Some Results on Interval Valued Fuzzy Neutrosophic Soft Sets ISSN

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1 Som Rsults on ntrval Valud uzzy Nutrosophi Soft Sts SSN rokiarani Dpartmnt of Mathmatis Nirmala ollg for Womn oimbator amilnadu ndia. R. Sumathi Dpartmnt of Mathmatis Nirmala ollg for Womn oimbator amilnadu ndia bstrat: his papr proposs th notion of intrval valud fuzzy nutrosophi soft sts and som of its oprations ar dfind. lso w hav haratrizd th proprtis of th intrval valud fuzzy nutrosophi soft st. Kywords: Soft sts uzzy nutrosophi soft st ntrval valud fuzzy nutrosophi st and intrval valud fuzzy nutrosophi soft st.

2 May Vol3 ssu 5. ntrodution: Problms in onomis nginring nvironmntal sin soial sin mdial sin and most of problms in vryday l hav various unrtaintis. o solv ths impris problms mthods in lassial mathmatis ar not always adquat. ltrnativly som kind of thoris suh as probability thory fuzzy st thory rough st thory soft st thory vagu st thory t. ar wll known mathmatial tools to dal with unrtaintis. n 999 Molodtsov [8] proposd th soft st thory as a nw mathmatial tool for daling with unrtaintis whih is fr from th dfiultis affting isting mthods. Prsntly works on soft st thory ar progrssing rapidly. Maji t al. [5] dfind and studid svral oprations on soft sts. h onpt of Nutrosophi st whih is a mathmatial tool for handling problms involving impris indtrminay and inonsistnt data was introdud by. Smarandah [0 ].n nutrosophi st indtrminay is quantid pliitly and truth-mmbrship indtrminay-mmbrship and falsity-mmbrship ar indpndnt. his assumption is vry important in many appliations suh as information fusion in whih w try to ombin th data from dfrnt snsors. Pabitra Kumar Maji [9] had ombind th Nutrosophi st with soft sts and introdud a nw mathmatial modl Nutrosophi soft st. Yang t al.[2] prsntd th onpt of intrval valud fuzzy soft sts by ombining th intrval valud fuzzy st and soft st modls. iang.y t al.[4] introdud intrval valud intutionisti fuzzy soft st whih is an intrval valud fuzzy tnsion of th intuitionisti fuzzy soft st thory. n this papr w ombin intrval valud fuzzy nutrosophi st and soft st and obtain a nw soft st modl whih is intrval valud fuzzy nutrosophi soft st. Som oprations and proprtis of intrval valud fuzzy nutrosophi soft st ar also studid. 2. Prliminaris: 2.. Dfinition [0]: Nutrosophi st on th univrs of disours X is dfind as = whr ] [ and 0 3. ntrnational ournal of nnovativ Rsarh and Studis Pag 386

3 May Vol3 ssu Dfinition [8]: Lt b th initial univrs st and E b a st of paramtrs.lt P dnots th powr st of. onsidr a non-mpty st E. pair is alld a soft st ovr whr is a mapping givn by : P Dfinition [0]: Lt b th initial univrs st and E b a st of paramtrs. onsidr a non-mpty st E. Lt P dnots th st of all nutrosophi sts of. h olltion is trmd to b th soft nutrosophi st ovr whr is a mapping givn by : P Dfinition [0]: nion of two Nutrosophi soft sts and ovr E is Nutrosophi soft st whr =. ; - ; - and is writtn as ~ =. ; 2.5. Dfinition [0]: ntrstion of two Nutrosophi soft sts and ovr E is Nutrosophi soft st whr =. = and is writtn as ~ = Dfinition []: uzzy Nutrosophi st on th univrs of disours X is dfind as = whr [0 ] and Dfinition []: Lt X b a non mpty st and nutrosophi soft sts. hn ar fuzzy ~ ma ma min ntrnational ournal of nnovativ Rsarh and Studis Pag 387

