Neutrosophic Vague Set Theory

Transcription

1 9 Netrosoph Vage Set Theory Shawkat lkhazaleh Department of Mathemats Falty of Sene and rt Shaqra Unersty Sad raba bstrat In 99 Ga and Behrer proposed the theory of age sets as an etenson of fzzy set theory. Vage sets are regarded as a speal ase of ontet-dependent fzzy sets. In 995 Smarandahe talked for the frst tme abot netrosophy and he defned the netrosoph set theory as a new mathematal tool for handlng problems nolng mprese ndetermnay and nonsstent data. In ths paper we defne the onept of a netrosoph age set as a ombnaton of netrosoph set and age set. We also defne and stdy the operatons and propertes of netrosoph age set and ge some eamples. Keywords Vage set Netrosophy Netrosoph set Netrosoph age set. knowledgement We wold lke to aknowledge the fnanal spport reeed from Shaqra Unersty. Wth or snere thanks and appreaton to Professor Smarandahe for hs spport and hs omments. Introdton Many sentsts wsh to fnd approprate soltons to some mathematal problems that annot be soled by tradtonal methods. These problems le n the fat that tradtonal methods annot sole the problems of nertanty n eonomy engneerng medne problems of deson-makng and others. There hae been a great amont of researh and applatons n the lteratre onernng some speal tools lke probablty theory fzzy set theory [] rogh set theory [9] age set theory [8] nttonst fzzy set theory [0 ] and nteral mathemats [ 4]. Crtal Reew. Volme X 05

2 0 Shawkat lkhazaleh Netrosoph Vage Set Theory Sne Zadeh pblshed hs lassal paper almost ffty years ago fzzy set theory has reeed more and more attenton from researhers n a wde range of sentf areas espeally n the past few years. The dfferene between a bnary set and a fzzy set s that n a normal set eery element s ether a member or a non-member of the set; t ether has to be or not. In a fzzy set an element an be a member of a set to some degree and at the same tme a non-member of the same set to some degree. In lassal set theory the membershp of elements n a set s assessed n bnary terms: aordng to a balent ondton an element ether belongs or does not belong to the set. By ontrast fzzy set theory permts the gradal assessment of the membershp of elements n a set; ths s desrbed wth the ad of a membershp fnton aled n the losed nt nteral [0 ]. Fzzy sets generalse lassal sets sne the ndator fntons of lassal sets are speal ases of the membershp fntons of fzzy sets f the later only take ales 0 or. Therefore a fzzy set n an nerse of dsorse X s a fnton : X [0 ] and sally ths fnton s referred to as the membershp fnton and denoted by μ (). The theory of age sets was frst proposed by Ga and Behrer [8] as an etenson of fzzy set theory and age sets are regarded as a speal ase of ontet-dependent fzzy sets. age set s defned by a trth-membershp fnton t and a falsemembershp fnton f where t () s a lower bond on the grade of membershp of dered from the edene for and f () s a lower bond on the negaton of dered from the edene aganst. The ales of t () and f () are both defned on the losed nteral [0 ]wth eah pont n a bas set where t () + f (). For more nformaton see [ ]. In 995 Smarandahe talked for the frst tme abot netrosophy and n 999 and 005 [4 6] defned the netrosoph set theory one of the most mportant new mathematal tools for handlng problems nolng mprese ndetermnay and nonsstent data. In ths paper we defne the onept of a netrosoph age set as a ombnaton of netrosoph set and age set. We also defne and stdy the operatons and propertes of netrosoph age set and ge eamples. Crtal Reew. Volme X 05

3 Shawkat lkhazaleh Netrosoph Vage Set Theory Prelmnares In ths seton we reall some bas notons n age set theory and netrosoph set theory. Ga and Behrer hae ntroded the followng defntons onernng ts operatons whh wll be sefl to nderstand the sbseqent dssson. Defnton. ([8]). Let be a age ale [ t f] where t f 0 and 0 t f. If t and f 0 (.e. ) then s alled a nt age ale. If 0 a zero age ale. t and f (.e ) then s alled Defnton. ([8]). Let and y be two age ales where [ t f ] and y ty f y. If t t and y (.e. [ t f] ty f y ). f f y then age ales and y are alled eqal Defnton. ([8]). Let be a age set of the nerse U. If U t and f 0 then s alled a nt age set where n. If U t 0 and n. f then s alled a zero age set where Defnton.4 ([8]). The omplement of a age set s denoted by t f s defned by f t. and Defnton.5 ([8]). Let and B be two age sets of the nerse U. If B B then the age set and B are alled eqal where n. U t f t f Defnton.6 ([8]). Let and B be two age sets of the nerse U. If U t t and f f then the age set are B B nlded by B denoted by B where n. Defnton.7 ([8]). The non of two age sets and B s a age set C wrtten as C B fntons are related to those of and B by whose trth-membershp and false-membershp f ma f f mn f f t ma t t C B. C B B Crtal Reew. Volme X 05

