Processing of Information with Uncertain Boundaries Fuzzy Sets and Vague Sets

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1 Processg of Iformato wth Ucerta odares Fzzy Sets ad Vage Sets JIUCHENG XU JUNYI SHEN School of Electroc ad Iformato Egeerg X'a Jaotog Uversty X'a PRCHIN bstract: - I the paper we aalyze the relatoshps ad propertes of some essetal theores betwee fzzy sets ad vage sets We fd that the certa bodares of the two kds of sets ca be descrbed by a sbterval of 0] The the method for measrg the smlarty of fzzy sets s developed based o the dea of vage sets Fally a form model for measrg the smlarty of two same type sets s proposed each of the two kds of sets metoed above It s proved that fzzy sets ad vage sets are eqvalet o measrg the smlarty of two same type sets accordg to ths form model d some propertes of ths model are also preseted ths paper Ths model s very sefl to settle tellget data formato processg ad fzzy cotrol problems Key-Words: - Fzzy sets; Vage sets; odary; Membershp fcto; Smlarty measre; Degree of smlarty Itrodcto Fzzy sets ad vage sets are extesos of classcal set theory They ca be sed the processg of formato wth certa bodares -3] The smlarty measre of certa sets s a mportat cocept of formato dsposal wth certa bodares ad t s also a mportat bass of processg certa formato 4-5] I ths paper we fd that all the certa bodares of the two kds of sets ca be descrbed by a sbterval of 0] Followg ths dea ad accordg to the dea of measrg the smlarty of vage sets 5-7] we propose the method of measrg the smlarty of fzzy sets d the a form model for measrg the smlarty of two same type sets s preseted each of the two kds of sets metoed above The form model satsfes reflexve ad symmetrc propertes et al It s proved that fzzy sets ad vage sets are eqvalet the form model d the deas ad techqes of ths paper ca also be easly employed to settle other problems of fzzy tellget cotrol ad fzzy clsterg aalyss 2 asc theory 2 asc theory of fzzy sets Let U be a o-empty set called the verse a fzzy set ~ of U s defed by a member fcto µ ~ : U 0] The core ad spport of fzzy set ~ are defed as follows ]: core µ ~ ) = U ) = } ) µ ~ spp µ ~ ) = U ~ ) > 0} 2) µ

2 The bodary rego of fzzy set ~ s defed as follows: N ~ = spp µ ~ ) core µ ~ ) 3) Defto 3 Let  be a vage set of verse U the bodary rego of  s defed as follows: ˆ N \ ˆ = U t ) < f ) } ˆ = ˆ ˆ < 22 asc theory of vage sets Let U be a fte ad o-empty verse U = } U a vage set  2 U s characterzed by a trth-membershp fcto t ˆ ad a false-membershp fcto t : U 0]  f : U 0]  f ˆ where t ) s a lower bodary o the grade of ˆ membershp of derved from the evdece for f ) s a lower bodary o the egato of ˆ derved from the evdece agast t ) f ) 2] The grade of membershp of ˆ ˆ the vage set  s boded to a sbterval ˆ ˆ t ) f )] of 0] Defto Let  be a vage set of verse U the postve rego of  s defed as follows: ˆ = U t ) } ˆ = Defto 2 Let  be a vage set of verse U the pper approxmato of follows: ˆ = U f ) } ˆ <  s defed as 23 Relatos amog µ ~ x) t x) ad f ˆ x) Fzzy set theory or vage set theory emphaszes the membershp relatos of elemets a set amely membershp degree The membershp fctos of fzzy sets ad vage sets are obtaed by the statstcs or expereces of experts ll membershp fcto vales of a fzzy set are a sgle vale of 0] Ths sgle vale cotas oly two kds of formato of a elemet x x U ) e spport degree ad egatve degree Whereas all membershp fcto vales of a vage set are a sbterval of 0] Ths sbterval cotas three kds of formato of a elemet x x U ) e spport degree egatve degree ad kow degree Vage sets are more accrate to descrbe some vage formato tha fzzy sets Theorem Let be a sbset of the verse of dscorse U for x U The fzzy sets ad vage sets the grade of membershp of x cold be descrbed by a sbterval a b] of 0] where a b 0] a s a lower bodary o the grade of membershp of x derved from the evdece for x ad b s a pper bodary o the grade of membershp of x derved from the evdece for x Proof I fzzy set the grade of membershp of x s µ x ) t s sed to create a ˆ 2

