On Similarity Measures of Fuzzy Soft Sets

Transcription

1 Int J Advance Soft Comput Appl, Vol 3, No, July ISSN ; Copyrght ICSRS Publcaton, www-csrsorg On Smlarty Measures of uzzy Soft Sets PINAKI MAJUMDAR* and SKSAMANTA Department of Mathematcs MUC Women s College, Burdwan, West-Bengal, Pn-734, INDIA pmajumdar@redffmalcom Department of Mathematcs, Vsva-Bharat, Santnketan, West-Bengal, Pn-7335, INDIA syamal_3@yahoocon Abstract In ths paper several smlarty measures of fuzzy soft sets are ntroduced The measures are examned based on the geometrc model, the set-theoretc approach and the matchng functon A comparatve study of these measures s done Keywords: uzzy soft set, Matchng functon, Proxmty measure, Smlarty measure Introducton Uncertanty s present n almost every sphere of our daly lfe Tradtonal mathematcal tools are not suffcent to handle all the practcal problems n felds such as medcal scence, socal scence, engneerng, economcs etc nvolvng uncertanty of varous types Zadeh [], n 965, was the frst to come up wth hs remarkable theory of fuzzy set for dealng these types of uncertantes where conventonal tools fal Hs theory brought a grand paradgmatc change n mathematcs Later there are theores namely the theory of ntutonstc fuzzy sets, vague sets, rough sets, nterval mathematcs etc to name a few, all are ntended to become a tool for handlng the uncertanty All these theores are successful to some extent n dealng wth the problems arsng due to the vagueness present n the real world But there are also cases where these theores faled to gve satsfactory results, possbly due to the nadequacy of the parameterzaton tool n them Then n 999, Molodtsov [6] ntated the theory of soft sets as a new mathematcal tool for dealng wth uncertanty Possble applcatons of soft set n varous problems such as smoothness of functons, game theory, operaton research, Reman ntegraton, Perron ntegraton, probablty theory, measurement theory, economcs, medcal scence etc are shown by Molodtsov [6] and others [,, 3] H aktas and N Cagman [] has shown that every fuzzy set and every rough set can be consdered as a soft set In that sense we can say that ths theory s much more general than ts predecessors * Correspondng author

2 Also n several problems t s often needed to compare two sets The sets may be fuzzy, may be vague etc We often nterested to know whether two patterns or mages are dentcal or approxmately dentcal or at least to what degree they are dentcal Several researchers lke Chen [3, 4, 5], L and Xu [], Hong and Km [7], CP Papps [8, 9] and many others [6, 8, 9] has studed the problem of smlarty measurement between fuzzy sets, fuzzy numbers and vague sets Recently P Majumdar and S K Samanta [4, 5] have studed the smlarty measure of soft sets and ntutonstc fuzzy soft sets Smlarty measures have extensve applcaton n several areas such as pattern recognton, mage processng, regon extracton, Psychology [7], handwrtng recognton [], decson makng [], codng theory etc The man purpose of ths paper s to ntroduce the concept of smlarty between fuzzy soft sets And that has been done usng three dfferent approaches A comparatve study was also done at the end Prelmnares In ths secton we brefly revew some defntons and examples whch wll be used n rest of the paper Defnton [] Let U be an ntal unversal set and let E be a set of parameters Let I U,( I [,]) denote the power set of all fuzzy subsets of U Let A E A par (, s called a fuzzy soft set over U, where s a mappng gven by : Example As an llustraton, consder the followng example Suppose a soft set (, descrbes attractveness of the shrts whch the authors are gong to wear U U the set of all shrts under consderaton { x x, x, x, } Let I be the collecton of all, 3 4 { e, e, e3, e4 fuzzy subsets of U Also let E {colorful, brght, cheap, warm} } A x x x x x x Let ( e ) {,,,, } ( e ) {,,,, }, ( e3 ) {,,,, }, x x ( e4 ) {,,,, } 3 So, the fuzzy soft set (, s a famly { ( e ),,,3,4 } of I U Defnton 3[] or two fuzzy soft sets (, over a common unverse U, we say that (, s a fuzzy soft subset of ( G, f ( ) A B, ( ) ε A, ( ε ) s a fuzzy subset of G (ε ) Defnton 4[] (Equalty of two fuzzy soft sets) Two soft sets (, over a common unverse U are sad to be fuzzy soft equal f (, s a fuzzy soft subset of ( G, and ( G, s a fuzzy soft subset of (, Defnton 5[] (Null fuzzy soft set) A soft set (, over U s sad to be null fuzzy soft set denoted by Φ, f ε A, ( ε ) null fuzzy set of U U I * Correspondng author

