A new method of measuring similarity between two neutrosophic soft sets and its application in pattern recognition problems

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1 Neutrosophi Sets and Systems, Vol. 8, A new method of measuring similarity between two neutrosophi soft sets and its appliation in pattern reognition problems Anjan Mukherjee, Sadhan Sarkar, Department of Mathematis, Tripura University, Suryamaninagar, Agartala-7990, Tripura,India. anjan00_m@yahoo.o.in Department of Mathematis, Tripura University, Suryamaninagar, Agartala-7990, Tripura,India. sadhan7_s@rediffmail.om Abstrat. Smarandahe in 995 introdued the onept of neutrosophi set and in 03 Maji introdued the notion of neutrosophi soft set, whih is a hybridization of neutrosophi set and soft set. After its introdution neutrosophi soft sets beome most effiient tools to deals with problems that ontain unertainty suh as problem in soial, eonomi system, medial diagnosis, pattern reognition, game theory, oding theory and so on. In this work a new method of measuring similarity measure and weighted similarity measure between two neutrosophi soft sets (NSSs) are proposed. A omparative study with existing similarity measures for neutrosophi soft sets also studied. A deision making method is established for neutrosophi soft set setting using similarity measures. Lastly a numerial example is given to demonstrate the possible appliation of similarity measures in pattern reognition problems. Keywords: Fuzzy sets, soft sets, neutrosophi sets, neutrosophi soft sets, similarity measure, pattern reognition. Introdution The onept of fuzzy set theory was initiated by Prof. L. A. Zadeh in965[0]. After its introdution several researhers have extended this onept iany diretions. The traditional fuzzy set is haraterized by the membership value or the grade of membership value. Sometimes it may be very diffiult to assign the membership value for a fuzzy set. To overome this diffiulty the onept of interval valued fuzzy sets was proposed by L.A. Zadeh in975[]. In some real life problems in expert system, belief system, information fusion and so on, we must onsider the truth-membership as well as the falsity-membership for proper desription of an objet in unertain, ambiguous environment. Neither the fuzzy sets nor the interval valued fuzzy sets is appropriate for suh a situation. Intuitionisti fuzzy setsintrodued by K. Atanassov[] in 986 and interval valued intuitionisti fuzzy setsintrodued by K. Atanassov and G. Gargov in 989[] are appropriate for suh a situation.but these do not handle the indeterminate and inonsistent information whih exists in belief system. F. Smarandahe in 995[6,7], introdued the onept of neutrosophiset,whih is a mathematial tool for handling problems involving impreise, indeterminay and inonsistent data. Soft set theory[7, ] has enrihed its potentiality sine its introdution by Molodtsov in 999. Using the onept of soft set theory P. K. Maji in 03[] introdued neutrosophi soft set. Neutrosophi sets and neutrosophi soft sets now beome the most useful mathematial tools to deal with the problems whih involve the indeterminate and inonsistent information. Similarity measure is an important topi in the fuzzy set theory. The similarity measure indiates the degree of similarity between two fuzzy sets. P. Z. Wang[8] first introdued the onept of similarity measure of fuzzy sets and gave a omputational formula. Siene then, similarity measure of fuzzy sets has attrated several researhers interest and has been investigated more. Domain of appliation of similarity measure of fuzzy sets are fuzzy lustering, image proessing, fuzzy reasoning, fuzzy neural network, pattern reognition, medial diagnosis, game theory, oding theory and several problems that ontain unertainties. S. M. Chen[5, 6] proposedsimilarity between vague sets, similarity measure of soft sets was studied by P. Majumderet al.