3 sbterval a b] of 0] amely a = t ) f )] be the vage membershp ˆ ˆ a b] = µ x) x)] I vage set the grade µ of membershp of x s boded by a sbterval t ˆ x) ˆ x)] of 0] amely f a b] = t ˆ x) ˆ x)] f Theorem 2 I fzzy sets ad vage sets ther certates of bodary regos ca be descrbed wth the lower approxmato sbtracted from the pper approxmato Proof I fzzy set ~ the lower approxmato of set ~ s ~ = U µ ~ ) = } ad the pper approxmato of set ~ s ~ = U µ ~ ) > 0} the the bodary rego of set ~ s ~ ~ \ I vage set  by sg Defto 3 the bodary rego of set  s ˆ \ ˆ 3 Smlarty measres betwee two same type sets of fzzy sets ad vage sets 3 Smlarty measre betwee vage sets 5-7] Let  ad ˆ be two vage sets the verse of dscorse U U = } U 2 vale of vage set  ad b = t ) ˆ ˆ f )] be the vage membershp vale of vage set ˆ The the degree of smlarty betwee the vage vales a ad evalated by fcto M as follows: M a b ) = b ca be S a ) S b ) t ˆ ) t ˆ ) f ˆ ) f ˆ ) 4 4) where S a ) = t ) f ) ad ˆ ˆ S b ) = t ) f ) ) ˆ ˆ Let vage sets  ad ˆ be defed as follows: ˆ = t ˆ = t ˆ ) f ˆ )] / t ˆ 2 ) f ˆ 2 )]/ 2 t )]/ ˆ ) f ˆ ˆ ) f ˆ )] / 2 t ˆ 2 ) f ˆ 2 )]/ t )]/ ˆ ) f ˆ The the degree of smlarty betwee the vage sets  ad ˆ ca be evalated by fcto VSmD as follows: VSmD ˆ ˆ) = 0 = M a b ) f ˆ = ˆ = Φ; else f codto; otherwse 5) 3

4 where codto s t ) 0 ad t ) 0 or ˆ = ) ˆ = ) ˆ = ) ˆ = ˆ t ) 0 ad t 0 at the same tme ˆ t ) f ad ˆ t ) f = 2 ) ˆ M a b ) s defed Eq4) or µ ~ ) = 0 ad µ ~ ) 0 = 2 ) M a b ) = µ ~ ) µ ~ ) a = µ ~ ) ad b = µ ~ ) 32 Smlarty measre betwee fzzy sets Let ~ ad ~ be two fzzy sets the verse of dscorse U U = } U 2 a = µ ~ ) be the fzzy membershp vale of set ~ ad b = µ ~ ) be the fzzy membershp vale of smlarty betwee the fzzy vales ~ The the degree of be evalated by fcto M as follows: where a ad b ca M a b ) = µ ~ ) µ ) 6) ~ Let fzzy sets ~ ad ~ of the verse of dscorse U be defed as follow: ~ = µ ~ ) / µ ~ 2 ) / 2 µ ~ ) / ~ = µ ~ ) / µ ~ 2 ) / 2 µ ~ ) / The the degree of smlarty betwee the fzzy sets ~ ad ~ ca be evalated by fcto FSmD as follows: FSmD ~ ~ ) = 0 = M a b ) f = = Φ; else f codto; otherwse 7) where codto s µ ~ ) 0 ad µ ~ ) = 0 33 form model of the smlarty measre betwee two same type sets of fzzy sets ad vage sets ased o Theorem ad the smlarty measres of fzzy sets ad vage sets we propose a form model for measrg the smlarty of two same type sets each of the two kds of sets metoed above The form model s defed as follows: f = = Φ; SmD ) = 0 else f codto; M a b ) otherwse = 8) where U s a fte ad o-empty set called the verse U = } ad are 2 two same type sets of U The codto the Eq8) s t ) = 0 ad t ) 0 or t ) 0 ad t ) = 0 at the same tme t ) f ) = ad t ) f ) = The degree of smlarty betwee sets ad ca be evalated by fcto SmD ) I fzzy sets SmD ) s FSmD ) I vage sets SmD ) s VSmD ) The M a b ) the Eq8) s defed as follows: M a b ) = S a ) S b ) t ) t ) f ) f ) 4 4

