Interval Valued Neutrosophic Soft Topological Spaces

Transcription

1 8 Interval Valued Neutrosoph Soft Topologal njan Mukherjee Mthun Datta Florentn Smarandah Department of Mathemats Trpura Unversty Suryamannagar gartala-7990 Trpura Indamal: Department of Mathemats Trpura Unversty Suryamannagar gartala-7990 Trpura Indamal: 3 Department of Mathemats Unversty of New Mexo Gallup USmal: bstrat In ths paper we ntrodue the onept of nterval valued neutrosoph soft topologal spae together wth nterval valued neutrosoph soft fner and nterval valued neutrosoph soft oarser topology We also defne nterval valued neutrosoph nteror and loser of an nterval valued neutrosoph soft set Some theorems and examples are tes Interval valued neutrosoph soft subspae topology are studed Some examples and theorems regardng ths onept are presented Keywords: Soft set nterval valued neutrosoph set nterval valued neutrosoph soft set nterval valued neutrosoph soft topologal spae Introduton In 999 Molodtsov [9] ntrodued the onept of soft set theory whh s ompletely new approah for modelng unertanty In ths paper [9] Molodtsov establshed the fundamental results of ths new theory and suessfully appled the soft set theory nto several dretons Maj et al [7] defned and studed several bas notons of soft set theory n 003 Pe and Mao [] ktas and Cagman [] and l et al [] mproved the work of Maj et al [7] The ntutonst fuzzy set s ntrodued by tanaasov [4] as a generalzaton of fuzzy set [5] where he added degree of non-membershp wth degree of membershp Neutrosoph set ntrodued by F Smarandahe n 995 [] Smarandahe [3] ntrodued the onept of neutrosoph set whh s a mathematal tool for handlng problems nvolvng mprese ndetermnay and nonstant data Maj [8] ombned neutrosoph set and soft set and establshed some operatons on these sets Wang et al [4] ntrodued nterval neutrosoph sets Del [6] ntrodued the onept of nterval-valued neutrosoph soft sets In ths paper we form a topologal struture on nterval valued neutrosoph soft sets and establsh some propertes of nterval valued neutrosoph soft topologal spae wth supportng proofs and examples Prelmnares In ths seton we reall some bas notons relevant to soft sets nterval-valued neutrosoph sets and nterval-valued neutrosoph soft sets Defnton : [9] Let U be an ntal unverse and be PU denotes the power set of U a set of parameters Let and Then the par overu where f s a mappng gven by f : PU f s alled a soft set Defnton : [3] neutrosoph set on the unverse of dsourse U s defned as x x x x x U where 0 are funtons suh that the U ondton: x U x x x 0 3 s satsfed Here x x x represent the truthmembershp ndetermnay-membershp and falstymembershp respetvely of the element x U From phlosophal pont of vew the neutrosoph set takes the value from real standard or non-standard subsets of 0 But n real lfe applaton n sentf and engneerng problems t s dffult to use neutrosoph set wth value from real standard or non-standard subset of 0 Hene we onsder the neutrosoph set whh takes the value from the subset of 0 Defnton 3: [4] n nterval valued neutrosoph set on the unverse of dsourse U s defned as x x x x x U where U Int 0 are funtons suh that the njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