4 May Vol3 ssu 5 ~ min min ma 2.8. Dfinition []: h omplmnt of a fuzzy nutrosophi soft st dnotd by and is dfind as = whr : P is a mapping givn by =< - > 3. ntrval Valud uzzy Nutrosophi Soft Sts: 3.. Dfinition: n intrval valud fuzzy nutrosophi st VNS in short on a univrs X is an objt of th form whr = X nt [0] = X nt [0] and = X nt [0] {nt[0] stands for th st of all losd subintrval of [0] satisfis th ondition X sup + sup + sup Dfinition: or an arbitrary st [0] w dfin inf and sup Dfinition: h union of two intrval valud fuzzy nutrosophi sts and is dnotd by whr { [sup sup ] [sup sup ] [inf inf ] } 3.4. Dfinition: h intrstion of two intrval valud fuzzy nutrosophi sts and is dnotd by whr ntrnational ournal of nnovativ Rsarh and Studis Pag 388

5 May Vol3 ssu 5 { [inf inf ] [inf inf ] [sup sup ] } 3.5. Dfinition: h omplmnt of intrval valud fuzzy nutrosophi sts and is dnotd by whr { / X} [ ] Not: and ar VNS Dfinition: Lt b an initial univrs and E b a st of paramtrs. VNS dnots th st of all intrval valud fuzzy nutrosophi sts of. Lt E. pair is an intrval valud fuzzy nutrosophi soft st ovr whr is a mapping givn by : VNS. Not: ntrval valud fuzzy nutrosophi soft st/sts is dnotd by VNSS/VNSSs Eampl: onsidr an intrval valud fuzzy nutrosophi soft st whr is a st of ths ars undr onsidration of th dision makr to purhas whih is dnotd by = { 2 3 } and is a paramtr st whr = { } = {pri milag ngin ompany}. h VNSS dsribs th attribut of buying a ar to th dision makr. ={< [0.60.8] [0.40.5] [0.0.2]> < 2 [0.80.9] [0.60.7] [0.050.]> < 3 [0.60.7] [ ] [ ]>} 2 ={< [0.70.8] [ ] [0.50.2]> < 2 [0.60.7] [0.40.5] [ ]> < 3 [0.50.7] [ ] [0.20.3]>} 3 ={< [ ] [0.40.5] [0.0.5]> < 2 [0.50.6] [ ] [ ]> < 3 [ ] [ ] [0.0.2]>} ntrnational ournal of nnovativ Rsarh and Studis Pag 389

6 May Vol3 ssu 5 4 ={< [ ] [ ] [0.050.]> < 2 [0.60.7] [0.0.9] [ ]> < 3 [ ] [0.50.6] [0.50.2]>} 3.8. Dfinition: Suppos that is an VNSS ovr is th fuzzy nutrosophi intrval valu st of paramtr thn all fuzzy nutrosophi intrval valu sts in VNSS ar rfrrd to as th fuzzy nutrosophi intrval valu lass of and is dnotd by thn w hav = { : } Dfinition: Lt b an initial univrs and E b a st of paramtrs. Suppos that E and b two VNSSs w say that is an intrval valud fuzzy nutrosophi soft subst of and only i. ii is an intrval valud fuzzy nutrosophi soft subst of that is for all and nd it is dnotd by. Similarly is said to b an intrval valud fuzzy nutrosophi soft supr st of is an intrval valud fuzzy nutrosophi soft subst of w dnot it by Eampl: ivn two VNSSs and = {h h 2 h 3 }. r is th st of houss. = { 2 } = {pnsiv bautul}; = { 2 3 } = {pnsiv bautul woodn} and ={<h [0.60.8] [0.40.5] [0.0.2]> <h 2 [0.80.9] [0.50.6] [0.050.]> <h 3 [0.60.7] [0.30.4] [ ]>} 2 ={<h [0.70.8] [ ] [0.50.2]> <h 2 [0.60.7] [0.40.5] [0.50.5]> <h 3 [0.50.7] [0.60.7] [0.20.3]>} ntrnational ournal of nnovativ Rsarh and Studis Pag 390

