Fuzzy Equivalence on Standard and Rough Neutrosophic Sets and Applications to Clustering Analysis

Fuzzy Equivalence on Standard and Rough Neutrosophic Sets and Applications to Clustering Analysis Nguyen Xuan Thao 1(&), Le Hoang Son, Bui Cong Cuong 3, Mumtaz Ali 4, and Luong Hong Lan 5 1 Vietnam National University of Agriculture, Hanoi, Vietnam nxthao@vnua.edu.vn VNU University of Science, Vietnam National University, Hanoi, Vietnam sonlh@vnu.edu.vn 3 Institute of Mathematics, Hanoi, Vietnam bccuong@math.ac.vn 4 University of Southern Queensland, Springfield Campus, 4300 Queensland, Australia Mumtaz.Ali@usq.edu.au 5 Thai Nguyen University of Education, Thai Nguyen, Vietnam lanlhbk@gmail.com Abstract. In this paper, we propose the concept of fuzzy equivalence on standard neutrosophic sets and rough standard neutrosophic sets. We also provide some formulas for fuzzy equivalence on standard neutrosophic sets and rough standard neutrosophic sets. We also apply these formulas for cluster analysis. Numerical examples are illustrated. AQ1 Keywords: Fuzzy equivalence Neutrosophic set Rough set Rough neutrosophic set Fuzzy clustering 1 Introduction In 199, Smarandache introduced neutrosophic set [1] NS is the generalization of fuzzy set [] and intuitionistic fuzzy set [3]. Over time, the subclass of the neutrosophic set [4 6] was proposed to capture more advantages in practical applications. In 014, Bui Cong Cuong introduced the concept of the picture fuzzy set [7]. After that, Son gave the applications of the picture fuzzy set in clustering problems in [ 19]. It is to be noted that the picture fuzzy set was regarded as a standard neutrosophic set. Rough set theory [0] is a useful mathematical tool for data mining, especially for redundant and uncertain data [1]. On the first time, rough set is established on equivalence relation. The set of equivalence classes of the universal set, obtained by an equivalence relation, is the basis for the construction of upper and lower approximation of the subset of the universal set. Recently, rough set has been developed into the fuzzy environment and obtained several interesting results [, 3]. Springer Nature Singapore Pte Ltd. 01 V. Bhateja et al. (eds.), Information Systems Design and Intelligent Applications, Advances in Intelligent Systems and Computing 67, https://doi.org/10.1007/97-91-10-751-4_

Fuzzy Equivalence on Standard and Rough Neutrosophic Sets 35 It has been realized that the combination of the neutrosophic set and rough set achieved more uncertainty in the analysis of sophisticated events in real applications [4 6]. Bui Cong Cuong et al. [7] firstly introduced some results of the standard neutrosophic soft theory. Later, Nguyen Xuan Thao et al. [, 9] proposed the rough picture fuzzy set and the rough standard neutrosophic set which are the results of approximation of the picture fuzzy set and standard neutrosophic set, respectively, with respect to a crisp approximation space. However, the fuzzy equivalence, a basic