2 9 ondton: x U 0 sup x sup x sup x 3 s satsfed In real lfe applatons t s dffult to use nterval valued neutrosoph set wth nterval-value from real standard or non-standard subset of Int 0 Hene we onsder the nterval valued neutrosoph set whh takes the nterval-value from the subset of Int 0 (where Int 0 denotes the set of all losed sub ntervals of 0 ) The set of all nterval valued neutrosoph sets on U s denoted by IVNS(U) Defnton 4: [6] Let U be an unverse set be a set of parameters and Let IVNs(U) denotes the set of all nterval valued neutrosoph sets of U Then the par f s alled an nterval valued neutrosoph soft set (IVNSs n short) over U where f s a mappng gven by f IVNsU The olleton of all nterval valued neutrosoph soft sets over U s denoted by IVNSs(U) Defnton 5: [6] Let U be a unverse set and be a f g B IVNSs U where set of parameters Let f IVNsU s defned by ( ) x x x x f a x U f a f a f a and g B IVNsU s defned by g( b) x x x x x U g b g b g b where x x x x x x Int 0 U Then f s alled nterval valued neutrosoph subset of f a f a f a g b g b g b for x () gb (denoted by f g B x ge x x ge x x ge x e x U x ge x nf sup x sup x sup ) f B and Where ff nf g e and sup g e x ff g e nf f e nf g e sup g e x nf and nf and g e ff ge sup ge () Ther unon denoted by f g B h C (say) s an nterval valued neutrosoph soft set overu where C B h C IVNS U s defned by and for e C h e h e h e h e x x x x x U where for x U x B x h e ge x B x ge x B x B x h e ge x B x ge x B x B x h e ge x B x ge x B () Ther nterseton denoted by f g B h C (say) s an nterval valued neutrosoph soft set of overu where C B and for e C h C IVNS U s defned by x x x x h e h e h e h e x U where for xu and e C x x x x x x he ge he ge and x x x h e g e (v) The omplement of f denoted by f s an nterval valued neutrosoph soft set over U and s defned as f f f IVNS U s defned by where x x sup x nf x x f a x U f a f a f a f a for a Defnton 6:[56] n IVNSs f over the unverse U s sad to be unverse IVNSs wth respet to f x x 00 x 00 f a f a f a x U a It s denoted by I njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

3 0 Defnton 7: n IVNSs f over the unverse U s sad to be null IVNSs wth respet to f x 00 f a x f a x f a x U a It s denoted by 3 Interval Valued Neutrosoph Soft Topologal In ths seton we gve the defnton of nterval valued neutrosoph soft topologal spaes wth some examples and results We also defne dsrete and ndsrete nterval valued neutrosoph soft topologal spae along wth nterval valued neutrosoph soft fner and oarser topology Let U be an unverse set be the set of U parameters be the set of all subsets of U IVNs(U) be the set of all nterval valued neutrosoph sets n U and IVSNs(U;) be the famly of all nterval valued neutrosoph soft sets over U va parameters n Defnton 3: Let be an element of IVNSs(U;) be the olleton of all nterval valued neutrosoph soft subsets of sub famly of s alled an nterval valued neutrosoph soft topology (n short IVNS-topology) on f the followng axoms are satsfed: () () k k f : k K f kk () If g h then g h The trplet s alled nterval valued neutrosoph soft topologal spae (n short IVNStopologal spae) over The members of are alled open IVNS sets (or smply open sets) Here : IVNS ( U) s defned as e x 00 : x U e xampl: Let U u u u3 e e e3 e4 e e e The tabular representaton of 3 gven by U e e u ([58][35][7]) ([47][3][3]) u ([47][34][]) ([69][][]) u 3 ([5][0][36]) ([68][4][3]) ([39][0][0]) ([48][][05]) ([49][3][4]) Table:Tabular representaton of The tabular representaton of s gven by U e e u ([00][][]) ([00][][]) u ([00][][]) ([00][][]) u 3 ([00][][]) ([00][][]) ([00][][]) ([00][][]) ([00][][]) Table:Tabular representaton of The tabular representaton of f s gven by U e e u ([7][48][3]) ([3][46][6]) u ([3][67][8]) ([05][58][4]) u 3 ([48][67][69]) ([03][47][8]) ([5][89][49]) ([03][69][7]) ([3][68][37]) Table3:Tabular representaton of f The tabular representaton of f s gven by U e e u ([47][57][49]) ([3][45][79]) u ([35][48][4]) ([46][35][5]) u 3 ([39][][67]) ([57][67][34]) ([37][58][]) ([3][35][68]) ([6][35][58]) Table4: Tabular representaton of f 3 Let f f f representaton of f 3 then the tabular s gven by njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