3 3 Defnton 6[] (Absolute fuzzy soft set) A soft set (, over U s sad to be absolute fuzzy soft set denoted by A ~, f ε A, ( ε ) U Defnton 7[] Unon of two soft sets (, over a common unverse U s the soft set ( H, C), where C A B, and e C, H ( e) ( e), e A B, G( e), e B A, ( e) G( e), e A B We denote the unon as (, U ( G, Defnton 8[4] Intersecton of two soft sets (, over a common unverse U s the soft set ( H, C), where C A B, and e C, H ( e) ( e) G (e) We denote the ntersecton as (, I ( G, 3 Smlarty measure of two fuzzy soft sets based on matchng functon In ths paper we redefne a fuzzy soft set for greater computatonal facltes We also assume that the unversal set U and the parameter set E are fnte Then we defne a fuzzy soft set as follows: U Defnton 3 Let U be an ntal unversal set and let E be a set of parameters Let I denote the collecton of all fuzzy subsets ofu A par (, s called a fuzzy soft set over U, where s a U mappng gven by : E I Actually defntons and 8 are the same because f we take any proper subset A of E and assgn the e- approxmaton ( e) e E \ A, then the fuzzy soft set (, and (, bear the same meanng Let U be the unverse and E, the set of parameters Then we can express a fuzzy soft set over U as a matrx We llustrate the process wth an example Consder the example Then the th (, j) entry of the matrx s the membershp value of e )( x ) f e A, and t s equal to f ( j e A, then we get a matrx called a fuzzy membershp matrx as below: * Correspondng author

4 Let  Then wth the above nterpretaton the fuzzy soft set 7 (, s 6 3 represented by the matrx A and we wrte (,  Clearly, the complement of (,, e C (, wll be represented by another matrx Bˆ where 5 ˆB Henceforth we wll denote a column of the fuzzy membershp matrx by the vector e ), or by smply e ), eg here ( e ) (5,9,,, ) n  ( Next we defne smlarty measure usng a matchng functon Defnton 3 Let (, be two fuzzy soft sets overu Then the smlarty between n r r { ( e ) G( e )} them, denoted by S(, or S, s defned by S(, S n r r {( ( e )) ( G( e )) } The followng s an example to llustrate the above defnton Example 33 Let (, be two fuzzy soft sets over U havng the fuzzy membershp matrx as follows: ( 7  and ˆB 3 3, 8 where U { x, x,,, } and the set of parameters s E { e, e, e3, e4} Then 4 r r { ( e ) G( e )} S(, S 67 4 r r {( ( e )) ( G( e )) } * Correspondng author

5 5 Proposton 34 Let (, be two fuzzy soft sets overu Then the followng holds: () S SG,, () (, ( G, S, () (, I ( G, Φ S and (v) f (, ( H, ( G,, then S S H, G Proof Trval 4 Smlarty measure based on set theoretc approach Let U { x, x,, xn} be the unversal set of elements and E { e, e,, em} be the unversal set of parameters Let ˆ (, and G ˆ ( G, be two fuzzy soft sets over ( U, Then ˆ { ( e ) P( U ); e E}, Gˆ { G( e ) P( U ); e E}, where ( e ) s called the e -th approxmaton of ˆ and G ( e ) s called the e -th approxmaton of Ĝ P(U ) be the collecton of all fuzzy subsets of U Let M ( ˆ, Gˆ ) ndcates the smlarty between the soft sets ˆ and Ĝ To fnd the smlarty between ˆ andĝ, frst we have to fnd the smlarty between ther e - approxmatons Let M ( ˆ, Gˆ ) denote the smlarty between the two eapproxmatons ( e ) and G ( e ) Defnton 4 Let us defne M G j G( e )( x ) I Then M, M ( ˆ, Gˆ ) max M ( ˆ, Gˆ ) j G n ( j Gj ) j ( ˆ, Gˆ ) n, where j ( e )( x j ) I and ( G ) j An example s gven to llustrate the above defnton Example 4 Consder the followng two fuzzy soft sets where U x, x, x, } and E e, e, e, }: { 3 e ˆ and Ĝ Then M 3, M 7, 3, 5 M 35 M 63 4 Hence M max{ M, M, M 3, M 4} M 7 j j { 3 * Correspondng author

6 6 Proposton 43 Let ˆ (, and G ˆ ( G, be two fuzzy soft sets over ( U, Then the followng condtons hold: () M M G,, () ˆ Gˆ M, () ˆ I Gˆ Φ M and (v) ˆ Hˆ Gˆ M M, G H, G Proof Can be easly proved from the defntons Note 44 Also here M does not mply ˆ Gˆ 5 Smlarty measure based on dstance We know that f A and B are two fuzzy sets and the dstance between them s d, then the smlarty between them can be defned as S Agan a fuzzy soft set s a collecton of ts + d e approxmatons whch are nothng but fuzzy sets Here we take the dstance between A and B as d ( A, max a b, where A a, a,, a ) and B b, b,, b ) are the two fuzzy ( n sets Then the smlarty between them wll be T ( A, + d ( A, ( n Now let (, { ( e ),,, n} { G( e j ), j,,, n} be two fuzzy soft sets where ( e ) s the e -th approxmaton of (, and G ( e ) s the e -th approxmaton of ( G, Let T (, denotes the smlarty between the e approxmatons e ) and G e ) So T (,, where d s the dstance between the e approxmatons ( e ) and G ( e ) + d Then the smlarty measure between (, wll be denoted by T (, and s defned by T (, mn (, T Example 5 Consder the followng two fuzzy soft sets, where U x, x, } and E e, e, }: { e3 ( ( { and G Then, d 4, d 9 and d 3 9 HenceT 7, T 53 and T 3 53 T mnt * Correspondng author