7 May Vol3 ssu 5 ={<h [ ] [ ] [ ]> <h 2 [0.820.] [ ] [ ]> <h 3 [0.70.8] [0.40.5] [0.0.5]>} 2 ={<h [0.80.9] [ ] [0.0.2]> <h 2 [0.70.8] [0.60.7] [ ]> <h 3 [0.60.7] [0.80.9] [0.0.25]>} 3 ={<h [ ] [ ] [0.050.]> <h 2 [0.60.7] [ ] [ ]> <h 3 [ ] [ ] [0.0.3]>} W obtain. 3.. Dfinition: Lt and b two VNSSs ovr a univrs and ar said to b intrval valud fuzzy nutrosophi soft qual and only and and w writ = Dfinition: h omplmnt of an NNSS is dnotd by and is dfind as = whr : VNSS is a mapping givn by for all and [ ] 3.3. Eampl: h omplmnt of th VNSS for th ampl 3.0 is givn as follows: = { not pnsiv hous = {<h [0.0.2] [0.50.6] [0.60.8]> <h 2 [0.050.] [0.40.5] [0.80.9]> <h 3 [ ] [0.60.7] [0.60.7]>} not bautul hous ntrnational ournal of nnovativ Rsarh and Studis Pag 39

8 May Vol3 ssu 5 = {<h [0.50.2] [ ] [0.70.8]> <h 2 [0.50.2] [0.50.6] [0.60.7]> <h 3 [0.20.3] [0.30.4] [0.50.7]>}} 3.4. Dfinition: h VNSS ovr is said to b a null VNSS dnotd by = [00] = [00] = [] Dfinition: h VNSS ovr is said to b a absolut VNSS dnotd by = [] = [] = [00]. Not: = and = 3.6. Dfinition: f and b two VNSSs ovr th univrs thn and is an VNSS dnotd by is dfind by = whr = that is [inf [inf [sup inf inf. sup ] ] ] 3.7. Dfinition: f and b two VNSSs ovr th univrs thn or is an VNSS dnotd by is dfind by = whr = that is ntrnational ournal of nnovativ Rsarh and Studis Pag 392

9 May Vol3 ssu 5 [sup [sup [inf sup inf. sup ] ] ] 3.8. horm: Lt and b two VNSS ovr. hn w dfin th following proprtis. i [] =. ii [] =. Proof: i Suppos that =. hn w hav [] = =. Sin = and = w hav =. ssum = = whr. [sup sup ] [sup sup ] [inf inf ] Sin = and = thn w hav = <. > and = <. >. hus hrfor ntrnational ournal of nnovativ Rsarh and Studis Pag 393

10 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 394 ] sup [sup ] sup [sup ] inf [inf W onsidr and = thn w hav = < > i. Sin thn and =. hus. ] sup [sup ] inf [inf ] inf [inf. ] inf [inf ] inf inf [ ] sup [sup. ] inf [inf ] sup [sup ] sup [sup hrfor n w obtain that and ar sam oprators. hus [] = Similarly w an prov ii.

11 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag horm: Lt and b thr VNSS ovr. hn w dfin th following proprtis. i [] = []. ii [] = [] Dfinition: h union of two VNSS and ovr a univrs is an VNSS whr =. ] sup [sup ] sup [sup ] inf [inf 3.2. Dfinition: h intrstion of two VNSS and ovr a univrs is an VNSS whr =. ] inf [inf

12 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 396 ] inf [inf ] sup [sup horm: Lt E b a st of paramtrs E is a null VNSS an absolut VNSS and and E two VNSS ovr thn i = ii = iii E = E iv E = v E = vi E = E horm: f and ar two VNSSs ovr thn w hav th following proprtis. i [] =. ii [] =. Proof: i ssum that = whr = and. ] sup [sup

