Fuzzy Logic and Zadeh Algebra

Transcription

1 Advances n Pure Matheatcs, 2017, 7, ISSN Onlne: ISSN Prnt: Fuzzy Logc and Zadeh Algebra Paavo Kukkuranen School of Engneerng Scence, Lappeenranta Unversty of Technology, Lappeenranta, Fnland How to cte ths paper: Kukkuranen, P (2017) Fuzzy Logc and Zadeh Algebra Advances n Pure Matheatcs, 7, eceved: May 22, 2017 Accepted: July 4, 2017 Publshed: July 7, 2017 Copyrght 2017 by author and Scentfc esearch Publshng Inc Ths work s lcensed under the Creatve Coons Attrbuton Internatonal Lcense (CC BY 40) Open Access Abstract In ths work we create a connecton between AFS (Axoatc Fuzzy Sets) fuzzy logc systes and Zadeh algebra Begnnng wth sple concepts we construct fuzzy logc concepts Sple concepts can be nterpreted seantcally The ebershp functons of fuzzy concepts for chans whch satsfy Zadeh algebra axos These chans are based on portant relatonshp condton (1) represented n the ntroducton where the bnary relaton of a sple concept s defned ore general n Defnton 210 Then every chan of ebershp functons fors a Zadeh algebra It deands a lot of prelnares before we obtan ths desred result Keywords AFS Fuzzy Logc Syste, Zadeh Algebra, Sple Concepts, Mebershp Functons 1 Introducton Startng wth sple concepts such as young people or tall people t s ( EM,,, The eleents are fuzzy possble to for AFS logc syste ( ) concepts constructed by sple concepts Notce that ( EM,, ) s a copletely dstrbutve lattce and s called the EI (expandng one set M) algebra over M So the AFS logc syste s a copletely dstrbutve lattce equpped wth the logcal negaton ' Let be a non-epty set For any EM, let : [ 0,1] be a ebershp functon of the concept Moreover, we assue that all the eleents n the set { EM} satsfy the three condtons, Defnton 217 Consder a bnary relaton of the concept For exaple, for any two persons x and y, ( xy, ) f and only f {(, ) (, ), x y} = x y x y age age where s a fuzzy concept old The exact defnton s represented n DOI: /ap July 7, 2017

2 P Kukkuranen Defnton 210 In Secton 2 all the results are known and can be found fro [1] Also the used exaples are there For Zadeh algebra axos we refer to [2] In Secton 3 t s proved new results But all the prelnares represented n Secton 2 are necessary to know for understandng these results and ther proofs The crucal condton s xz yx (1) for sple concepts and for all the pars ( xz, ) and (, ) fact, the condton deternes a chan ( ), Lea 31 (c) Let EM : [ 0,1] { } yx n In be a set of ebershp functons of the concept of the AFS fuzzy logc EM,,, corres- syste ( ) Accordng to Proposton 34 a chan ( ) EM pondng to the chan ( ) satsfes the seven Zadeh algebra axos and EM then fors soe Zadeh algebra, Proposton 35 These are the two an results Observng that by Lea 31 (a) the condton (1) ples the condton (2) needed n Proposton 34 x z y z (2) In the concluson t s llustrated the research otvaton and contrbuton of ths paper 2 Prelnares 21 Lattces In ths subsecton we refer to [3], pages 1, 2, 6, 8, 9, 10, 119 and [1], pages 61-64, 67, 77 Defnton 21 A partally ordered set or a poset s a set n whch a bnary relaton s defned satsfyng the followng condtons (P1)-(P3): (P1) For all x, x x (P2) If x y and y x, then x= y (P3) x y and y z, then x z Let (P4) Gven x and y, ether x y or y x A poset whch satsfes (P4) s sad to be lnearly ordered and s called a chan Let be a subset of a poset P Denote the least upper bound of by lub e sup and the greatest lower bound of by glb e nf Defnton 22 A lattce L s a poset P where any two of whose eleents x and y have glb or a eet denoted by x y, and lub or a jon denoted by x y A lattce L s coplete f each of ts subsets has lub and glb n L It s clear that any nonvod coplete lattce contans a least eleent 0 and a greatest eleent 1 In any lattce L (or a poset), the operatons and satsfy the followng laws, whenever the expressons are refered to exst: 354