component in the standard neutrosophic set for the approximation and inference processes, has not been defined yet. In this paper, we introduce the concept of fuzzy equivalence for the standard neutrosophic set and the rough standard neutrosophic set. Some examples of the fuzzy equivalence for those sets and application on clustering analysis are also given. The rest of the paper is organized as follows: The rough standard neutrosophic set and fuzzy equivalence are recalled in Sect. 3. Sections 3 and 4 propose the concept of fuzzy equivalence for two standard neutrosophic sets. In Sect. 5, we give an application of clustering and Sect. 6 draws the conclusion. AQ Preliminary Definition 1 [7]. Let U be a universal set. A standard neutrosophic set (SNS) A on the U is A ¼ fðu l A ðuþ g A ðuþ c A ðuþþju Ug, where l A ðuþ is called the degree of positive membership of u in A, g A ðuþ is called the degree of indeterminate/neutral membership of u in A, and c A ðuþc A ðuþis called the degree of negative membership of u in A, where l A ðþ u g A ðþc u A ðuþ ½0 1Š satisfy the following condition: 0 l A ðuþþg A ðuþþc A ðuþ1 u U: The family of all standard neutrosophic sets in U is denoted by SNSðUÞ. Definition [, 9]. For a given A SNSðUÞ, the mappings RP RP : SNSðUÞ! SNSðUÞ, in which RPðAÞ ¼ f u l RPðAÞ ðuþ g RPðAÞ ðuþ c RPðAÞ ðþ u ju Ug RPðAÞ ¼ fðu l RPðAÞ ðuþ g RPðAÞ ðþ u c RPðAÞ ðuþju Ug AQ3 where l RPðAÞ ðuþ ¼ _ vrs ðuþl A ðþ v g RPðAÞ ðuþ ¼ ^vrs ðuþg A ðþ v c RPðAÞ ðuþ ¼ ^vrs ðuþc A ðþ v and

36 N. X. Thao et al. l RPðAÞ ðuþ ¼ ^vrs ðuþl A ðuþ g RPðAÞ ðþ¼^vrs u ðuþg A ðþ v ð Þ ¼ _ vrs ðuþc A ðþ v c RPðAÞ u are called to the upper and lower standard neutrosophic approximation operators, respectively, and the pair RPðAÞ ¼ RPðAÞ RPðAÞ is referred as the rough standard neutrosophic set of A w.r.t the approximation space ðu RÞ,orA is called roughly defined on the approximation space ðu RÞ. The collection of all rough standard neutrosophic sets defined on the approximation space ðu RÞ is denoted by RSNSðUÞ. Definition 3 [1] A mapping e : ½0 1Š! ½0 1Š is a fuzzy equivalence if it satisfies the following conditions: (e1) ea ð bþ ¼ eb ð aþ a b in ½0 1Š (e) eð1 0Þ ¼ 0 (e3) eb ð bþ ¼ 1 b in ½0 1Š (e4) If a a 0 b 0 b, then ea ð bþea ð 0 b 0 Þ: Note that (e4) satisfies iff ea ð cþminfea ð bþ eb ð cþg for all a b c and a b c ½0 1Š: 3 Fuzzy Equivalence on Standard Neutrosophic Set Definition 4 A mapping E : SNSðUÞSNSðUÞ! ½0 1Š is a fuzzy equivalence if it satisfies the following conditions: (E1) EA ð BÞ ¼ EB ð AÞ for all A B SNSðUÞ (E) EA ð BÞ ¼ 0iffA ¼ 1 U and B ¼ 0 U (E3) EA ð AÞ ¼ 1 A SNSðUÞ, (E4) EA ð CÞminfEA ð BÞ EB ð CÞg for all A B C SNSðUÞ satisfy A B C. Example 1 Let U ¼ fx 1 x... x n g be a universal set and A B SNSðUÞ. A mapping E : SNSðUÞSNSðUÞ! ½0 1Š, where < EA ð BÞ ¼ 1 P n minfl A ðx i Þl B ðx i Þgþminfg A ðx i Þg B ðx i Þgþð1maxfc A ðx i Þc B ðx i ÞgÞ n iff A 6¼ B : 1 iff A ¼ B is a fuzzy equivalence of A and B. Indeed, conditions (E1), (E), (E3), and (E4) are obvious.