4 Shawkat lkhazaleh Netrosoph Vage Set Theory Defnton.8 ([8]). The nterseton of two age sets and B s a age set C wrtten as C B whose trth-membershp and false-membershp fntons are related to those of and B by f mn f f ma f f t mn t t C B. C B B In the followng we reall some defntons related to netrosoph set gen by Smarandahe. Smarandahe defned netrosoph set n the followng way: Defnton.9 [6] netrosoph set on the nerse of dsorse X s defned as = {< T ( ) I ( ) F ( ) > X } where T I F: X ] 0 [ and 0 T ( ) I ( ) F ( ). Smarandahe eplaned hs onept as t follows: "For eample netrosoph log s a generalzaton of the fzzy log. In netrosoph log a proposton s T tre I ndetermnate and F false. For eample let s analyze the followng proposton: Pakstan wll wn aganst Inda n the net soer game. Ths proposton an be ( ) whh means that there s a possblty of 60% that Pakstan wns 0% that Pakstan has a te game and 0% that Pakstan looses n the net game s. Inda." Now we ge a bref oerew of onepts of netrosoph set defned n [8 5 7]. Let S and S be two real standard or non-standard sbsets then S S { s s s S and s S } S s s S { } S S { s s s S and s S } S S { s. s s S and s S } S s s S { }. Defnton.0 (Contanment) netrosoph set s ontaned n the other netrosoph set B B f and only f nf T nf T spt sp T B B nf I nf I sp I sp I B B for all X. nf F nf F sp F sp F B B Crtal Reew. Volme X 05

5 Shawkat lkhazaleh Netrosoph Vage Set Theory Defnton. The omplement of a netrosoph set s denoted by and s defned by T T I I   F F for all  X. Defnton. (Interseton) The nterseton of two netrosoph sets and B s a netrosoph set C wrtten as C = B whose trth-membershp ndetermnay-membershp and falsty-membershp fntons are related to those of and B by T T T C B I I I C B for all X. F F F C B Defnton. (Unon) The non of two netrosoph sets and B s a netrosoph set C wrtten as C = B whose trth-membershp ndetermnay-membershp and falsty-membershp fntons are related to those of and B by T T T T T C B B I I I I I C B B F F F F F for all X. C B B Netrosoph Vage Set age set oer U s haraterzed by a trth-membershp fnton t and a false-membershp fnton f t : U 0 and f : U 0 respetely where t s a lower bond on the grade of membershp of whh s dered from the edene for f s a lower bond on the negaton of dered from the edene aganst and t f. The grade of membershp of of 0. The age ale t f ndates that the eat grade of n the age set s bonded to a sbnteral t f Crtal Reew. Volme X 05

6 4 Shawkat lkhazaleh Netrosoph Vage Set Theory membershp µ of maybe nknown bt t s bonded by where t f. Let U be a spae of ponts t µ f (objets) wth a gener element n U denoted by. netrosoph sets (Nsets) n U s haraterzed by a trth-membershp fnton an ndetermnay-membershp fnton I and a falsty-membershp fnton F. T ; I and F are real standard or nonstandard sbsets of 0. It an be wrtten as: T ( ) I ( ) F ( ) : U T ( ) I ( ) F ( ) 0. There s no restrton on the sm of T ; I and F so: 0 sp T ( ) sp I ( ) sp F ( ). By sng the aboe nformaton and by addng the restrton of age set to netrosoph set we defne the onept of netrosoph age set as t follows. Defnton. netrosoph age set dsorse X wrtten as T (S n short) on the nerse of = {< T () I () F () > X} whose trth-membershp ndetermnay-membershp and falsty-membershp fntons s defned as where T ( ) T T I ( ) I I F ( ) F F T F F T and 0 T I F when X s ontnos a S an be wrtten as T I F / X. X When X s dsrete a S an be wrtten as n T I F / X. In netrosoph log a proposton s T tre I ndetermnate and F false sh that: 0 sp T ( ) sp I ( ) sp F ( ). N N N Crtal Reew. Volme X 05

7 Shawkat lkhazaleh Netrosoph Vage Set Theory 5 lso age log s a generalzaton of the fzzy log where a proposton s T tre and F false sh that: t f he eat grade of membershp µ of maybe nknown bt t s bonded by. t µ f For eample let s analyze the Smarandahe's proposton sng or new onept: Pakstan wll wn aganst Inda n the net soer game. Ths proposton an be as t follows: T I and F whh means that there s possblty of 60% to 90% that Pakstan wns 0% to 40% that Pakstan has a te game and 40% to 60% that Pakstan looses n the net game s. Inda. Eample. Let U be a set of nerse we defne the S as follows: Defnton. Let be a S of the nerseu where U then T ( ) I ( ) 0 0 F ( ) 0 0 s alled a nt S where n. Let be a S of the nerseu where U T ( ) 0 0 I ( ) F ( ) then s alled a zero S where n. Defnton. The omplement of a S by T ( ) T T I ( ) I I F ( ) F F s denoted by and s defned Crtal Reew. Volme X 05

