International Journal of Mathematical Archive-5(1), 2014, Available online through ISSN

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1 ntrnational Journal of Mathmatical rchiv vailabl onlin through SSN ON -CU UZZY NEUROSOPHC SO SES. rockiarani* &. R. Sumathi* *Dpartmnt of Mathmatics Nirmala Collg f womn Coimbatamilnadu ndia. Rcivd on: ; Rvisd & ccptd on: SRC n this papr w study th notions of -cut and strong cut of an uzzy Nutrosophic soft st. Som rlatd proprtis hav bn stablishd with countr ampls. lso w hav dfind disjunctiv sum and dfrnc of two fuzzy nutrosophic soft sts and thir charactrizations ar discussd. Kywds: uzzy Nutrosophic st uzzy Nutrosophic soft st Disjunctiv sum and dfrnc -cut and strong cut of an uzzy Nutrosophic soft st. MSC 2000: E NRODUCON Many this hav bn dvlopd f uncrtaintis including th thy of probability thy of fuzzy sts thy of intuitionistic fuzzy sts and thy of rough sts and so on. lthough many nw tchniqus hav bn dvlopd as a rsult of ths this yt dficultis ar still thr. h maj dficultis aris du to inadquacy of paramtrs. h novl notion of soft st was initiatd by Molodtsov[6] in his class of sts is a compltly nw mthod f modling uncrtainty and had a rich potntial f application in svral dirctions. his so- calld soft st thy is fr from th dficultis affcting isting mthods. h fuzzy st was introducd by Zadh [13] in 1965 whr ach lmnt had a dgr of mmbrship. h intuitionistic fuzzy st S f sht on a univrs X was introducd by K.tanaasov[2] in 1983 as a gnralization of fuzzy st whr bsids th dgr of mmbrship and th dgr of non mmbrship of ach lmnt. h concpt of Nutrosophic st which is a mathmatical tool f handling problms involving imprcis indtrminacy and inconsistnt data was introducd by. Smarandach [11]. Pabitra Kumar Maji [10] had combind th Nutrosophic st with soft sts and introducd a nw mathmatical modl Nutrosophic soft st. n [9] Nog and Sut hav dfind disjunctiv sum and dfrnc of two fuzzy soft sts. h notions of - cut soft st and - cut strong soft st of a fuzzy soft st hav bn put fward in thir wk. n [5] Manoj a t al. hav dfind disjunctiv sum and dfrnc -cut soft st and cut strong soft st of an ntuitionistic uzzy soft sts. n th prsnt study w hav dfind disjunctiv sum and dfrnc -cut and strong cut of an uzzy Nutrosophic soft st. 2. PRELMNRES Dfinition: 2.1[11] Nutrosophic st on th univrs of discours X is dfind as = whr ] [and Dfinition: 2.2 [10] Lt U b th initial univrs st and E b a st of paramtrs. Lt PU dnots th powr st of U. Considr a non-mpty st E. pair is calld a soft st ovr U whr is a mapping givn by : PU Dfinition: 2.3[10]Lt U b th initial univrs st and E b a st of paramtrs. Considr a non-mpty st E. Lt PU dnots th st of all nutrosophic sts of U. h collction is trmd to b th soft nutrosophic st ovr U whr is a mapping givn by : PU. Crsponding auth:. R. Sumathi* *Dpartmnt of Mathmatics Nirmala Collg f womn Coimbatamilnadu ndia. sumathi_raman2005@yahoo.co.in ntrnational Journal of Mathmatical rchiv- 51 Jan

2 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan Dfinition: 2.4 [10] Union of two Nutrosophic soft sts and ovr U E is Nutrosophic soft st whr C = C. ; - H = ; - ; and is writtn as = HC. Dfinition: 2.5 [10] ntrsction of two Nutrosophic soft sts and ovr U E is Nutrosophic soft st whr C = C. H = and is writtn as = H C. Dfinition: 2.6 [1] uzzy Nutrosophic st on th univrs of discours X is dfind as = whr [0 1] and Dfinition: 2.7 [1] Lt U b th initial univrs st and E b a st of paramtrs. Considr a non-mpty st E. Lt P U dnots th st of all fuzzy nutrosophic sts of U. h collction is trmd to b th fuzzy nutrosophic soft st ovr U whr is a mapping givn by : PU. hroughout this papr uzzy Nutrosophic soft st is dnotd by NS st / NSS. Dfinition: 2.8 [1] fuzzy nutrosophic soft st is containd in anothr nutrosophic st. i.. X. Dfinition: 2.9 [1] h complmnt of a fuzzy nutrosophic soft st dnotd by c and is dfind as c = c whr c : PU is a mapping givn by c = < c = c = 1- = > Dfinition: 2.10 [1]Lt X b a non mpty st and = = ar fuzzy nutrosophic soft sts. hn = = ma min ma min c min ma Dfinition: 2.11 fuzzy nutrosophic soft st ovr th univrs U is said to b mpty fuzzy nutrosophic soft st with rspct to th paramtr = 0 = 0 = 1 U.t is dnotd by 0N. Dfinition: 2.12 NS st ovr th univrs U is said to b univrs NS st with rspct to th paramtr = 1 = 1 = 0 U.t is dnotd by 1 N. Not: c 0 = N 1 c and N 1 N = 0 N Dfinition: 2.13[1] i E is calld absolut uzzy Nutrosophic soft st ovr U = 1N f any E. W dnot it by UE ii E is calld rlativ null uzzy Nutrosophic soft st ovr U = Not: W dnot φ E by φ and U E by U 3. NEW OPERONS O UZZY NEUROSOPHC SO SES 0N f and E. W dnot it by φe. Dfinition 3.1: Disjunctiv sum of uzzy Nutrosophic soft sts Lt and b two fuzzy nutrosophic soft sts ovr U E. W dfin th disjunctiv sum of and as th fuzzy nutrosophic soft st H C ovr writtn as U E = H C whr C = ϕ and C U. H = mamin min 2014 JM. ll Rights Rsrvd 264