[8, 9, 0] and W.K. Min[3], NaimCagman and Irfan Deli[4] introdued similarity measure for intuitionisti fuzzy soft sets, severalsimilarity measures for interval-valued fuzzy soft sets were studied by A. Mukherjee and S. Sarkar[4]. Said Broumi and FlorentinSmarandahe[3] introdued the onept of several similarity measures of neutrosophi sets and Jun Ye[9] introdued the onept of similarity measures between interval neutrosophi sets. Reently A. Mukherjee and S. Sarkar[5] introdued several methods of similarity measure for neutrosophi soft sets Pattern reognition problem has been one of the fastest growing areas during the last two deades beause of its usefulness and fasination. The main objetive of pattern reognition problems is supervised or unsupervised

2 Neutrosophi Sets and Systems, Vol. 8, lassifiation of unknown patterns. Among the various frameworks in whih pattern reognition problem has been traditionally formulated the statistial approah has been most intensively studied and used in pratie. In this paper a new method of measuring degree of similarity and weighted similarity measure between two neutrosophi soft set is proposed and some basi properties of similarity measure also are studied. A deisioaking method is established based on the proposed similarity measure. An illustrative numerial example is given to demonstrate the appliation of proposed deisioaking method in a supervised pattern reognition problem that is on the basis of the knowledge of the known pattern our aim is to lassify the unknown pattern. The rest of the paper is organized as --- setion : some preliminary basi definitions are given in this setion. In setion 3 similarity measures, weighted similarity measure between two NSSs is defined with examples and some basi properties are studied. In setion 4 a deisioaking method is established with an example in a pattern reognition problem. In Setion 5 a omparative study of similarity measures between existing and proposed method is given. Finally in setion 6 some remarks of the proposed similarity measure between NSSs and the proposed deisioaking method are given. Preliminaries and related works In this setion we briefly review some basi definitions related to neutrosophi soft sets whih will be used in the rest of the paper.. Definition[0] Let X be a non empty olletion of objets denoted by x. Then a fuzzy set (FS for short) in X is a set of ordered pairs having the form x, x : x X where the funtion : X 0,, is alled the membership funtion or grade of membership (also degree of ompatibility or degree of truth) of x in.the interval M = 0, is alled membership spae.. Definition[] Let D[0, ] be the set of losed subintervals of the interval [0, ]. An interval-valued fuzzy set in X, X and Card(X) = n, is an expression A given by A ( x, M ( x )) : x X M : X D 0,., where A.3 Definition[] Let X be a non empty set. Then an intuitionisti fuzzy set (IFS for short) A is a set having the form A={(x, A (x), A (x)): xx} where the funtions A : X[0,] and A : X[0,] represents the degree ofmembership and the degree of non-membership A respetively of eah element xx and 0 A (x)+ A (x) for eah xx..4 Definition[7,] Let U be an initial universe and E be a set of parameters. Let P(U) denotes the power set of U and