7 7 Proposton 5 Let (, be two fuzzy soft sets over ( U, Then the followng holds: () T TG,, () (, ( G, T, () ˆ Hˆ Gˆ T T, for any soft set ( H, over ( U,, G H, G Note 53 The followng property does not hold here: () ˆ I Gˆ Φ T 6 A comparatve study In ths paper we have dscussed three types of smlarty measure of fuzzy soft sets based on the matchng functon S, the set-theoretc approach M and the geometrc modelt Some of the propertes are common to all of them but few propertes are exclusve for partcular measures The table gven below gves a comparson between the three measures rom ths table we can have an dea about sutablty of a partcular measure for a partcular applcaton Table : Comparson table of three types of measures of smlarty Property S M T X X G, Y Y Y (, ( G,, Y N Y X G (, I ( G, Φ, N Y N X G ( Y Y Y, ( H, ( G, X X H, G Here n the table, S denotes the smlarty measure based on a matchng functon, M denotes the measure based on set theoretc approach and T denotes the measure based on geometrc model 7 Concluson Ths paper ntroduces the noton of smlarty between two fuzzy soft sets We have ntroduced three measures of smlarty for comparng two fuzzy soft sets We have studed few propertes of these three measures and at the end compared the propertes of all of them The smlarty measures have natural applcatons n the feld of pattern recognton, feature extracton, regon extracton, mage processng, codng theory etc Acknowledgement The present work s partally supported by Specal Assstance Programme (SAP) of UGC, New Delh, Inda [Grant No 5/8/DRS/4 (SAP-I)] * Correspondng author

8 8 References [] H Aktas and N Cagman, Soft sets and Soft Groups, Informaton Scence, 7(7), pp [] DG Chen et al, Some notes on the Parameterzaton Reducton of Soft Sets, Proceedng of the Second Internatonal Conference on Machne Learnng and Cybernetcs, X an, -5 Nov 3 [3] SM Chen, et al, A comparson of smlarty measures of fuzzy values, uzzy sets and systems, 7(995),pp [4] SM Chen, Measures of Smlarty between vague sets, uzzy sets and systems, 74(995) 7-3 [5] SM Chen, Smlarty Measures between vague sets and between elements, IEEE Transactons on System, Man and Cybernetcs (Part, Vol7,No(), (997), pp53-68 [6] P Grzegorzewsk, Dstances between ntutonstc fuzzy sets and/or nterval-valued fuzzy sets based on the Hausdorff metrc, uzzy sets and systems, 48(4),pp [7] DH Hong and CA Km, Note on Smlarty measure between vague sets and elements, Informaton Scences, 5(999),pp [8] J Kacprzyk, Multstage uzzy Control, Wley, Chchester, (997) [9] LT Kόczy and T Domonkos, uzzy rendszerek, Typotex, () [] L and ZY Xu, Smlarty measure between vague sets, Chnese Jr of Software, Vol, No(6), (), pp 9-97 [] W Y Leng and SM Shamsuddn, Wrter Identfcaton for chnese handwrtng, Int J Advance Soft Comput Appl, Vol, No, (), pp4-73 [] PK Maj, et al, uzzy soft-sets, The Jr of uzzy Math, Vol 9, No(3), (), pp [3] PK Maj, et al, Soft Set Theory, Computers and Math wth Appl, 45(3),pp [4] P Majumdar and S K Samanta, Smlarty measure of soft sets, New Mathematcs & Natural Computaton, Vol4,No(), (8),pp - [5] P Majumdar and S K Samanta, On Dstance based Smlarty measure between Intutonstc uzzy Soft sets, Anusandhan, Vol,No (), (),pp 4-5 [6] D Molodtsov, Soft set theory frst results, Computers Math wth Appl 37(999), pp9-3 [7] RM Nosofsky, Choce, Smlarty, and the Context Theory of Classfcaton, Jr of Exp Psychology: Learnng, Memory, and Cognton, Vol (984),pp 4-4 [8] CP Papps, Value approxmaton of fuzzy systems varables, uzzy sets and systems, 39(99),pp -5 [9] CP Papps and NI Karacaplds, A comparatve assessment of measures of smlarty of fuzzy values, uzzy sets and systems, 56(993),pp 7-74 [] J Wllams and N Steele, Dfference, dstance and smlarty as a bass for fuzzy decson support based on prototypcal decson classes, uzzy sets and systems, 3(),pp [] LAZadeh, uzzy Sets, Informaton and Control, 8(965) * Correspondng author