4 U e e u ([7][58][4]) ([3][46][79]) u ([3][68][8]) ([05][58][4]) u 3 ([38][67][69]) ([03][67][38]) ([5][89][49]) ([03][69][68]) ([3][68][58]) Table5:Tabular representaton of f 3 4 Let f f f representaton of f 4 then the tabular s gven by U e e u ([47][47][39]) ([3][45][6]) u ([35][47][4]) ([46][35][5]) u 3 ([49][][67]) ([57][47][4]) ([37][58][]) ([3][35][7]) ([6][35][37]) Table6:Tabular representaton of f 4 Here we observe that the sub-famly 3 4 f f f f of s a IVNS-topology on the neessary three axoms of topology and as t satsfes s a IVNS-topologal spae But the sub-famly f f of s not an IVNS-topology on 4 f f f as the unon does not belong to Defnton 33: s every IVNS-topology on must ontans the sets and so the famly forms a IVNS-topology on The topology s alled ndsrete IVNS-topology and the trplet s alled an ndsrete nterval valued neutrosoph soft topologal spae (or smply ndsrete IVNS-topologal spae) Defnton 34: Let denotes the famly of all IVNSsubsets of Then we observe that satsfes all the axoms of topology on Ths topology s alled dsrete nterval valued neutrosoph soft topology and the trplet s alled dsrete nterval valued neutrosoph soft topologal spae (or smply dsrete IVNS-topologal spae) Theorem 35: Let : I be any olleton of IVNStopology on Then ther nterseton s also a IVNS-topology on Proof: () Sne Hene () Let k I for eah I I f : k K be an arbtrary famly of nterval valued neutrosoph soft sets where f k for eah k K Then for eah I I f k for k K and sne for eah I a a IVNS-topology therefore k kk Hene f k kk I () Let f for eah I I f f then f f for eah I Sne for eah s an IVNS-topology therefore for eah I Hene f f Hene I Thus I I f f I satsfes all the axoms of topology forms a IVNS-topology But unon of IVNStopologes need not be a IVNS-topology Let us show ths wth the followng example xampl6: In exampl the sub famles 3 f and 4 f are IVNS-topologes n But ther unon 3 4 f f s not a IVNS-topology n Defnton 37: Let be an IVNS-topologal spae over n nterval valued neutrosoph soft njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

5 subset f of s alled nterval valued neutrosoph soft losed set (n short IVNS-losed set) f ts omplement f s a member of xampl8: Let us onsder exampl then the IVNSlosed sets n are U e e u ([7][57][58]) ([3][78][47]) u ([][67][47]) ([][89][69]) u 3 ([36][9][5]) ([3][68][68]) ([0][9][39]) ([05][89][48]) ([4][79][49]) Table7:Tabular representaton of U e e u ([] [00][00]) ([] [00][00]) u ([] [00][00]) ([] [00][00]) u 3 ([] [00][00]) ([] [00][00]) Table8:Tabular representaton of ([] [00][00]) ([] [00][00]) ([] [00][00]) U e e u ([3][6][7]) ([6][46][3]) u ([8][34][3]) ([4][5][05]) u 3 ([69[34][48]) ([8][36][03]) ([49][][5]) ([6][4][03]) ([37][4][3]) Table9:Tabular representaton of f U e e u ([49][35][47]) ([79][56][3]) u ([4][6][35]) ([5][57][46]) u 3 ([67][89][39]) ([34][34][57]) ([][5][37]) ([68][57][3]) ([58][57][6]) Table0:Tabular representaton of f U e e u ([4][5][7]) ([79][46][3]) u ([8][4][3]) ([4][5][05]) u 3 ([69][34][38]) ([38][34][03]) ([49][][5]) ([68][4][03]) ([58][4][3]) Table:Tabular representaton of f 3 U e e u ([39][36][47]) ([6][56][3]) u ([4][36][35]) ([5][57][46]) u 3 ([67][89][49]) ([4][36][57]) ([][5][37]) ([7][57][3]) ([37][57][6]) Table:Tabular representaton of f 4 are the IVNS-losed sets n Theorem 39: Let be an IVNS-topologal spae over Then are IVNS-losed sets rbtrary nterseton of IVNS-losed sets s IVNS-losed set 3 Fnte unon of IVNS-losed sets s IVNS-losed set therefore Proof: Sne Let k are IVNS-losed sets f : k K be an arbtrary famly of IVNS-losed sets n f f k kk and let njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