A E. Then the pair (F,A) is alled a soft set over U, where F is a mapping given byf:a P(U)..5 Definition[6,7] A neutrosophi set A on the universe of disourse X is defined as A {( x, T ( x), I ( x), F ( x)), x X} where T, I, F : A A A X ] 0, [ and 0 T ( x) I ( x) F ( x) 3. A A A From philosophial point of view, the neutrosophi set takes the value from real standard or non-standard subsets of ] 0, [. But in real life appliation in sientifi and engineering problems it is diffiult to use neutrosophi set with value from real standard or non-standard subset of ] 0, [. Hene we onsider the neutrosophi set whih takes the value from the subset of [0,] that is 0 T ( x) I ( x) F ( x) 3. A A A.6 Definition[] Let U be the universe set and E be the set of parameters. Also let A E and P(U) be the set of all neutrosophi sets of U. Then the olletion (F, A) is alled neutrosophi soft set over U, where F is a mapping given by F: A P(U)..7 Definition[] Let (F,E ) and (G,E ) be two neutrosophi soft sets over the ommon universe U, where E, E are two sets of parameters. Then (F,E ) is said to be neutrosophi soft subset of (G,E ) if E E and T ( e) ( x) TG ( e) ( x), IF ( e) ( x) IG( e) ( x), FF ( e) ( x) FG ( e) ( x), e E, x U. If (F,E ) be neutrosophi soft subset of (G,E ) then it is denoted by F, E G, E..8 Definition[] Let E = {e, e, e 3,...,e m } be the set of parameters, then the set denoted by E and defined by E = { e, e, e 3,..., e m }, where e i = not e i, i is alled NOT set of the set of parameters E. Where and different operators..9 Definition[] The omplement of a neutrosophi soft set (F,E) denoted by ( FE, ) is defined as ( FE, ) = ( F, E), where F F : E PU ( ) is amapping given by ( ) neutrosophi soft omplement with

3 Neutrosophi Sets and Systems, Vol. 8, T F F ( x) F( x), I I F ( x) F( x) and F TF( x). F ( x) 3 Similarity measure for neutrosophi soft sets(nsss) In this setion we have proposed a new method for measuring similarity measure and weighted similarity measure for NSSs and some basi properties are also studied. 3. Similarity measure U = {x, x, x 3,...,x n } be the universe of disourse and E = {e, e, e 3,...,e m } be the set of parameters and (N,E), (N,E) be two neutrosophi soft sets over U(E). Then the similarity measure between NSSs (N,E) and (N,E) is denoted by Sim( N, N ) and is defined as follows : Sim( N, N ) 3 ( )( ) ( )( ) T x e T x e N N 3mn I ( x )( e ) I ( x )( e ) F ( x )( e ) F ( x )( e ) () 3. Theorem If Sim( N, N ) be the similarity measure between two NSSs (N,E) and (N,E) then (i) 0 Sim( N, N ) Sim( N, N ) Sim( N, N ) (ii) (iii) Sim( N, N ) (iv) Proof: If ( N, E) ( N, E) ( N, E) then 3 Sim( N, N ) Sim( N, N ) 3 3 (i) Obvious from definition 3.. (ii) Obvious from definition 3.. (iii) Obvious from definition 3. (iv) Let U = {x, x, x 3,..., x n } be the universe of disourse and E = {e, e, e 3,...,e m } be the set of parameters and (N,E), (N,E), (N 3,E) be three neutrosophi soft sets over U(E), suh that ( N, E) ( N, E) ( N, E).Now by definition of 3 neutrosophi soft sub sets (Maji, 03) we have T ( x )( e ) T ( x )( e ) T ( x )( e ) N N N 3 I ( x )( e ) I ( x )( e ) I ( x )( e ) N N N 3 F ( x )( e ) F ( x )( e ) F ( x )( e ) N N N 3 T ( x )( e ) T ( x )( e ) T ( x )( e ) T ( x )( e ), 3 3 I ( x )( e ) I ( x )( e ) I ( x )( e ) I ( x )( e ), 3 3 F ( x )( e ) F ( x )( e ) F ( x )( e ) F ( x )( e ) 3 3 T ( x )( e ) T ( x )( e ) I ( x )( e ) I ( x )( e ) 3 3 F ( x )( e ) F ( x )( e ) T ( x )( e ) T ( x )( e ) 3 3 I ( x )( e ) I ( x )( e ) F ( x )( e ) F ( x )( e ) N N3 N N3 3 T ( x )( e ) T ( x )( e ) I ( x )( e ) I ( x )( e ) 3 3 F ( x )( e ) F ( x )( e ) 3 T ( x )( e ) T ( x )( e ) 3 3 I ( x )( e ) I ( x )( e ) F ( x )( e ) F ( x )( e ) T ( x )( e ) T ( x )( e ) I ( x )( e ) I ( x )( e ) N N3 N N3 3mn F ( x )( e ) F ( x )( e ) 3 T ( x )( e ) T ( x )( e ) N N3 N N3 3mn I ( x )( e ) I ( x )( e ) F ( x )( e ) F ( x )( e ) 3 3 Sim( N, N ) Sim( N, N ) [ By equation () ] Weighted similarity measure Let U = {x, x, x 3,...,x n } be the universe of disourse and E = {e, e, e 3,...,e m } be the set of parameters and (N,E), (N,E) be two neutrosophi soft sets over U(E). Now if we onsider weights w i of x i (i =,,3,...,n) then the weighted similarity measure between NSSs (N,E) and (N,E) is denoted by WSim( N, N ) is proposed as follows : WSim( N, N ) w 3 T ( x )( e ) T ( x )( e ) i N N 3m I ( x )( e ) I ( x )( e ) F ( x )( e ) F ( x )( e ) Where w 0, i partiular if we take... (), i =,,3,...,n and w i WSim( N, N ) Sim( N, N ). n i i w. In, i =,,3,......,n then n 3.4 Theorem Let If WSim( N, N ) be the similarity measure between two NSSs (N,E) and (N,E) then

4 Neutrosophi Sets and Systems, Vol. 8, (i) 0 WSim( N, N ) (ii) WSim N N WSim N N (, ) (, ) (iii) WSim( N, N ) (iv) Proof: If ( N, E) ( N, E) ( N, E) then 3 WSim( N, N ) WSim( N, N ) 3 3 (i) Obvious from definition 3.3. (ii) Obvious from definition 3.3. (iii) Obvious from definition 3.3. (iv) Similar to proof of (iv) of theorem Example Here we onsider example 3.3 of [5]. Let U={x,x,x 3 } be the universal set and E={e,e, e 3 } be the set of parameters. Let (N,E) and (N,E) be two neutrosophi soft sets over U suh that their tabular representations are as follows: Table : tabular representation of (N,E) (N,E) e e e 3 x (0.,0.4,0.7) (0.5,0.,0.3) (0.4,0.,0.3) x (0.7,0.0,0.4) (0.0,0.4,0.8) (0.5,0.7,0.3) x 3 (0.3,0.4,0.3) (0.6,0.5,0.) (0.5,0.7,0.) Table : tabular representation of (N,E) (N,E) e e e 3 x (0.3,0.5,0.4) (0.4,0.3,0.4) (0.5,0.,0.) x (0.7,0.,0.5) (0.,0.4,0.7) (0.5,0.6,0.3) x 3 (0.3,0.3,0.4) (0.7,0.5,0.) (0.6,0.6,0.) Now by definition 3. similarity measure between (N,E) and (N,E) is given by Sim(N,N ) = Example Let U={x,x,x 3 } be the universal set and E={e,e, e 3 } be the set of parameters. Let (A,E) and (A,E) be two neutrosophi soft sets over U suh that their tabular representations are as follows: Table 3: tabular representation of (A,E) (A,E) e e e 3 x (0.,0.,0.) (0.,0.,0.) (0.,0.,0.) x (0.3,0.,0.) (0.,0.,0.3) (0.7,0.,0.) x 3 (0.9,0.3,0.) (0.,0.,0.) (0.,0.3,0.8) Table 4: tabular representation of (A,E) (A,E) e e e 3 x (0.9,0.9,0.8) (0.8,0.7,0.9) (0.9,0.8,0.9) x (0.9,0.8,0.8) (0.8,0.8,0.9) (0.,0.9,0.9) x 3 (0.,0.9,0.9) (0.8,0.8,0.9) (0.8,0.9,0.) Now by definition 3. similarity measure between (A,E) and (A,E) is given by Sim(A,A ) = Definition Let (N,E) and (N,E) be twonsssover the universe U. Then (N,E) and (N,E) are said be α - similar, denoted by ( N, E) ( N, E) if and only if Sim(N,N ) > α for α (0,). We all the two NSSs signifiantly similar if Sim(N,N ) > Definition Let (N,E) and (N,E) be twonsss over the universe U. Then (N,E) and (N,E) are said be substantially-similar if Sim(N,N ) > 0.95 and is denoted by (N,E) (N,E). 3.9 Definition In example 3.5 Sim(N,N ) = 0.9> 0.5, therefore (N,E) and (N,E) are signifiantly similar. Again in example 3.6 Sim(A,A ) = 0.3 < 0.5, therefore (A,E) and (A, E) are not signifiantly similar. 3.0 Theorem Let (N,E) and (N,E) be two neutrosophi soft sets over the universe U and N, E and N, E be their omplements respetively. Then i. if Sim( N, N) then ( Sim N, N), ( 0 ). ii. if WSim( N, N) then ( WSim N, N), 0. Proof : Straight forward from definition.7, 3. and 3.3.