3 P Kukkuranen (L1) x x= x, x x= x (L2) x y = y x, x y = y x (L3) x ( y z) = ( x y) z, x ( y z) = ( x y) z (L4) x ( x y) = x ( x y) = x Conversely, any syste L wth the two bnary operatons satsfyng (L1) - (L4) s a lattce Moreover, x y s equvalent to each of the condtons x y = x and x y = y If a poset P (or a lattce) has an 0, then 0 x = 0 and 0 x= x for all x P If P has a unversal upper bound I, then x I = x and x I = I for all x P Defnton 23 A lattce L s dstrbutve f and only f the condtons x ( y z) = ( x y) ( x z) x ( y z) = ( x y) ( x z) hold n L In fact, these condtons are equvalent f they are vald Defnton 24 [1], pages 77, 116 or [3], page 119 Let L be a coplete lattce Then L s called a copletely dstrbutve lattce f t satsfes the extended dstrbutve laws: for any faly { aj L I, j J} where I and J are non-epty ndexng sets, the followng equatons are vald a = af ( ) j I j J f J I I a = af ( ) j I j J f J I I 22 A Survey to Sple Concepts and Ther Operatons In ths subsecton we approach to sple concepts and ther operatons because t s necessary to for the dea what do sple concepts ean The exact defnton wll be represented n Defnton 213 All these are based on [1], pages 113,114 Consder the set of four people x 1, x 2, x 3, x 4 and a sple concept har colour By ntuton, we ay set: x 1 has har black wth nuber 6 and x2, x3, x 4 wth nubers 4,6,3 So, the nubers ply the order x4 > x2 > x3 = x1 whch can be nterpreted as follows: Movng fro rght to left, the relatonshp states how strongly the har colour resebles black colour More exactly, x > xj eans that the har of x s closer to the black colour than the colour of the har whch x j has Let M be a set of fuzzy or Boolean concepts on the set For each M we assocate to a sngle feature For exaple 1 : old people s a fuzzy concept but 2 : ale s a Boolean concept In fact, M s a set of sple concepts In general let A M and denote by a conjugaton of the A 355

4 P Kukkuranen concepts on A Correspondngly eans a dsjuncton A Exaple 25 Let 1 : old people, 2 : ale, 3 : tall people Then 1 2: old ales and 1+ 3: old or tall people Further, : old or tall ales However, eans the sae Ths s because for any person x the degree of x belongng to the fuzzy concept represented by s always less than or equal to the degree of x belongng to the fuzzy concept represented by 1 2 or 2 3 Therefore the forer s ncludng n both of the latter ones 1 2 or AFS Fuzzy Logc Syste All the defntons and the propostons wth ther proofs are represented n [1], pages For a oent we gve up the assupton that M conssts only of sple concepts Let M be a non-epty set The set EM s defned by EM = A M, I, I s any non-epty ndexng set I A where the eleents of EM or (dsjuncton) and and (conjuncton) are expressed seantcally wth equvalent to, Defnton 26 Let M be a non-epty set A bnary relaton on EM defned as follows: for ( I A ), ( ) j EM j J B ( ) ( A ) j J Bj I (1) A, I, Bh, h J such that A Bh, (2) B j, j J, Ak, k I such that Bj Ak s an equvalence relaton and we defne EM as the quotent set EM Proposton 27 Let M be a non-epty set Then ( EM,, ) fors a copletely dstrbutve lattce under the bnary copostons and defned as follows: for any I ( A ) ( ) j J Bj, EM, = + I A j J Bj k I J Ck I A j J Bj where the dsjont unon I = I A j J Bj I, j J A Bj J eans that every eleent n I and every eleent n J are always regarded as dfferent eleents n I J Therefore for any k I J, Ck = Ak f k I, and Ck = Bk f k J The proof of the proposton can be found fro [1] To be a dstrbutve lattce eans that for any γη,, EM, s 356