3 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan H = mamin 1 min 1 H = minma ma. Eampl: 3.2 Lt U = a b c} ; E = { } = { } E ; = { } E. = { 1 = {a b c } 2 = {a b c } 4 = {a b c }} ={ 1 = {a b c } 2 = {a b c } 3 = {a b c }} hn = H C whr C = = { 1 2 } and H C = {H 1 = {a mamin min mamin min minma ma0.60.5} b mamin min mamin min minma ma0.10.1} c mamin min mamin min minma ma0.10.2}} {H 2 = {a mamin min mamin min minma ma0.10.7} b mamin min mamin min minma ma0.30.0} c mamin min mamin min minma ma0.50.3}} {H 1 = {a ma ma min b ma ma min c min min min } {H 2 = {a ma ma min b ma ma min c min min min } H 1 = {a b c } H 2 = {a b c } Proposition: 3.3 Lt and b two NSS ovr U E. hn th following rsults hold. i = ii H C = H C Proof: i = { U } = { U } Lt = H C whr C = ϕ and C U. H = mamin min H = mamin 1 min 1 H = minma ma Lt = K D whr D = ϕ and D U. K = mamin min K = mamin 1 min 1 K = minma ma JM. ll Rights Rsrvd 265

4 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan rom 1 and 2 it follows that H C = K D hrf = Proof of ii can b don in a similar way. Proposition: 3.4 i ϕ = ii U = Proof: i = { U } ϕ = { 001 U } C Lt ϕ = H whr U. w hav H = ma min ϕ min ϕ = ma min 1 min 0 = ma 0 = H = ma min 1 ϕ min ϕ 1 = ma min 1 min 0 1 = ma 0 = H = minma ϕ ma ϕ = min ma 0 ma 1 = min ma 1 = hrf H = { U }= t follows that ϕ = ii = { U } U = { 110 U } ϕ Lt = H whr U. W hav H = mamin U min U = mamin 0 min 1 = ma0 = H = mamin 1 U min U 1 = ma min 0 min 1 1 = ma 0 1 = 1 H = minma U ma U = min ma 1 ma 0 = min ma 1 = hrf H = { 1 U }= c t follows that U = C Dfinition: 3.5 Dfrnc of uzzy Nutrosophic soft sts Lt and b two fuzzy nutrosophic soft sts ovr U E. W dfin th dfrnc of and as th fuzzy nutrosophic soft st H C ovr U E writtn as Θ = H C whr C = ϕ and C U. H = min H = min 1 H =ma 2014 JM. ll Rights Rsrvd 266

5 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan Eampl: 3.6 Using th valus of ampl 3.2 w calculat th dfrnc valus and w obtain th rsult H C as H 1 ={a b c } H 2 ={a b c } Proposition: 3.7 i Θ ϕ = ii Θ U = ϕ Proof: i Lt ϕ = H H = min ϕ = min 1 = whr U. w hav H = min 1 ϕ = min 1 = H = ma ϕ = ma 0 = hrf H = { U }= Θ ϕ t follows that = ii Lt Θ U = H whr U w hav H = min U = min 0 = 0 H = min 1 U = min 0 = 0 H = ma U = ma 1 = 1 hrf H = { U }= ϕ. t follows that U Θ 4. -CU UZZY NEUROSOPHC SO SES = ϕ Dfinition: 4.1 [-cut of an uzzy Nutrosophic soft st] Lt b an NSS ovr U E. W dfin th cut of NSS dnotd by as th soft st whr = {: ; U [01] + + 3}. Eampl: 4.2 Lt U = a b c} ; E = { } = { } E. Lt us considr an NSS as = { 2 ={a b c } 3 ={a b c } 4 ={a b c }} Lt = 0.3; = 0.4 = 0.5 ; [01]hn = = { ={ac} ={a b c} ={a b}} Dfinition: 4.3 [-strong cut of an uzzy Nutrosophic soft st] Lt b an NSS ovr U E. W dfin th cut strong soft st of NSS dnotd by + as th soft st + whr + = {: > > < ; U [01] + + 3} JM. ll Rights Rsrvd 267