6 3 Now for eah k Thus k and k K k K k k f f f f k K so kk f s IVNS-losed set 3 Let IVNS-losed sets n f g f Hene f f : 3 n be a famly of and let n g f n Now for 3 n g f f n and n so Thus g s IVNS-losed set f Hene Defnton 30: Let and IVNS-topologal spaes over f mples f be two Iah then s alled nterval valued neutrosoph soft fner topology than and s alled nterval valued neutrosoph soft oarser topology than xampl: In exampl and 36 s nterval valued neutrosoph soft fner topology than 3 and 3 s alled nterval valued neutrosoph soft oarser topology than Defnton 3: Let be a IVNS-topologal spae over and ß be a subfamly of Ivery element of an be express as the arbtrary nterval valued neutrosoph soft unon of some elements of ß then ß s alled an nterval valued neutrosoph soft bass for the IVNS-topology xampl3: In exampl for the IVNStopology 3 4 f f f f the 3 subfamly ß= f f f of s a nterval valued neutrosoph soft bass for the IVNS-topology 4 Some Propertes of Interval Valued Neutrosoph Soft Topologal In ths seton some propertes of nterval valued neutrosoph soft topologal spaes are ntrodued Some results on IVNSInt and IVNSCl are also ntodued Defnton 4: Let be a IVNS-topologal spae and let f IVNSS ( U ; ) The nterval valued neutrosoph soft nteror and loser of f s denoted by IVNSInt(f ) and IVNSCl(f ) are defned as IVNSInt f g : g f and g : f g respetvely INVNSCl f xample 4: Let us onsder exampl and take an IVNSS f 5 as U e e u ([8][36][8]) ([4][46][4]) u ([6][45][7]) ([6][57][7]) u 3 ([58][56][58]) ([4][46][5]) ([6][78][34]) ([4][5][5]) ([5][58][4]) Table3:Tabular representaton of f 5 5 Now IVNSInt f f and IVNSCl f f 5 Theorem 43: Let be a IVNS-topologal spae and f g IVNSS U ; then the followng propertes hold IVNSInt f f f g IVNSInt f IVNSInt g 3 IVNSInt f 4 f IVNSInt f f 5 IVNSInt IVNSInt f IVNSInt f 6 IVNSInt IVNSInt U U Proof: Straght forward f g mples all the IVNS-open sets ontaned n f also ontaned n g e * * * * f : f f g : g g njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

7 4 e * * * * f : f f g : g g e IVNSInt f IVNSInt g * * 3 IVNSInt f f : f f * * It s lear that f : f f So IVNSInt f 4 Let f then by () IVNSInt f f Now sne f and f f Theorem 44: Let be a IVNS-topologal spae and f g IVNSs U ; then the followng propertes hold f IVNSCl f f g IVNSCl f IVNSCl g 3 IVNSCl f 4 f IVNSCl f f 5 IVNSCl IVNSCl f IVNSCl f 6 IVNSCl IVNSCl U U Proof: straght forward Theorem 45: Let be an IVNS-topologal spae on and let f ; g IVNSs U Then the followng propertes hold IVNSInt f g IVNSInt f IVNSInt g Proof: IVNSInt f g IVNSInt f IVNSInt g 3 IVNSCl f g IVNSCl f IVNSCl g 4 IVNSCl f g IVNSCl f IVNSCl g 5 IVNSInt f IVNSCl f 6 IVNSCl f IVNSInt f By theorem 4 () IVNSInt f f Therefore and IVNSInt g g Thus * * f g : g g IVNSInt f IVNSInt f IVNSInt g f g Hene e f IVNSInt f IVNSInt f IVNSInt g IVNSInt f g Thus IVNSInt f f () Conversly let IVNSInt f f gan sne f g f By theorem 4 () IVNSInt f g IVNSInt f Sne by (3) IVNSInt f Therefore f Smlarly 5 By (3) IVNSInt f IVNSInt f g IVNSInt g Hene By (4) IVNSInt IVNSInt f IVNSInt f IVNSInt f g IVNSInt f IVNSInt g () 6 We know that U Usng () and () we get By (4) IVNSInt IVNSInt U U IVNSInt f g IVNSInt f IVNSInt g Sne f f g By theorem 4 () IVNSInt f IVNSInt f g Smlarly IVNSInt g IVNSInt f g Hene IVNSInt f g IVNSInt f IVNSInt g 3 Smlar to 4 Smlar to 5 IVNSInt f g : g f g : f g IVNSCl f njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

