 × 0.18 + 0.3𝑛 × 0.27+ 0.6𝑛 × 0.25+ 0.8𝑛 × 0.16+ 0.9𝑛 × 0.14) 1 𝑛 , (0.4𝑛 × 0.18+ 0.9𝑛 × 0.27 + 0.7𝑛 × 0.25+ 0.7𝑛 × 0.16 + 0.5𝑛 × 0.14) 1 𝑚) = { 0.667, 0.818 𝑖𝑓 𝑛 = 2, 𝑚 = 3 0.696,0.832 𝑖𝑓 𝑛 = 3, 𝑚 = 5 0.717,0.862 𝑖𝑓 𝑛 = 4, 𝑚 = 8 0.783,0.851 𝑖𝑓 𝑛 = 10, 𝑚 = 14 0.824,0.89 𝑖𝑓 𝑛 = 20, 𝑚 = 30 0.875,0.84 𝑖𝑓 𝑛 = 70, 𝑚 = 50. Theorem 4.3. Let 𝐴𝑖 = (𝐽𝐴𝑖, 𝐾𝐴𝑖)(𝑖 = 1,2, … 𝑘) be a family of Beal’s fuzzy sets and let ∈= (∈1, ∈2. . ∈𝑘) 𝑇 be a weighted vector of 𝐴𝑖 with ∈𝑖> 0 𝑎𝑛𝑑 ∑ ∈𝑖= 1𝑘 𝑖=1 . Then Beal’s fuzzy weighted power mean (𝐴1, 𝐴2, …𝐴𝑘) is a Beal’s fuzzy set. Theorem 4.4. Let 𝐴𝑖 = ( 𝐽𝐴𝑖 , 𝐾𝐴𝑖)(𝑖 = 1,2, . . 𝑘) be a value of Beal’s fuzzy sets 𝐴 = (𝐽𝐴, 𝐾𝐴) be a Beal’s fuzzy set and ∈= (∈1, ∈2, … ∈𝑘) 𝑇 be a weighted vector of 𝐴𝑖with ∑ ∈𝑖= 1𝑘 𝑖=1 . Then 𝐵𝐹𝑊𝑃𝐴(𝐴1⊕𝐴2⊕… .𝐴𝑘⊕𝐴) ≥ 𝐵𝐹𝑊𝑃𝐴(𝐴1⨂𝐴2⨂…𝐴𝑘⨂𝐴). Theorem 4.5. Let 𝐴𝑖 = (𝐽𝐴𝑖, 𝐾𝐴𝑖)(𝑖 = 1,2, … , 𝑘) be a value of Beal’s fuzzy sets, 𝐴 = (𝐽𝐴, 𝐾𝐴) be a Beal’s fuzzy set, and ∈= (∈1, ∈2, … ∈𝑘) 𝑇 be a weighted vector of 𝐴𝑖with ∑ ∈𝑖= 1.𝑘 𝑖=1 Then (𝑖)𝐵𝐹𝑊𝑃𝐴 (𝐴1⊕𝐴2⊕… .𝐴𝑘⊕𝐴) ≥ 𝐵𝐹𝑊𝑃𝐴(𝐴1, 𝐴2, …𝐴𝐾) ⊗ 𝐴. (𝑖𝑖)𝐵𝐹𝑊𝑃𝐴 (𝐴1, 𝐴2, …𝐴𝑘) ⊕ 𝐴 ≥ 𝐵𝐹𝑊𝑃𝐴(𝐴1, 𝐴2, … , 𝐴𝑘)⨂𝐴. Theorem 4.6. Let 𝐴𝑖 = (𝐽𝐴𝑖, 𝐾𝐴𝑖) and 𝐵𝑖 = (𝐽𝐵𝑖 , 𝐾𝐵𝑖)(𝑖 = 1,2, … , 𝑘) be two values of Beal’s fuzzy sets and ∈= (∈1, ∈2, … ∈𝑘) 𝑇 be a weighted vector of them with ∑ ∈𝑖= 1𝑘 𝑖=1 . Then (𝑖)𝐵𝐹𝑊𝑃𝐴(𝐴1⨁𝐵1, 𝐴2⨁𝐵2, … , 𝐴𝑘⊕𝐵𝑘) ≥ (𝐴1⨂𝐵1, 𝐴2⨂𝐵2, … , 𝐴𝑘⨂𝐵𝑘). (𝑖𝑖)𝐵𝐹𝑊𝑃𝐴 (𝐴1, 𝐴2, … , 𝐴𝑘)⨁𝐵𝐹𝑊𝑃𝐴 (𝐵1, 𝐵2, … , 𝐵𝑘) (𝑖𝑖𝑖)𝐵𝐹𝑊𝑃𝐴 (𝐴1, 𝐴2, … , 𝐴𝑘)⨂𝐵𝐹𝑊𝑃𝐴 (𝐵1, 𝐵2, … , 𝐵𝑘) = ((∑ ∈𝑖 𝐽𝐴𝑖 𝑛𝑘 𝑖=1 ) 1 𝑛 (∑ ∈𝑖 𝐽𝐵𝑖 𝑛𝑘 𝑖=1 ) 1 𝑛 √∑ ∈𝑖 𝐾𝐴𝑖 𝑚 + ∑ ∈𝑖 𝐾𝐴𝑖 𝑚 − ∑ ∈𝑖 𝐾𝐴𝑖 𝑚 ∑ ∈𝑖 𝐾𝐴𝐵𝑖 𝑚 ) 𝑛 𝑖=1 𝑘 𝑖=1 𝑘 𝑖=1 𝑘 𝑖=1 Application of Beal’s fuzzy set in pattern recognition In this section, we propose similarity measures that can be dignity in commander’s chosen in military office, medical diagnosis of disease. Plant leaf diseases classifications, construction material selections and other multi-attribute decision-making problems. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 401 https://internationalpubls.com The following contribution showed to accept on unknown pattern using the new idea of similarity theorem. Example 4.7 Let us taken three unknown patterns 𝑃𝑖(𝑖 = 1,2,3) which are represented by the Beal’s fuzzy set (𝑚 = 5, 𝑛 = 5)𝑃𝑖(𝑖 = 1,2,3) in the characteristic as: 𝑅 = {𝑟1, 𝑟2, 𝑟3}. 𝑃1 = 〈𝑟1, 0.5,0.6〉, 〈𝑟2,0.1,0.7〉, 〈𝑟3, 0.4, 0.7〉 𝑃2 = 〈𝑟1,0.1,0.8〉, 〈𝑟2, 0.8,0.8〉, 〈𝑟3, 0.7, 0.6〉 𝑃3 = 〈𝑟1,0.5,0.7〉, 〈𝑟2, 0.6,0.6〉, 〈𝑟3, 0.7, 0.5〉 Consider, as unknown pattern 𝑃 ∈ 𝐵4 5 (𝑅) that will be recognized, where: 𝑃 = 〈𝑟1,0.9,0.7〉, 〈𝑟2, 0.8,0.7〉, 〈𝑟1, 0.6,0.8〉. Then the proposed similarity techniques 𝑆1 𝑎𝑛𝑑 𝑆2 which have been evaluated from P to 𝑃𝑖(𝑖 = 1,2,3) are given in the following table. From the number point of given presented in the previous data, we know the similarity measures between 𝑃2 and 𝑃 are the biggest or highest one. Similarity measure between 𝑃𝑖(𝑖 = 1,2,3) and P Similarity Measure (𝑃1, 𝑃) (𝑃2, 𝑃) (𝑃3, 𝑃) 𝑆1(𝑃𝑖, 𝑃) 0.7180 0.8487 0.7589 𝑆2(𝑃𝑖, 𝑃) 0.7679 0.8382 0.7625 Among the calculated measures, (𝑃2, 𝑃) is the largest one. 5. Conclusion In the Beal’s fuzzy set, the complement, necessity, possibility and various arithmetic operators are explained and various results related to properties of these operations have been demonstrated in this article. 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