On Union and Intersection of Fuzzy Soft Set
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1 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 On Unon and Interseton of uzzy Soft Set Trdv Jyot Neog Dusmanta Kumar Sut. Researh Sholar Department of Mathemats MJ Unversty Shllong Meghalaya emal : trdvjyot@gmal.om. ssstant Professor Department of Mathemats Jorhat Insttute of Sene and Tehnology Jorhat ssam emal : sutdk00@yahoo.om bstrat Molodtsov ntrodued the theory of soft sets whh an be seen as a new mathematal approah to vagueness. Maj et al. have further ntated several bas notons of soft set theory. They have also ntrodued the onept of fuzzy soft set a more generalzed onept whh s a ombnaton of fuzzy set and soft set. They ntrodued some propertes regardng fuzzy soft unon nterseton omplement of a fuzzy soft set DeMorgan Laws et. These results were further revsed and mproved by hmad and Kharal. They defned arbtrary fuzzy soft unon and nterseton and proved DeMorgan Inlusons and DeMorgan Laws n uzzy Soft Set Theory. In ths paper we gve some propostons on fuzzy soft unon and nterseton wth proof and examples. Usng the defnton of arbtrary fuzzy soft unon and nterseton proposed by hmad and Kharal we are gvng two more propostons wth proof and examples. We further gve the proof of DeMorgan Laws for a famly of fuzzy soft sets n a fuzzy soft lass proposed by hmad and Kharal and verfy these laws wth examples. Key words: Soft Set uzzy Soft Set uzzy Soft lass.. Introduton. In order to deal wth many omplated problems n the felds of engneerng soal sene eonoms medal sene et nvolvng unertantes lassal methods are found to be nadequate n reent tmes. Molodstov [7] ponted out that the mportant exstng theores vz. Probablty Theory uzzy Set Theory Intutonst uzzy Set Theory Rough Set Theory et. whh an be onsdered as mathematal tools for dealng wth unertantes have ther own dffultes. e further ponted out that the reason for these dffultes s possbly the nadequay of the parameterzaton tool of the theory. In 999 he proposed a new mathematal tool for dealng wth unertantes whh s free of the dffultes present n these theores. e ntrodued the novel onept of Soft Sets and establshed the fundamental results of the new theory. e also showed how Soft Set Theory s free from parameterzaton nadequay syndrome of uzzy Set Theory Rough Set Theory Probablty Theory et. Many of the establshed paradgms appear as speal ases of Soft Set Theory. In 00 P.K.Maj R.Bswas and.r.roy [6] studed the theory of soft sets ntated by Molodstov. They defned equalty of two soft sets subset and super set of a soft set omplement of a soft set null soft set and absolute soft set wth examples. Soft bnary operatons lke ND OR and also the operatons of unon nterseton were also defned. In 005 Pe and Mao [8] and hen et al. [] mproved the work of Maj et al. [ 6]. In 008 M.Irfan l eng eng Xaoyan LuWon Keun Mn M.Shabr [] gave some new notons suh as the restrted nterseton the restrted unon the restrted dfferene and the extended nterseton of two soft sets along wth a new noton of omplement of a soft set. In reent tmes researhes have ontrbuted a lot towards fuzzfaton of Soft Set Theory. Maj et al. [5] ntrodued some propertes regardng fuzzy soft unon nterseton omplement of a fuzzy soft set DeMorgan Law et. These results were further revsed and mproved by hmad and Kharal []. They defned arbtrary fuzzy soft unon and nterseton and proved DeMorgan Inlusons and DeMorgan Laws n uzzy Soft Set Theory. IJT SPT-OT 0 valable onlne@ 60