5 P Kukkuranen γ ( η) = ( γ ) ( γ η) γ ( η) = ( γ ) ( γ η) A copletely dstrbutve lattce s defned n Defnton 24 Because ( EM,, ) s such a lattce t guarantees the exstance of the EM eleents ( I A ) and ( ) j J Bj ( EM,, ) as follows: I We can also defne the order n Bj, j J, Ak, k I such that Bj Ak A j J Bj Further, as a (dstrbutve) copletely lattce ( EM,, ) s also a coplete lattce The lattce ( EM,, ) s called the EI (expandng one set M ) algebra over M Proposton 28 Let M be a set and g: M M be a ap satsfyng g( g( ) ) = for all M If the operator g : EM EM s defned as follows I g = g( ) = g( ) A I A I A Then for any αβ, EM, g has the followng propertes: g (1) ( α ) g = α, (2) ( ) g g g α β = α β, ( ) g g g α β α β g g (3) α β α β Therefore the operator g =, s an order reversng nvoluton n the EI algebra ( EM,, ) The operator g defnes the negaton of the concept : g( ) Then = ( g( ) ) = g( g( ) ) = = Then Let ( ) I A EM = = I A I A stands for the logcal negaton of ( EM,,,' ) = s called an AFS fuzzy logc syste Exaple 29 Let 1 : old people, 2 : tall people, 3 : ales Then = = ( ) ( ) ( ) ( ) ( ) = + = = = = + where : old ales or tall people and : not old and not tall people or not tall ales EM,,,' can be regarded as a copletely The AFS fuzzy logc syste ( ) 357

6 P Kukkuranen dstrbutve lattce It s also a coplete lattce But ths lattce s equpped wth the logcal negaton We conclude that the coplexty of huan concepts s a drect result of the cobnatons of a few relatvely sple concepts In fact, soe sutable sple concepts play the sae role as used n lnear vector spaces and we can regard the as a bass 24 elatons, Sple and Coplex Concepts For ths subsecton we refer to [1], pages 124, 125 Defnton 210 Let be any concept on the unverse of dscourse s called the bnary relaton of the concept f satsfes: xy,, ( xy, ) f and only f x belongs to concept at soe extent or x s a eber of and the degree of x belongng to s larger or equal to that of y, or x belongs to concept at soe degree and y does not at all Exaple 211 Let fuzzy concept : old and Therefore ( xx, ) {(, ) (, ), x y} = x y x y age age eans that x belongs to at soe degree and that ( xx, ) eans that x does not belong to at all If for the two persons x and y, age x = 30 and age y = 20 then ( xy, ) but ( yx, ) Exaple 212 Let fuzzy concept : har black and defne n the correspondng way as above By huan ntuton, we assue that for the three persons x, = 1, 2, 3 the degree of s the followng: blackx > black 1 x > black 2 x3 but the fourth person x 4 has no hars Then ( x1, x2),( x1, x3),( x2, x3) and ( x4, x4) but ( x, x4 ) See Defnton 213 (2) Defnton 213 Let be a set and be a bnary relaton on s called a sub-preference relaton on f for xyz,,, x y, satsfes the followng condtons: (1) f ( xy, ), then ( xx, ), (2) f ( xx, ) and ( yy, ), then ( xy, ), (3) f ( xy, ) and ( yz, ), then ( xz, ), (4) f ( xx, ) and ( yy, ), then ether ( xy, ) or ( yx, ) We defne that a concept on s sple f s a sub-preference relaton on Otherwse s called a coplex concept on Exaple 214 Let : old people The concept s sple For exaple f for the persons x 1, x 2, x 3 we have agex > age 1 x > age 2 x 3 s a sub-preference relaton on the set { x1, x2, x 3} where s the bnary relaton defned n Exaple 211 Observe that the latter of the assuptons of (2) n Defnton 213 s not vald and so the condton (2) s vald In general t s known that all eleents belongng to a sple concept at soe degree are coparable and are arranged n a lnear order, that s, they for a chan In above we can thnk shortly that x x x > > Further, there exsts a par of dfferent eleents belongng to a coplex 358