6 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan Eampl: 4.4 Lt U = a b c} ; E = { } = { } E.Considr an NSS as = { 2 = a b c } 3 ={a b c } 4 ={a b c }} Lt = 0.3; = 0.4 = 0.5 ; [01] hn = = { = {} ={a b c} ={a}} Proposition: 4.5 Lt and b two NSS ovr U E. hn th following rsults hold f all [01]. i + + ii = and + iii Proof: i = + = and + + = + + Lt. hn and U; Lt us assum that thr ar [0 1] such that Now = hn thr ist = { 0 0 U f atlast on such that i and 0 < 0 < 0 > his is a contradiction sinc U w hav hus f all [01] and h rvrs inclusion is not valid which is clar from th following ampl. Eampl: 4.6 Lt U = a b c} ; E = { } = { 1 2 } E ; = { } E. = { 1 = {a b c } 2 = {a b c } = { 1 = {a b c } 2 = {a b c } 3 = {a b c }} Hr = ={ = {c} ={ac}} = ={ = {a b c} ={ac} ={c}} 2014 JM. ll Rights Rsrvd 268

7 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan t is clar that ut as a = 0.6 a = a = 0.3 a = a = a = 0.5 hus a > a a > a a < a 2 c 2 Similarly > > c 2 c 2 c 2 2 c < c 2 ii Lt = H C. hn C = C. ; - H = ; - ; ; - H = ; - ma ; ; - H = ; - ma ; H = min ; ; ; - - Now = HC = H C whr C = C. H { : U } { : U } = { : U ma ma min } Lt H f som C. hn ma ma min 2014 JM. ll Rights Rsrvd 269

8 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan JM. ll Rights Rsrvd 270 hus HC.1 Convrs part: C and whr C = = = C K ; - ; - ; K ε = Lt K f som C thn U U U } min ma ma : { } : { } : { H hus K H C. HC. 2 rom 1 & 2 =. h proof of scond rsult is similar. iii Lt = HC.. hn C = C. H = min H = min H = ma

9 . rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan Now = H C = H C whr C = and C. H = { : U } Lt H f som C. hn H min and H min and H ma and and and. hus H C 1 Convrs part: = = KC whr C = and C. K = Lt K f som C. and and. min min ma H 2 rom 1 and 2 th rsult is provd. h proof of scond rsult is similar. Proposition: 4.7 Lt b a NSS ovr U E and λ µ δ [01] thn th following rsults hold. i + ii λ µ δ λµδ + λµδ+ Proof: Lt b a NSS ovr U E thn + = {: > > < ; U [01] + + 3} {: ; U [01] + + 3} = hrf + h proof of scond rsult is similar JM. ll Rights Rsrvd 271

10 5. CONCLUSON. rockiarani* &. R. Sumathi*/ On -Cut uzzy Nutrosophic Soft Sts / JM- 51 Jan W hav mad an invstigation on disjunctiv sum and dfrnc oprats. lso w hav dfind -cut and strong cut of an uzzy Nutrosophic soft st. urthr wk in this rgard would b rquird to study whthr th notions put fward in this papr yild a fruitful rsult. REERENCES [1].rockiarani.R.SumathiJ.Martina Jncy uzzy Nutrosophic Soft opological Spacs JM [2] K.tanassov ntuitionistic fuzzy sts in V.Sgurv d. vii KRS Sssion Sofia Jun 1983 cntral Sci. and chn. Library ulg.cadmy of Scincs [3] M.a.J.Nog and D.K.Sut study on som oprations of fuzzy soft sts ntrnational Journal of Mathmatics rnds and chnology- Volum3 ssu [4] M.rfan li.ng X.Liu W.K.Min and M.Shabir On som nw oprations in soft st thy Comput. Math ppl [5] Manoj a.n.nog. D.K.Sut Som Nw oprations of uzzy Nutrosophic soft sts JSCE Volum 2 ssu 4 Sptmbr [6] D.Molodtsov Soft st hy - irst Rsults Comput.Math.ppl [7] P.K.Maji R. iswas ans.r.roy uzzy soft sts Journal of uzzy Mathmatics Vol 9 no.3 pp [8] P.K.Maji R. iswas ans.r.roy ntuitionistic uzzy soft sts h journal of fuzzy Mathmatics Vol [9].J.Nog D.K.Sut Som Nw Oprations of uzzy soft sts J. Math. Comput. Sci No [10] Pabitra Kumar Maji Nutrosophic soft st nnals of uzzy Mathmatics and nfmatics Volum 5 No [11].Smarandach Nutrosophy and Nutrosophic Logic irst ntrnational Confrnc on Nutrosophy Nutrosophic Logic St Probability and Statistics Univrsity of Nw Mico allup NM US [12].Smarandach Nutrosophic st a gnrialization of th intuituionistics fuzzy sts ntr. J. Pur ppl.math [13] L..Zadh uzzy Sts nfm and Control Sourc of suppt: Nil Conflict of intrst: Non Dclard 2014 JM. ll Rights Rsrvd 272