5 where the degree of smlarty abot elemet 9) betwee sets ad s evalated by M a b ) I M a b ) a = t ) f )] ad b = t ) f )] by applyg Theorem ) S a ) = t ) f ) S b ) = t ) f ) thereto: f If s a vage set the t ) = t ) ad ) = f )) If s a fzzy set the t ) = µ ) ad f ) = µ ) t ) ad f ) are smlar to t ) ad f ) Obvosly these measre methods of fzzy sets ad vage sets are eqvalet o measrg the smlarty of two same type sets ths form model Eq 8) The larger the vale of SmD ) the more the smlarty betwee the sets ad The smaller the vale of SmD ) the less the smlarty betwee the sets ad Ths form model Eq 8) satsfes some propertes as follows: Theorem 3 I fzzy sets or vage sets M a b ) satsfes some propertes: ) M a b ) 0] 2) M a b ) = M b a ) 3) M a b ) = 0 a = 00] ad b = ] or a = ] ad b = 00] where The proof s omtted Theorem 4 I fzzy sets or vage sets SmD ) 0] Proof The coclso s clear accordg to Theorem 3 ad Eq 8) Theorem 5 I fzzy sets or vage sets SmD ) satsfes ) reflexve: SmD ) = ; 2) symmetrc: SmD ) = SmD ) ; Theorem 6 I fzzy sets or vage sets SmD ) = 0 f ad oly f t ) = 0 ad t ) 0 or t ) 0 ad t ) = 0 meawhle t ) f ) = ad Where t ) f ) = for all = 2 a = t ) f )] b = t ) f )] Proof The coclso s a drect reslt of Eq 8) For ay sbsets U SmD ) does t satsfy the trastvty 4 Coclso I ths paper we aalyze these relatos betwee some cocepts of fzzy sets ad vage sets ad preset that the certa bodares of the two kds of sets ca be descrbed by a sbterval of 0] The the method for measrg the smlarty of fzzy sets s developed based o the dea of vage sets ased o the Theorem we proposed a form model for measrg the smlarty of two same type sets of fzzy oes or vage oes Ths 5

6 form model satsfes reflexve ad symmetrc propertes et al It s proved that fzzy sets ad vage sets are eqvalet o measrg the smlarty of two same type sets ths form model Ths model s very sefl to hadlg certa reasog ad fzzy tellget cotrol problems ckowledgemets Ths work was spported part by the Natoal Natral Scece Fodato of Cha der the grat No ad Natral Scece Fodato of Hea Provce der the grat No Refereces: ] L Zadeh Fzzy sets Iformato ad Cotrol 9658: ] WL Ga ad DJ ehrer Vage sets IEEE Tras Systems Ma Cyberetcs ): ] YYYao comparatve stdy of fzzy sets ad rogh sets Joral of formato sceces 99809: ] H Lee-Kwag YS Sog ad KM Lee Smlarty measre betwee fzzy sets ad betwee elemets Fzzy sets ad Systems 99462: ] SM Che Measres of smlarty betwee vage sets Fzzy Sets ad Systems 99574: ] DH Hog ad C Km ote o smlarty measres betwee vage sets ad betwee elemets Iformato Sceces 9995: ] F L ad ZY X Smlarty measre betwee vage sets Software Joral 20026): bstract Chese) 6