2 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 In ths paper we gve the proof of some propostons ntrodued by hmad and Kharal [] and support them wth examples. We further gve some more propostons regardng fuzzy soft unon and nterseton and support these propostons wth proof and examples. Defnton. [7] par E s alled a soft set over U f and only f s a mappng of E nto the set of all subsets of the set U. In other words the soft set s a parameterzed famly of subsets of the set U. Every set E from ths famly may be onsdered as the set of - elements of the soft set E or as the set of - approxmate elements of the soft set. Example. Let U { } be the set of four ars under onsderaton and E { e ostly e Beautful e uel Effent e ModernTehnology e 5 Luxurous} be the set of parameters and { e e e } E. Then { e { }e { }e { }} s the soft set representng the attratveness of the ar whh Mr. X s gong to buy. We an represent ths soft set n a tabular form as shown below [].Ths style of representaton wll be useful for storng a soft set n a omputer memory. U e e e Defnton. [5] par s alled a fuzzy soft set over U where : P U s a mappng from nto P U. Example. Let U { } be the set of four ars under onsderaton and E { e ostly e Beautful e uel Effent e ModernTehnology e 5 Luxurous} be the set of parameters and { e e e } E. Then { e { /0.7 /0. /0. /0.6} e { /0.8 /0.6 /0. /0.5} e { /0. /0. /0.7 /0.}} s the fuzzy soft set representng the attratveness of the ar whh Mr. X s gong to buy. Defnton. [] Let U be a unverse and E a set of attrbutes. Then the par U E denotes the olleton of all fuzzy soft sets on U wth attrbutes from E and s alled a fuzzy soft lass. Defnton. [5] or two fuzzy soft sets and G B n a fuzzy soft lass U E we say that s a fuzzy soft subset of G B f B or all G and s wrtten as G B. IJT SPT-OT 0 valable onlne@ 6
3 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Example. Let { } the set of four ars under onsderaton and { e ostly e Beautful e uel Effent e ModernTehnology e Luxurous} U be E 5 be the set of parameters { e e e } E and B { e e e e 5 } E. Then { e { /0.7 /0. /0. /0.6} e { /0.8 /0.6 /0. /0.5} e { /0. /0. /0.7 /0.}} s the fuzzy soft set representng the attratveness of the ar whh Mr. X s gong to buy and G B { Ge { /0.7 /0. /0. /0.7} Ge { /0.9 /0.6 /0.5 /} Ge { /0. /0. /0.8 /0.} Ge 5 { /0. /0. /0.7 /0.}} s the fuzzy soft set representng the attratveness of the ar whh Mr. Y s gong to buy. ere B and for all G. Thus G B. Defnton 5. [5] The omplement of a fuzzy soft set s denoted by and s defned by where : P U s a mappng gven by σ σ for all σ. Example. Let U { } be the set of four ars under onsderaton and E { e ostly e Beautful e uel Effent e ModernTehnology e 5 Luxurous} be the set of parameters and { e e e } E. Then { e { /0.7 /0. /0. /0.6} e { /0.8 /0.6 /0. /0.5} e { /0. /0. /0.7 /0.}} s the fuzzy soft set representng the attratveness of the ar whh Mr. X s gong to buy. ere { e { /0. /0.9 /0.8 /0.} e { /0. /0. /0.9 /0.5} e { /0.9 /0.8 /0. /0.7}} Defnton 6. [5] Unon of two fuzzy soft sets and G B n a soft lass U E s a fuzzy soft set where B and f x B G f x B G f x B G B. and s wrtten as IJT SPT-OT 0 valable onlne@ 6
4 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Example 5. Let { } U be the set of four ars under onsderaton and { e ostly e Beautful e uel Effent e ModernTehnology e Luxurous} E 5 be the set of parameters { e e e } E and B { e e e e 5 } E. We onsder the fuzzy soft sets { e { /0.9 /0. /0. /0.6} e { / /0 /0.9 /0.5} e { /0.8 /0. /0.7 /0.6}} and G B { Ge { /0.7 /0. /0. /0.7} Ge { /0.9 /0.6 /0.5 /} Ge { /0. /0. /0.8 /0.} Ge 5 { /0. /0. /0.7 /0.}} G B where B {e e e e 5 } and Then { e { /0.9 /0. /0. /0.7} e { / /0.6 /0.9 /} e { /0.8 /0. /0.8 /0.6} e 5 { /0. /0. /0.7 /0.