Theorem 1 Let U ¼ fx 1 x... x n g be a universal set and A B SNSðUÞ. Let a mapping E : PFSðUÞPFSðUÞ! ½0 1Š is defined by < EA ð BÞ ¼ 1 P n t 1 ðl A ðx i Þl B ðx i ÞÞþt ðg A ðx i Þg B ðx i ÞÞþð1sðc A ðx i Þc B ðx i ÞÞÞ n iff A 6¼ B : 1 iff A ¼ B where t 1 t are t-norm on [0, 1] and s is a t-conorm on [0, 1] then, EA ð BÞ is a fuzzy equivalence of A and B. 4 Fuzzy Equivalence on Rough Standard Neutrosophic Set Here, we propose a fuzzy equivalence of the rough standard neutrosophic sets. Let U ¼ fx 1 x... x n g, A B SNSðUÞ, and RPðAÞ ¼ RPA RPA, RPðBÞ ¼ RPB RPB. For all xi U, denote l EA ðx i Þ ¼ l RPA ðx i Þl RPA ðx i Þ g EA ðx i Þ ¼ g RPA ðx i Þg RPA ðx i Þ c EA ðx i Þ ¼ c RPA ðx i Þc RPA ðx i Þ l RPA ðx i Þþl RPA ðx i Þ l AE ðx i Þ ¼ g RPA ðx i Þþg RPA ðx i Þ g AE ðx i Þ ¼ c RPA ðx i Þþc RPA ðx i Þ c AE ðx i Þ ¼ l E ¼ 1 n g E ¼ 1 n c E ¼ 1 n X n X n X n Fuzzy Equivalence on Standard and Rough Neutrosophic Sets 37 l RPA ðx i Þl RPB ðx i Þ 9 þ lrpa ðx i Þl RPB ðx i Þ þ jl EA ðx i Þl EB ðx i Þj >< 1 >= 4 >: j l AEðx i Þl BE ðx i Þj > g RPA ðx i Þg RPB ðx i Þ 9 þ grpa ðx i Þg RPB ðx i Þ þ jg EA ðx i Þg EB ðx i Þj >< 1 >= 4 >: j g AEðx i Þg BE ðx i Þj > c RPA ðx i Þc RPB ðx i Þ 9 þ crpa ðx i Þc RPB ðx i Þ þ jc EA ðx i Þc EB ðx i Þj >< 1 >= 4 : >: j c AEðx i Þc BE ðx i Þj >

3 N. X. Thao et al. Theorem The mapping E : SNSðUÞSNSðUÞ! ½0 1Š is defined by EA ð B Proof We verify the conditions for EA ð BÞ (E1) is obvious. (E) A ¼ 1 U B ¼ 0 U. Then, Þ ¼ l E þ g E þ c E is a fuzzy equivalence: 3 l EA ðx i Þ ¼ 0 g EA ðx i Þ ¼ 0 c EA ðx i Þ ¼ 0 l AE ðx i Þ ¼ 1 g AE ðx i Þ ¼ 0 g AE ðx i Þ ¼ 0 l EB ðx i Þ ¼ 0 g EB ðx i Þ ¼ 0 c EB ðx i Þ ¼ 0 l BE ðx i Þ ¼ 0 g BE ðx i Þ ¼ 0 g BE ðx i Þ ¼ 1 So that l E ¼ g E ¼ c E ¼ 0 and EA ð BÞ ¼ l E þ g E þ c E 3 ¼ 0 (E3) is obvious. (E4) Note that, if 0 a b c 1, then ja cj ja bj and 1 ja cj jc bj 0 so that 0 1 ja cj 1 ja bj 1 and 0 1 ja cj 1 jc bj 1. Because A B C, then RPA RPB, RPA RPB, and RPB RPC, RPB RPC. Hence, EA ð CÞ minfea ð CÞ EB ð CÞg. Now, let U ¼ fx 1 x... x n g, A B SNSðUÞ, and RPðAÞ ¼ RPA RPA, RPðBÞ ¼ RPB RPB. For all xi U, denote E ða BÞðx i Þ ¼ l RPA ðx i Þl RPA ðx i Þjþjg RPA ðx i Þg RPB ðx i Þ þjcrpa ðx i Þc RPB ðx i Þ EA ð BÞðx i Þ ¼ l RPA ðx i Þl RPB ðx i Þ þ g RPA ðx i Þg RPB ðx i Þ þ c RPA ðx i Þc RPB ðx i Þ: Theorem 3 The mapping E : SNSðUÞSNSðUÞ! ½0 1Š is defined by EA ð BÞ ¼ 1 X n 1 EA ð BÞðx i ÞþEA ð BÞðx i Þ n is a fuzzy equivalence. Proof Similar to proof of Theorem. 5 An Application to Clustering Analysis Example Suppose there are three types of products D ¼ fd 1 D D 3 gand ten regular customers U ¼ fu 1 u... u 10 g. R is an equivalence and U=R ¼ fx 1 ¼ fu 1 u 3 u 9 g