7 P Kukkuranen concept at soe degree such that ther degrees n ths coplex concept are ncoparable Exaple 215 The set conssts of dsjont sets Y : ales and Z : feales The concept : beautful s sple on Y and on Z : {( 1, 2) ( 1, 2), x } 1 x2 {( 1, 2) ( 1, 2), y y } = x x x x Y Y b b = y y y y Z Z b b 1 2 However, f we apply to the whole set t s a coplex concept because the degrees of the eleents x and y ay be ncoparable: If x Y and y Z, then ( xy, ) and If y Y and x Z, then ( yx, ) In ths case ( xx, ) and ( yy, ) ples that both ( xy, ) and ( yx, ) The condton (4) n Defnton 213 s not satsfed and so s a coplex concept 25 The AFS Fuzzy Logc and Coherence Mebershp Functons For ntroducton to characterstc and ebershp functons we refer to [4], page 255, and [5], pages Defntons 216 and 217 can be found fro [1], pages 128, 130 We frst becoe acquanted wth concepts fuzzy sets and ebershp functons Let and x, A Defne a characterstc functon for the set A as follows: { } f : 0,1 f A A 1, x A = 0, otherwse A 0< A < 1 s also possble We call for the set A (a) a crsp set, f ts characterstc functon s fa : { 0,1}, (b) a fuzzy set, f ts extended characterstc functon or a ebershp functon s A : [ 0,1] Therefore for every eleent x there s a ebershp degree ( x ) A [ 0,1] The set of pars Consder an extended case 0 f ( x) 1, that s, f ( x) {(, A ( )) } A= x x x deternes copletely the fuzzy set A The characterstc functon of a crsp set A s a specal case of a ebershp functon A : [ 0,1] Defnton 216 [1] Let, M be sets and 2 M be the power set of M Let : 2 M ( M,, ) s called an AFS structure f satsfes the followng axos: (1) ( x1, x2), ( x1, x2) ( x1, x1), x, x, x, x, x, x x, x x, x (2) ( ) ( ) ( ) ( ) ( ) We agan return to the case that M s a set of sple concepts Let be a set of objects and M be a set of sple concepts on : 2 M xy, s defned as follows: for any ( ) 359

8 P Kukkuranen ( xy) { M( xy) }, =,, 2 M where s the bnary relaton of sple concepts M defned n Defnton 210 (t was defned ore general than for sple concepts) It s proved n [1] that ( M,, ) Defnton 217 [1] Let ( M,, ), the set A ( x) x, A M s an AFS structure be an AFS structure of a data set For s defned as follows: ( ) = {, (, ) } A x y y xy A For EM, let : [ 0,1] be the ebershp functon of the concept x EM s called a set of coherence ebershp functons of the { } ( ) AFS fuzzy logc syste ( EM,,,' ) and the AFS structure ( M,, ) followng condtons are satsfed: (1) For αβ, EM, f α β for any x x = EM (2) For, η ( ) η ( x) 0 =, f the n lattce ( EM,,,' ), then ( x) ( x), f A ( x) I A, f A ( x) A ( y) (3) For x, y, A M, η = EM A x y A x = then ( x) = 1 η ( ) ( ) ; f ( ) η η = for all I then α, then eark: It s portant to see that M conssts of sple eleents 26 Zadeh Algebra We refer to [2] Defnton 218 Suppose that ( L,, ) s a coplete dstrbutve lattce Let = { : } be the set of all functons fro to L Assue that L L the lattce operatons the least upper bound and the greatest lower bound on L are extended pontwse for the functons on L Further, defne the extree constant functons L, 0 : x and 1 : x for all x, where and are the least and the greatest eleents of L, respectvely A unary operaton η on L satsfes the nvoluton property for any a L, and η s extended pontwse for the functons on L Then Z ( L,,, η,, ) ( ) L, e, ( ) ηη = for any = 01 s called Zadeh algebra f t satesfes the followng condtons: (Z1) The operatons and are coutatve on L (Z2) The operatons and are assocatve on L (Z3) The operatons and are dstrbutve on L (Z4) The neutral eleents of the operatons and are 0 and 1, respectvely, e, for all L and for all x, ( )( x) = ( x) ( 1 )( x) = ( x) (Z5) For any functon L and for all x, there exsts η ( ) L such that η( ( x) ) = ( x) η ( ( x) ) η( ( x) ) = ( x) ng (Z6) 0 1 β 0 and , e, η s order revers- 360

9 P Kukkuranen (Z7) Zadeh algebra fulfls the Kleene condton: for any functon ν, L and for any xy,, ( x) ( x) ν ( y) ν ( y), where and ν are the logcal negatons of and ν, respectvely 3 Connecton between Coherence Mebershp Functons of the AFS Fuzzy Logc Syste ( EM,,,') and Zadeh Algebra Lea 31 Let ( I A ) and ( ) j J Bj be eleents n EM where concepts are sple and let be a non-epty set If every relaton satsfes the condton for pars ( xz, ) and ( yx, ), then (a) ( x, z) ( y, z) (b) there exsts A n the set { A I} that s, k xz yx (1) such that Ak Bj for every B j, I A j J Bj (c) Let α = I( A ), β ( ) j J Bj =, = ( ), be δ ν V Cν the eleents n EM such that the condton (a) s satsfed for all pars αβ βδ Then α β δ (, ),(, ), Proof Assue that the condton (1) holds (a) Let ( xz, ) Then ( yx, ) Because s sple, s a subpreference relaton and by Defnton 213 (3) t s transtve Ths ples that ( yz, ) and (a) s vald (b) Let ( xz, ) { A ( xz, ) } = { j } (, ) (, ) yx = B yx Because s sple ( xy, ) and ( yz, ) are defned Because ( xz, ) ( yx, ) we conclude that ( xz, ) ( yx, ) A n the set { A I} A and so there exsts k such that for every B j s k Bj If there exsts B r whch does not contan A k then Br A does not hold Ths s a contradcton Accordng to dscusson after Proposton 27 we have (c) Consder a par (, ) ( C ) δ vv v I A j J Bj βδ, where β ( ) j J Bj = We wll prove that β δ = and, that s, 361