13 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 397 ] sup [sup ] inf [inf Sin = thn w hav = = whr = < > for all and = =. n ] inf [inf ] inf [inf ] sup [sup

14 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 398 ] sup [sup Sin = and = thn w hav =. Suppos that = D whr D = = and w tak D. ] inf [inf ] inf [inf ] inf [inf ] inf [inf

15 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 399 ] sup [sup ] sup [sup hrfor and ar sam oprators. hus [] =. Similarly w an prov ii horm: f and and b thr VNSSs ovr thn w hav th following proprtis. i [] = []. ii [] = []. iii [] = [][ ]. iv [] = [][ ]. Proof: Suppos that = S whr S = and S. ] inf [inf

16 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 400 ] inf [inf ] sup [sup Sin {] = S. Suppos S = K whr = S = thn w hav K ] inf [inf ] inf [inf ] inf [inf ] inf [inf

17 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 40 K ] inf [inf ] inf [inf ] inf [inf ] inf [inf K ] sup [sup ] sup [sup ] sup [sup ] sup [sup ssum that = RV whr V = V. R ] inf [inf

18 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 402 R ] inf [inf R ] sup [sup Sin {] = RV. Suppos RV = LW whr W = V = thn w hav L ] inf [inf ] inf [inf ] inf [inf ] inf [inf

19 May Vol3 ssu 5 ntrnational ournal of nnovativ Rsarh and Studis Pag 403 L ] inf [inf ] inf [inf ] inf [inf ] inf [inf L ] sup [sup ] sup [sup ] sup [sup ] sup [sup hrfor K = L K = L and K = L. hrfor K and L ar th sam oprators. Similarly w an prov ii iii and iv. 4. onlusions: Soft st thory in ombination with th intrval valud fuzzy nutrosophi st has bn proposd as th onpt of th intrval-valud fuzzy nutrosophi soft st. W hav studid som nw oprations and proprtis on th VNSS. s far as futur dirtions ar

20 May Vol3 ssu 5 onrnd w hop that our approah will b usful to handl svral ralisti unrtain problms. ntrnational ournal of nnovativ Rsarh and Studis Pag 404

21 May Vol3 ssu 5 Rfrns:.. rokiarani. R. Sumathi. Martina ny uzzy Nutrosophi Soft opologial Spas M K. tanassov ntuitionisti fuzzy sts in V.Sgurv d. vii KRS Sssion Sofia un 983 ntral Si. and hn. Library ulg.admy of Sins M.ora..Nog and D.K.Sut study on som oprations of fuzzy soft sts ntrnational ournal of Mathmatis rnds and hnology- Volum3 ssu iang.y ang.y hn.q Liu. ang. ntrval valud intuitionisti fuzzy soft sts and thir Proprtis omputrs and Mathmatis with appliations pp: P. K. Maji R. iswas ans. R. Roy Soft st thory omputrs and Mathmatis with appliations P. K. Maji R. iswas ans.r.roy uzzy soft sts ournal of uzzy Mathmatis Vol 9 no.3 pp P. K. Maji R. iswas ans. R. Roy ntuitionisti uzzy soft sts h journal of fuzzy Mathmatis Vol D.Molodtsov Soft st hory - irst Rsults omput.math.ppl Pabitra Kumar Maji Nutrosophi soft st nnals of uzzy Mathmatis and nformatis Volum 5 No Smarandah Nutrosophy and Nutrosophi Logi irst ntrnational onfrn on Nutrosophy Nutrosophi Logi St Probability and Statistis nivrsity of Nw Mio allup NM 8730 S Smarandah Nutrosophi st a gnrialization of th intuituionistis fuzzy sts ntr.. Pur ppl. Math X.. Yang.N.Lin. Y. Yang Y.Li D Yu ombination of intrval valud fuzzy sts and soft st omputr and Mathmatis with appliations L..Zadh uzzy Sts nform and ontrol ntrnational ournal of nnovativ Rsarh and Studis Pag 405