8 6 Shawkat lkhazaleh Netrosoph Vage Set Theory Eample. Consderng Eample. we hae: Defnton.5 Let then the S Defnton.6 Let then the S and B be two Ss of the nerse U. If U T T I I and F = F B B B and B are alled eqal where n. and B be two Ss of the nerse U. If U T T I I and B B F F B are nlded by B denoted by B where n. Defnton.7 The non of two Ss C B and B s a S C wrtten as whose trth-membershp ndetermnay-membershp and false-membershp fntons are related to those of and B by T C () = [ma(t T B ) ma(t T B )] I C () = [mn(i I B ) mn(i I B )] F C () = [mn(f F B ) mn(f F B )]. Defnton.8 The nterseton of two Ss as H B and false-membershp fntons are related to those of and B s a S H wrtten whose trth-membershp ndetermnay-membershp and B by T H () = [mn(t T B ) mn(t T B )] I H () = [ma(i I B ) ma(i I B )] F H () = [ma(f F B ) ma(f F B )]. Eample. Let U be a set of nerse and let S and B defne as follows: Crtal Reew. Volme X 05

9 Shawkat lkhazaleh Netrosoph Vage Set Theory B Then we hae C B where C Moreoer we hae H B where H Theorem. Let P be the power set of all S defned n the nerse X. Then P ; s a dstrbte latte. Proof Let B C be the arbtrary Ss defned on X. It s easy to erfy that (dempoteny) B B B B (ommtatty) ( B) C ( B C)( B) C ( B C) (assoatty) and ( B C) ( B) ( C) ( B C) ( B) ( C) (dstrbtty). Crtal Reew. Volme X 05

10 8 Shawkat lkhazaleh Netrosoph Vage Set Theory 4 Conlson In ths paper we hae defned and stded the onept of a netrosoph age set as well as ts propertes and ts operatons gng some eamples. 5 Referenes []. Kmar S.P. Yada S. Kmar Fzzy system relablty analyss sng based arthmet operatons on L R type nteral aled age sets n Internatonal Jornal of Qalty & Relablty Management 4 (8) (007) [] D. H. Hong C. H. Cho Mltrtera fzzy deson-makng problems based on age set theory n Fzzy Sets and Systems 4 (000) 0. [] F. Smarandahe Netrosoph set a generalsaton of the nttonst fzzy sets n Inter. J. Pre ppl. Math. 4 (005) [4] F. Smarandahe Netrosophy: Netrosoph Probablty Set and Log mer. Res. Press Rehoboth US 05 p [5] Florentn Smarandahe nfyng feld n logs. Netrosophy: Netrosoph probablty set and log meran Researh Press Rehoboth 999. [6] H. Bstne P. Brllo Vage sets are nttonst fzzy sets n Fzzy Sets and Systems 79 (996) [7] H. Wang F. Smarandahe Y. Q. Zhang and R. Snderraman Interal Netrosoph Sets and Log: Theory and pplatons n Comptng Hes Phoen Z 005. [8] J. Wang S.Y. L J. Zhang S.Y. Wang On the parameterzed OW operators for fzzy MCDM based on age set theory n Fzzy Optmzaton and Deson Makng 5 (006) 5 0. [9] K. tanasso Inttonst fzzy sets n Fzzy Sets and Systems 0 (986) [0] K. tanasso Operators oer nteral aled nttonst fzzy sets n Fzzy Sets and Systems 64 (994) [] L. Zho W.Z. W On generalzed nttonst fzzy rogh appromaton operators n Informaton Sene 78 () (008) [] L.. Zadeh Fzzy sets n Informaton and Control 8 (965) 8 5. [] M. B. Gorzalzany method of nferene n appromate reasonng based on nteral-aled fzzy sets n Fzzy Sets and Systems (987) 7. Crtal Reew. Volme X 05

11 Shawkat lkhazaleh Netrosoph Vage Set Theory 9 [4] S. M. Chen nalyzng fzzy system relablty sng age set theory n Internatonal Jornal of ppled Sene and Engneerng () (00) [5] S. M. Chen Smlarty measres between age sets and between elements IEEE Transatons on Systems Man and Cybernets 7 () (997) [6] F. Smarandahe Netrosoph set - generalzaton of the nttonst fzzy set IEEE Internatonal Conferene on Granlar Comptng (006) 8-4. [7] W. L. Ga D.J. Behrer Vage sets IEEE Transatons on Systems. Man and Cybernets () (99) [8] Z. Pawlak Rogh sets n Internatonal Jornal of Informaton and Compter Senes (98) Crtal Reew. Volme X 05