10 P Kukkuranen f j J Bj vv Cv yw vy (condton (1)) epeatng the proof of (b) we obtan the followng: Let { } ( yw, ) = Bj ( yw, ), ( vy, ) = { Cv ( vy, ) } and there exsts l Bj j J such that Bl Cv for all C v Ths proves β δ In the sae way we can prove that α β αβ, and then we conclude that α β δ Then ( yw, ) ( vy, ) B n the set { } for a par ( ) = If the condton Lea 32 Let ( ) holds for every sple con c ept where : [ 0,1] I A EM xz yx A then ( x) ( y) s the ebershp functon of the concept Proof Let M be sple concepts and Assue that the condton holds for every ( xz, ) { A ( xz, ) } = ( ) = { (, ) } A x z xz A xz yx A By Lea 31 (a) x z y z also holds Then ( xz, ) ( yz, ) and so A ( xz, ) A ( yz, ) It follows that A ( x) A ( y) Let A = These exst because EM s a copletely lattce By x y for every I : 0,1 s Defnton 217 (3) ( ) ( ) where [ ] a ebershp functon Let = ( ) sup I = A I Also exsts and we obtan ( x) ( y) Lea 33 Assue that the bnary relatons concepts satsfy the condton for pars ( xz, ),( yx, ) xz yx Let α = ( ) and β ( ) j J Bj I A of sple = be eleents n EM Then α β and ebershp functons α and β satsfy the Kleene condton ( x) ( x) ( y) ( y) α α β β Proof By Lea 31 (b), α β On the other hand, by Lea

11 P Kukkuranen Because α β ( x) β ( y) β, by Defnton 217 (1), α ( x) β ( x) ( y) ( y) ( y) ( x) ( x) ( x) ( x) We obtan β β β β α α α Proposton 34 Let { } [ 0,1 ] = : [ 0,1] be a set of ebershp functons of the AFS fuzzy logc syste ( EM,,, ) The EM eleents are of the for = I A Let be bnary relatons of sple concepts If the condton x, z y, z ( ) ( ) s vald for every sple concept A, I and xyz,, then the ebershp functons satsfy the condtons (Z1) - (Z7) of Zadeh algebra n Defnton 218 Proof We verfy the Zadeh algebra axos: The frst three axos (Z1) - (Z3) are clear (Z1) The operatons and are coutatve on [ 0,1] ( x) ( x) = ( x) ( x) ( x) ( x) = ( x) ( x) (Z2) The operatons and are assocatve on [ 0,1] (Z3) The operatons and are dstrbutve on [ 0,1] (Z4) The neutral eleents of the operatons and are : 0 [ 0,1 ] 0 ( x) 0 and : 1 [ 0,1 ], ( ) 1 x 1 ( x) = 0 ( x) = ( x) ( x) = ( )( x) 0 0 ( x) = 1 ( x) = 1( x) ( x) = ( 1 )( x) (Z5) For any : [ 0,1] and for all x ton [ ] [ ] ( ) = g( ) η: 0,1 0,1, η where the operaton g : EM EM s defned by I g = g( ) A I A, there exsts a unary opera- Let M be sple concepts Observe that we need ths assupton for Defnton 217 used bellow: n Defnton 216 and Defnton 217 the defnton of ( xy, ) deands to be sple Accordng to Proposton 28 g s an order reversng nvoluton and g( g( ) ) = but n ths case need not be sple We obtan ( ( )) g( ) ( ) g g( ) η η = η = ( ) = 363