5 Neutrosophi Sets and Systems, Vol. 8, Appliation of similarity measure of NSSs in pattern reognition problem In this setion we developed an algorithm for pattern reognition problem in neutrosophi soft set setting using similarity measure. A numerial example is given to demonstrate the effetiveness of the proposed method. Steps of algorithm are as follows: Step: onstrut NSS(s) N (i =,,3...,n)as ideal ˆ i pattern(s). Step: onstrut NSS(s) M ˆ j (j =,,3...,m)for sample pattern(s) whih is/are to be reognized. Step3: alulate similarity measure between NSS(s) for ideal pattern(s) and sample pattern(s). Step4: reognize sample pattern(s) under ertain predefined onditions. 4. Example In order to demonstrate the appliation of the proposed method of measuring similarity between NSSs, we onsider the medial diagnosis problem disussed in example 5. [5] as a supervised pattern reognition problem. In this example our proposed method is applied to determine whether an ill person having some visible symptoms is suffering from aner or not suffering from aner. We first onstrut an ideal NSS (known pattern) for aner disease and NSS(sample pattern) for the ill person(s) and we also assume that if the similarity measure between these two NSSs is greater than or equal to 0.75 then the ill person is possibly suffering from the diseases. Let U be the universal set, whih ontains only two elements x = severe and x = mild i.e. U={x,x }. Here the set of parameters E is a set of ertain visible symptoms. Let E = {e,e,e 3,e 4,e 5 }, where e = headahe, e = fatigue, e 3 = nausea and vomiting, e 4 = skin hanges, e 5 = weakness. Step : onstrut an ideal NSS ( ˆN,E) for illness (aner) whih an be done with the help of medial expert. Table 5: tabular representation of NSS ( ˆN,E) for aner. ( ˆN,E) e e e 3 x (0.6,0.,0.3) (0.7,0.3,0.4) (0.4,0.3,0.6) x (0.4,0.,0.) (0.3,0.,0.) (0.,0.,0.4) e 4 e 5 (0.8,0.,0.3) (0.5,0.3,0.) (0.3,0.,0.4) (0.,0.,0.3) Step : onstrut NSSs for ill persons (patients) X and Y. Table 6: tabular representation of NSS ( ˆM,E) for patient X. ˆM,E) ( e e e 3 x (0.7,0.3,0.4) (0.8,0.,0.5) (0.4,0.,0.5) x (0.3,0.,0.3) (0.,0.,0.3) (0.3,0.,0.3) e 4 e 5 (0.8,0.,0.) (0.5,0.3,0.) (0.3,0.,0.3) (0.,0.,0.) Table 7: tabular representation of NSS ( ˆM, E) for patient Y. ( ˆM,E) e e e 3 x (0.,0.5,0.8) (0.,0.0,0.8) (0.8,0.6,0.) x (0.9,0.6,0.7) (0.7,0.5,0.6) (0.7,0.6,0.) e 4 e 5 (0.,0.5,0.8) (0.9,0.6,0.8) (0.8,0.7,0.9) (0.8,0.7,0.7) Step 3: By definition 3. similarity measure between ( ˆN,E) and ( ˆM,E) is given by Sim( Nˆ, Mˆ ) 0.9 and similarity measure between ( ˆN,E) and ( ˆM,E) is given by Sim( Nˆ, Mˆ ) Step 4: Sine Sim( Nˆ, Mˆ ) 0.9> 0.75 therefore patient X is possibly suffering from aner. Again sine Sim( Nˆ, Mˆ ) 0.54 < 0.75 therefore patient Y is possibly not suffering from aner. The result obtained here is same as the result obtained in [5]. 5 Comparison of different similarity measures for NSSs In this setion effetiveness of the proposed method is demonstrated by the omparison between the proposed similarity measure and existing similarity measures in NSS setting. Here we onsider NSSs of examples 3.5, 3.6 and 4. for omparison of similarity measures as given in table 8.