Fuzzy Equivalence on Standard and Rough Neutrosophic Sets 39 X ¼ fu u 7 u 10 g X 3 ¼ fu 4 g X 4 ¼ fu 5 u g X 5 ¼ fu 10 gg. This division can be based on age or income. We consider that each customer evaluates the product by the linguistic labels {Good, Not-Rated, Not-Good}. Thus, each customer is a neutrosophic set on the products set (Table 1). Now, we can look at similar levels of customer groups in order to strategically sell products. Therefore, one can consider the sets of equivalence classes of customers for the three product categories above. In Table, we obtain a rough neutrosophic information system, in which for each X i, the upper line is RPX i and the lower line is RP X i ði ¼ 1 :: 5Þ. Table 1. A neutrosophic information system U D 1 D D 3 u 1 (0., 0.3, 0.5) (0.15, 0.6, 0.) (0.4, 0.05, 0.5) u (0.3, 0.1, 0.5) (0.3, 0.3, 0.3) (0.35, 0.1, 0.4) u 3 (0.6, 0, 0.4) (0.3, 0.05, 0.6) (0.1, 0.45, 0.4) u 4 (0.15, 0.1, 0.7) (0.1, 0.05, 0.) (0., 0.4, 0.3) u 5 (0.05, 0,, 0.7) (0., 0.4, 0.3) (0.05, 0.4, 0.5) u 6 (0.1, 0.3, 0.5) (0., 0.3, 0.4) (1, 0, 0) u 7 (0.5, 0.3, 0.4) (1, 0, 0) (0.3, 0.3, 0.4) u (0.1, 0.6, 0.) (0.5, 0.3, 0.4) (0.4, 0, 0.6) u 9 (0.45, 0,1, 0.45) (0.5, 0.4, 0.3) (0., 0.5, 0.3) u 10 (0.05, 0.05, 0.9) (0.4, 0., 0.3) (0.05, 0.7, 0.) Table. A rough neutrosophic information system D 1 D D 3 X 1 (0.6, 0, 0.4) (0.3, 0.05, 0.) (0.4,0.05,0.3) (0., 0, 0.5) (0.15, 0.05, 0.6) (0.1,0.05,0.5) X (0.3, 0.05, 0.4) (1, 0, 0) (0.35, 0.1, 0.) (0.05, 0.05, 0.9) (0.3, 0.1, 0.3) (0.05, 0.1, 0.4) X 3 (0.15, 0.1, 0.7) (0.1, 0.05, 0.) (0., 0.4, 0.3) (0.15, 0.1, 0.7) (0.1, 0.05, 0.) (0., 0.4, 0.3) X 4 (0.1, 0., 0.) (0.5, 0.3, 0.3) (0.4, 0, 0.5) (0.05, 0., 0.7) (0., 0.3, 0.4) (0.05, 0, 0.6) X 5 (0.1, 0.3, 0.5) (0., 0.3, 0.4) (1, 0, 0) (0.1, 0.3, 0.5) (0., 0.3, 0.4) (1, 0, 0) We calculate the similarity relations on fx 1 X X 3 X 4 X 5 g (based on Theorem ) as follows:

40 N. X. Thao et al. 3 1:0000 0:111 1:0000 R 1 ¼ 0:770 0:70 1:0000 6 7 4 0:7 0:7750 0:745 1:0000 5 0:7069 0:6347 0:7000 0:719 1:0000 Use the maximum tree method for fuzzy clustering analysis. Firstly, the Kruskal method is used to draw the largest tree: 3! 0:745 4! 0:775! 0:719 1! 0:7 5 The tree implies that for a ½0 1Š, we can classify the U=R ¼ fx 1 X X 3 X 4 X 5 g by fuzzy equivalence EX i X j a where Xi X j U=R as follows: þ If 0 a 0:745 then it has a cluster fx 1 X X 3 X 4 X 5 g: þ If 0:745\a 0:775 then we have two clustersfx 3 g fx 1 X X 4 X 5 g: þ If 0:775\a 0:719 then we have three clusters fx 3 g fx 4 g fx 1 X X 5 g: þ If 0:719\a 0:7 then we have four clusters fx 3 g fx 4 g fx g fx 1 X 5 g: þ If 0:7\a 1 then we have five clusters fx 3 g fx 4 g fx g fx 1 g fx 5 g: Now, we calculate the similarity relations on fx 1 X X 3 X 4 X 5 g (based on Theorem 3) as follows: 3 1:0000 0:79 1:0000 R ¼ 0:54 0:167 1:0000 6 7 4 0:13 0:563 0:5 1:0000 5 0:5 0:7667 0:775 0:51 1:0000 Use the maximum tree method for fuzzy clustering analysis. Firstly, the Kruskal method is used to draw the largest tree: 3! 0:5 4! 0:51! 0:563 1! 0:13 5 The tree implies that for a ½0 1Š, we can classify the U=R ¼ fx 1 X X 3 X 4 X 5 g by fuzzy equivalence EX i X j a, where Xi X j U=R as follows: þ If 0 a 0:5 then it has a cluster fx 1 X X 3 X 4 X 5 g: þ If 0:5\a 0:51 then we have two clustersfx 3 g fx 1 X X 4 X 5 g: þ If 0:51\a 0:563 then we have three clusters fx 3 g fx 4 g fx 1 X X 5 g: þ If 0:563\a 0:13 then we have four clusters fx 3 g fx 4 g fx g fx 1 X 5 g: þ If 0:13\a 1\a 1 then we have five clusters fx 3 g fx 4 g fx g fx 1 g fx 5 g:

Fuzzy Equivalence on Standard and Rough Neutrosophic Sets 41 The clustering analysis on the rough standard neutrosophic set is analogously done. We find that the clustering result by Theorems and 3 is giving the same clustering results. But theoretically, computation using Theorem 3 is simpler than Theorem. 6 Conclusions We have introduced the preliminary results of the fuzzy equivalences on the standard neutrosophic set and the rough standard neutrosophic set. Using these definitions, we can perform clustering analysis on the datasets of neutrosophic sets. Further studies regarding this research can be expanded of fuzzy equivalence of topological spaces and metrics. With that, we can also build fuzzy equivalent matrix models for clustering problems for real applications. Acknowledgements. We acknowledge the seminar series of Prof. Bui Cong Cuong at the Hanoi University of Science and Technology, 016 017, for supporting fruitful ideas. AQ4 References 1. Smarandache, F.: Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 105 p., 199 http://fs.gallup.unm. edu/ebook-neutrosophics6.pdf(last edition online).. Zadeh, L. A.: Fuzzy Sets. Information and Control (3) (1965) 33 353. 3. Atanassov, K.: Intuitionistic Fuzzy Sets. Fuzzy set and systems 0 (196) 7 96. 4. Wang, H., Smarandache, F., Zhang, Y.Q. et al: Interval NeutrosophicSets and Logic: Theory and Applications in Computing. Hexis, Phoenix, AZ (005). 5. Wang, H.,Smarandache, F., Zhang, Y.Q.,et al., Single Valued NeutrosophicSets. Multispace and Multistructure 4 (010) 410 413. 6. Ye, J.: A Multi criteria Decision-Making Method Using Aggregation Operators for Simplified Neutrosophic Sets. Journal of Intelligent & Fuzzy Systems 6 (014) 459 466. 7. Cuong, B.C.: Picture Fuzzy Sets. Journal of Computer Science and Cybernetics 30(4) (014) 409 40.. Cuong, B.C., Son, L.H., Chau, H.T.M.: Some Context Fuzzy Clustering Methods for Classification Problems. Proceedings of the 1st International Symposium on Information and Communication Technology (010) 34 40. 9. Son, L.H., Thong, P.H.: Some Novel Hybrid Forecast Methods Based On Picture Fuzzy Clustering for Weather Nowcasting from Satellite Image Sequences. Applied Intelligence 46 (1) (017) 1 15. 10. Son, L.H., Tuan, T.M.: A cooperative semi-supervised fuzzy clustering framework for dental X-ray image segmentation. Expert Systems With Applications 46 (016) 30 393. 11. Son, L.H., Viet, P.V., Hai, P.V.: Picture Inference System: A New Fuzzy Inference System on Picture Fuzzy Set. Applied Intelligence (017) https://doi.org/10.1007/s1049-016-056-1. 1. Son, L.H.: A Novel Kernel Fuzzy Clustering Algorithm for Geo-Demographic Analysis. Information Sciences 317 (015) 0 3. 13. Son, L.H.: Generalized Picture Distance Measure and Applications to Picture Fuzzy Clustering. Applied Soft Computing 46 (016) 4 95.