12 P Kukkuranen Therefore η s an nvoluton Here s not necessary sple Let α β g α be eleents n EM, and snce g s order reversng, ( ) g( β) Usng Defnton 217 (1) t s ( ) ( x) ( ) ( x) Therefore and η s order reversng g α g β ( α) = ( ) g g( ) = α β ( β) η η (Z6) 0( x) 1( x) for all x (Z7) The Kleene condton For any functon [ ] xy, we have βα, 0,1 and for any β( y) β( y) β( y) β( x) α( x) α( x) α( x) whch s proved n Lea 33 Proposton 35 Let { } [ 0,1 ] = : [ 0,1] be a set of ebershp functons ξ of the AFS fuzzy logc syste EM,,, The EM eleents are of the for ( ) Let = I A be bnary relatons of sple concepts If the condton xz yx s vald for every sple concept A, I and xyz,, then (a) Functons (b) Any chan ( ) EM Proof We conclude for a chan correspondng to the chan ( ) EM consttutes soe Zadeh algebra [ ] ( 0,1,,, η,, 0 1) (a) By Lea 31 the eleents fors a chan ( ) By Defnton EM 217 (1) k l k l (b) Lea 31 (a) ples that x z y z and n Proposton 34 t s proved that every chan ( ) EM (Z7) satsfes (Z1) - 4 Conclutons Sple concepts for chans The eleents of any chan for a bass n AFS EM,,, wth operatons dsjuncton, conjuncton fuzzy logc syste ( ) and the logcal negaton The eleents are of the for ( ) I A where sple concepts A are defned n Defnton 213 and operatons n ( EM,,, ) are defned n Proposton 27 ( EM,, ) s a copletely dstrbutve lattce By eans of the bnary relatons defned n Defnton 210 we construct the condton ( xz, ) ( yx, ) whch ples the condton ( x, z) ( y, z) Here s a non-epty set and xyz,, on and are sple concepts Then the condtons const- 364

13 P Kukkuranen of the fuzzy concepts for chans ( ) ; second, every chan EM ( ) fors a Zadeh algebra These results are represented n Propostons EM 34 and 35 and we can use the as startng ponts to contnue theoretcal consderatons The other way to contnue the nvestgatons s to utlze drectly the condtons above: there are two knds of successve events The frst one ples the second one or they have no connecton In the latter case the second event only follows the frst one although they are ndependent of each other In the frst case t s possble to apply to the condtons (above) (1) or (2) represented n the ntroducton In Exaple 21, [2], the set { : [ 0,1] } of ebershp functons fors a Zadeh algebra More exactly, f I = [ 0,1] then ( I,, ) s a coplete dstrbutve lattce Further, I = { : [ 0,1] } s the set of ebershp functons The operatons and are extended pontwse on I Now Z = { I, 01,,,, } s a Zadeh algebra wth 0 ( x ) = 0, 1 ( x ) = 1, and the logcal negaton of s ( x) = ( 1 )( x) In ths paper we have consdered ore general ebershp functons and constructed Zadeh algebras tute the two thngs: frst, the ebershp functons : [ 0,1] n ( EM,,, ) eferences [1] Lu, D and Pedrycz, W (2004) Axoatc Fuzzy Set Theory and Its Applcatons Studes n Fuzzness and Soft Coputng 2009, Sprnger-Verlag, Berln Hedelberg [2] Mattla, JK (2012) Zadeh Algebra as the Bass of Lukasewcz Logcs Lappeenranta Unversty of Technology, Departent of Matheatcs and Physcs [3] Brkhoff, G (1995) Lattce Theory Aercan Matheatcal Socety 3rd Edton, Eght Prntng 1995, Colloqu Publcatons, ISBN Prnted n the Unted States of Aerca [4] Mattla, JK (2002) Suean logkan oppkrja Johdatus suean ateatkkaan Kolas uusttu panos, Dark Oy, ISBN , Vantaa [5] Negota, LV and alescu, DA (1975) Applcatons of Fuzzy Sets to Systes Analyss Brkhauser Verlag, ISBN , Basel und Stuttgart Subt or recoend next anuscrpt to SCIP and we wll provde best servce for you: Acceptng pre-subsson nqures through Eal, Facebook, LnkedIn, Twtter, etc A wde selecton of journals (nclusve of 9 subjects, ore than 200 journals) Provdng 24-hour hgh-qualty servce User-frendly onlne subsson syste Far and swft peer-revew syste Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the nuber of cted artcles Maxu dssenaton of your research work Subt your anuscrpt at: Or contact ap@scrporg 365