8 5 6 Smlar to 5 qualty does not hold n theorem 44 () (4) Let us show ths by an example xample 46: Let U u u e e e 3 e e The tabular representaton of s gven by U e e u ([58][35][7]) ([39][][0]) u ([46][34][]) ([48][3][]) Table4:Tabular representaton of The tabular representaton of s gven by U e e u ([00] [] []) ([00] [] []) u ([00] [] []) ([00] [] []) Table5:Tabular representaton of The tabular representaton of f s gven by U e e u ([7][48][3]) ([5][79][37]) u ([][67][7]) ([03][58][4]) Table6:Tabular representaton of f Clearly f s a IVNS-topology on Let us now take two nterval valued neutrosoph soft sets g and h as U e e u ([6][49][4]) ([5][79][38]) u ([][67][8]) ([0][59][4]) Table7:Tabular representaton of g U e e u ([07][58][3]) ([5][8][67]) u ([][68][37]) ([03][68][5]) Table8:Tabular representaton of h Now g h f IVNSInt g h IVNSInt f f IVNSInt h lso IVNSInt g IVNSInt g IVNSInt h Thus IVNSInt f g IVNSInt f IVNSInt g Therefore equalty does not hold for () By theorem 44 (5) IVNSCl g IVNSl g Smlarly IVNSl h Therefore IVNSCl g IVNSCl h lso IVNSCl g h IVNSCl g h IVNSInt g h IVNSInt f f Thus IVNSCl f g IVNSCl f IVNSCl g Therefore equalty doesnot hold n (4) 5 Interval Valued Neutrosoph Soft Subspae Topology In ths seton we ntrodue the onept of nterval valued neutrosoph soft subspae topology along wth some examples and results Theorem 5: Let be an IVNS-topologal spae on f Then the and olleton f : f g g an IVNS-topology on Proof: () Sne f f f s therefore and f f f () Let k f k f f g where k K Now f k K Then g for eah k k k f f g f g f kk kk kk g k as eah g k (sne kk () Let f f f then f f g and f f g where g g njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