4 N. X. Thao et al. 14. Son, L.H.: Measuring Analogousness in Picture Fuzzy Sets: From Picture Distance Measures to Picture Association Measures. Fuzzy Optimization and Decision Making (017) https://doi.org/10.1007/s10700-016-949-5. 15. Son, L.H.: DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets. Expert systems with applications 4 (015) 51 66. 16. Thong, P.H., Son, L.H., Fujita, H.: Interpolative Picture Fuzzy Rules: A Novel Forecast Method for Weather Nowcasting. Proceeding of the 016 IEEE International Conference on Fuzzy Systems (016) 6 93. 17. Thong, P.H., Son, L.H.: A Novel Automatic Picture Fuzzy Clustering Method Based On Particle Swarm Optimization and Picture Composite Cardinality. Knowledge-Based Systems 109 (016) 4 60. 1. Thong, P.H., Son, L.H.: Picture Fuzzy Clustering for Complex Data. Engineering Applications of Artificial Intelligence 56 (016) 11 130. 19. Thong, P.H., Son, L.H.: Picture Fuzzy Clustering: A New Computational Intelligence Method. Soft Computing 0(9) (016) 3544 356. 0. Pawlak, Z.: Rough Sets. International Journal of Computer and Information Sciences 11 (5) (19) 341 356. 1. Fodor, J., Yager, R. R.: Fuzzy Set Theoretic Operations and Quantifers. Fundermentals of Fuzzy Sets. Klwuer (000).. Dubois, D., Prade, H.: Rough Fuzzy Sets and Fuzzy Rough Sets. International Journal of General Systems 17 (1990) 191 09. 3. Yao, Y.Y: Combination of Rough and Fuzzy Sets Based on a level sets. Rough sets and Data mining: analysis for imprecise data. Kluwer Academic Publisher, Boston (1997) 301 31. 4. Broumi, S. and Smarandache, F.: Rough neutrosophic sets. Italian Journal of Pure and Applied Mathematics, N.3, (014) 493 50. 5. Broumi, S. and Smarandache, F.: Lower and upper soft interval valued neutrosophic rough approximations of an IVNSS-relation, Sisom& Acoustics, (014) pages. 6. Broumi, S. and Smarandache, F.: Interval Valued Neutrosophic Soft Rough Set, International Journal of Computational Mathematics. Volume 015 (015), Article ID 3919, 13 pages http://dx.doi.org/10.1155/015/3919. 7. Cuong, B. C., Phong, P. H. and Smarandache, F.: Standard Neutrosophic Soft Theory: Some First Results. Neutrosophic Sets and Systems 1 (016) 0 91.. Thao, N. X., Dinh, N. V.: Rough Picture Fuzzy Set and Picture Fuzzy Topologies. Journal of Science computer and Cybernetics 31 (3) (015) 45 54. 9. Thao, N.X., Cuong. B. C., Smarandache, F.: Rough Standard Neutrosophic Sets: An Application on Standard Neutrosophic Information Systems. International Conference on Communication, Management and Information Technology, in press.