6 Neutrosophi Sets and Systems, Vol. 8, Table 8: omparison of different similarity measures NSSs (N,N ) (A,A ) ( Nˆ, Mˆ ) ˆ ˆ Similarity ( N, M ) measure based on Hamming distane Set theoreti approah Proposed method Table 8 shows that eah method has its oweasuring but the results of similarity measures by proposed method are emphati over the other. Conlusions In this paper we proposed a new method of measuring degree of similarity and weighted similarity between two neutrosophi soft sets and studied some properties of similarity measure. Based on the omparison between the proposed method and existing methods introdued by Mukherjee and Sarkar[5], proposed method has been found to give strong similarity measure. A deisioaking method is developed based on similarity measure. Finally a fititious numerial example is given to demonstrate the appliation of similarity measure of NSSs in a supervised pattern reognition problem. Next researh work is to develop the appliation in other fields. Referenes [] K. Atanassov, Intuitionisti fuzzy sets, Fuzzy Sets and Systems 0 (986) [] K. Atanassov, G. Gargov, Interval valued intuitionisti fuzzy sets, Fuzzy Sets and Systems 3 (989) [3] Said Broumi, FlorentinSmarandahe, Several Similarity Measures of Neutrosophi Sets, Neutrosophi Sets and Systems (03) [4] NaimCagman, Irfan Deli, Similarity measure of intuitionisti fuzzy soft sets and their deisioaking, arxiv : vI [ math.lo] 3jan 03. [5] S. M. Chen, Measures of similarity between vague sets, Fuzzy Sets and Systems 74(995) 7 3. [6] S. M. Chen, Similarity measures between vague sets and between elements, IEEE Transations on System, Man and Cybernetis (Part B), 7() (997) [7] D.Molodtsov, Soft set theory first results, Computers and Mathematis with Appliation 37 (999) 9 3. [8] PinakiMajumdar and S. K. Samanta, Similarity measure of soft sets, New Mathematis and Natural Computation 4 () (008). [9] PinakiMajumdar and S. K. Samanta, On similarity measures of fuzzy soft sets, International Journal of Advane Soft Computing and Appliations 3 ( ) July 0. [0] PinakiMajumder and S. K. Samanta, On distane based similarity measure between intuitionisti fuzzy soft sets, Anusandhan () (00) [] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Computers and Mathematis with Appliations 45 (4-5) (003) [] P. K. Maji, Neutrosophi soft set, Annals of Fuzzy Mathematis and Informatis 5() (03) [3] Won Keun Min, Similarity in soft set theory, Applied Mathematis Letters 5 (0) [4] A. Mukherjee and S. Sarkar, Similarity measures of interval-valued fuzzy soft sets, Annals of Fuzzy Mathematis and Informatis, 8(3) [5] A. Mukherjee and S. Sarkar, Several Similarity Measures of Neutrosophi Soft Sets and its Appliation in Real Life Problems, Annals of Pure and Applied Mathematis 7() (04) 6. [6] F. Smarandahe, Neutrosophi Logi and Set, mss [7] F. Smarandahe, Neutrosophi set, a generalisation of the intuitionisti fuzzy sets, Inter. J. Pure Appl. Math. 4 (005) [8] P. Z. Wang, Fuzzy sets and its appliations, Shanghai Siene and Tehnology Press, Shanghai 983 in Chinese. [9] Jun Ye, Similarity measures between interval neutrosophi sets and their multiriteria deision making method, Journal of Intelligent and Fuzzy Systems, 03, DOI:0.333/IFS-074. [0] L.A. Zadeh, Fuzzy set, Information and Control 8 (965) [] L.A. Zadeh, The onept of a linguisti variable and its appliation to approximate reasoning-, Information Sienes 8 (975) Reeived: Marh, 05. Aepted: Marh 6, 05