9 6 Now f f f g f g f g g f (sne g g as g g ) Defnton 5: Let be an IVNS-topologal spae on and f Then the IVNS-topology f g : g f s alled nterval valued neutrosoph soft subspae topology and f f s alled nterval valued neutrosoph soft subspae of xample 53: Let us onsder the IVNS-topology 3 4 f f f f as n exampl and an IVNSS f : U e e u ([46][67][35]) ([57][46][03]) u ([3][36][57]) ([68][45][3]) u 3 ([57][46][34]) ([45][79][67]) ([35][58][3]) ([58][57][3]) ([3][79][57]) Table9:Tabular representaton of f Then f f : U e e u ([00][][]) ([00][][]) u ([00][][]) ([00][][]) u 3 ([00][][]) ([00][][]) ([00][][]) ([00][][]) ([00][][]) Table0:Tabular representaton of f g f f : U e e u ([6][67][3]) ([3][46][6]) u ([3][67][58]) ([05][45][4]) u 3 ([47][46][69]) ([03][79][68]) ([5][58][49]) ([03][69][7]) ([3][79][57]) Table:Tabular representaton of g g f f : U e e u ([46][67][49]) ([3][46][79]) u ([3][48][57]) ([46][45][5]) u 3 ([37][46][67]) ([45][79][67]) ([35][58][3]) ([3][57][68]) ([3][79][38]) Table:Tabular representaton of g 3 3 g f f : U e e u ([6][68][4]) ([3][46][79]) u ([3][68][58]) ([05][45][4]) u 3 ([37][46][69]) ([03][79][68]) ([5][58][49]) ([03][69][68]) ([3][79][58]) Table3:Tabular representaton of g 4 4 g f f : U e e u ([5][58][49]) ([5][58][49]) u ([03][69][68]) ([03][69][68]) u 3 ([3][79][58]) ([3][79][58]) 3 ([35][58][3]) ([3][57][7]) ([3][79][57]) Table4:Tabular representaton of g 4 4 Then f g g g s an nterval valued neutrosoph soft subspae njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal

10 7 topology for and f f s alled nterval valued neutrosoph soft subspae of Theorem 54: Let be an IVNS-topologal spae on f f g f g ß be an IVNS-bass for and Then the famly ß = : ß s an IVNS-bass for subspae topology f h be arbtrary then there exsts Proof: Let f an IVNSS g suh that h f g Sne ß s a bass for therefore there exsts a sub olleton : I of ß suh that g Now I h f g f I Sne f ß f therefore ß f s an IVNS-bass for the subspae topology f Conluson In ths paper we ntrodue the onept of nterval valued neutrosoph soft topology Some bas theorem and propertes of the above onept are also studed IVN nteror and IVN loser of an nterval valued neutrosoph soft set are also defned Interval valued neutrosoph soft subspae topology s also studed In future there wll be more researh work n ths onept takng the bas defntons and results from ths artle Referenes [] H ktas N Cagman Soft Sets and soft groups Inform S 77(007) [] M I l F Feng X Lu W K Mn and M Shabr On some new operatons n soft set theory Computers and Mathemats wth pplatons 57(9)(009) [3]I rokaran I R Sumath J Martna Jeny Fuzzy neutrosoph soft topologal spaes Internatonal Journal of Mathematal rhve4(0)( 03) 5-38 [4] K tanassov Intutonst fuzzy sets Fuzzy Sets and Systems 0(986) [5] S Broum I Del and F Smarandahe Relatons on Interval Valued Neutrosoph Soft Sets Journal of New Results n Sene 5 (04) -0 [6] I Del Interval-valued neutrosoph soft sets and ts deson makng Kls 7 ralık Unversty Kls Turkey I [7] P K Maj R Bswas and R Roy Soft Set Theory Computers and Mathemats wth pplatons 45(003) [8] P K Maj Neutrosoph soft set nnals of Fuzzy Mathemats and Informaton 5()(03) [9] D Molodtsov Soft set theory-frst results Computers and Mathemats wth pplatons 37(4-5)(999) 9-3 [0] njan Mukherjee joy Kant Das bhjt Saha Interval valued ntutonst fuzzy soft topologal spaes nnals of Fuzzy Mathemats and Informats 6 (3) ( 03) [] D Pe D Mao From soft sets to nformaton systems Pro I Int Conf Granular Comput (005) 67-6 [] F Smarandahe Neutrosoph Log and Set mss [3] F Smarandahe Neutrosoph set- a generalsaton of the ntutonst fuzzy sets nt J Pure ppl Math 4(005) [4] H Wang F Smarandahe YQ Zhang R Sunderraman Interval Neutrosoph Sets and log: Theory and pplatons n Computng Hexs; Neutrosoph book seres No: [5] L Zadeh Fuzzy sets Informaton and Control 8 (965) Reeved: September epted: Otober 5 04 njan Mukherjee Mthun Datta Florentn Smarandahe Interval Valued Neutrosoph Soft Topologal