}} Defnton 7. [5] Interseton of two fuzzy soft sets and G B n a soft lass U E s a fuzzy soft set where B and or G as both are same fuzzy set and s wrtten as G B. hmad and Kharal [] ponted out that generally or G may not be dental. Moreover n order to avod the degenerate ase he proposed that defnton as follows. B must be non-empty and thus revsed the above Defnton 8. [] Let and G B be two fuzzy soft sets n a soft lass U E wth B φ.then Interseton of two fuzzy soft sets and G B n a soft lass U E s a fuzzy soft set where B and G. We wrte G B. Example 6. or the two fuzzy soft sets and G B gven n Example 5 G B where B { e e e } and { e { /0.7 /0. /0. /0.6} e { /0.9 /0 /0.5 /0.5} e { /0. /0. /0.7 /0.}}. Some Propostons on uzzy Soft Unon and Interseton. hmad and Kharal [] gave some propostons on fuzzy soft unon and nterseton. ere we gve the proof of those propostons along wth some addtonal propostons. Let and be three fuzzy soft sets n a soft lass U E. IJT SPT-OT 0 valable onlne@ 6
5 ommutatve Property. Proof. Let Where and gan let Where and learly and. Thus. Let Where and f f f gan let Where and f f f. Thus and ene. ssoatve Property. Proof. Let and Where and nd gan let and Where and nd Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol IJT SPT-OT 0 valable onlne@ 6 ISSN:9-609
6 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 It s lear that and Thus Let and and Where.. ase I. When.. ase II When f f.. f ase III When Now When by Thus.. When by Thus..5 When by Thus..6 gan Let and Where and..7 ase I When f f..8 f ase II When..9 IJT SPT-OT 0 valable onlne@ 65
7 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 ase III When When by 8 Thus.e...0 When by 8 Thus.e... When by 8 Thus.e... Now when by nd by 8 When When by by 8 { ` } by by 9 When IJT SPT-OT 0 valable onlne@ 66
8 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 { } When { } by by When When by and by 8 When by 5 by 0 When by 6 nd by Thus Idempotent Property. IJT SPT-OT 0 valable onlne@ 67
9 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Proof. Let Where and Thus Let Where and Thus v bsorpton Property. Proof. Let and f Where and f f lso and Now f nd f Thus Let and Where and lso and Thus v Dstrbutve Property. Proof. Let and Where Let where IJT SPT-OT 0 valable onlne@ 68
10 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 ase I. When ase - II When ase - III When Let and 5 5 Where 5 When ase I. 5 When 5 When ase II. 5 When 5 When ase III. 5 When 5 When IJT SPT-OT 0 valable onlne@ 69
11 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 It s lear from above that Ths an be proved n a smlar way. v Proof. Let where and Now and Thus. The other result an also be proved n a smlar way. Let where f nd f f f - Now and f Thus Thus. The other result an also be proved n a smlar way. v Proof. Let. Then Now let and. Then as nd as Thus Let. Then and Now let. Then as nd as Thus IJT SPT-OT 0 valable onlne@ 70
12 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Defnton 9. [] Let I { I} be a famly of fuzzy soft sets n a fuzzy soft lass U E.Then the unon of fuzzy soft sets n I s a fuzzy soft set where and for all f Where ϕ f Example 7. Let U { } be the set of four ars under onsderaton and E { e ostly e Beautful e uel Effent e ModernTehnology e 5 Luxurous} be the set of parameters and { e e e } E { e e } E { e e e e } E. We onsder three fuzzy soft sets and as follows. { e { /0.7 /0. /0. /0.6} e { /0.8 /0.6 /0. /0.5} e { /0. /0. /0.7 /0.}} { e { /0. /0.6 /0. /0.6} e { /0.8 /0. /0. /0.5}} { e { /0. /0. /0.5 /0} e { /0. /0.9 /0.5 /0.5} e { /0. /0. /0.6 /0.} e { /0.8 /0. /0.5 /0.}} Thus where { e e e e } and { e { /0.7 /0.6 /0.5 /0.6} e { /0.8 /0.9 /0.5 /0.5} e { /0. /0. /0.7 /0.} e { /0.8 /0. /0.5 /0.5}} Defnton 0. [] Let I { I} be a famly of fuzzy soft sets n a fuzzy soft lass U Ewth ϕ.then the nterseton of fuzzy soft sets n I s a fuzzy soft set where and for all IJT SPT-OT 0 valable onlne@ 7
13 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Example 8. U Let { } be the set of four ars under onsderaton and { e ostly e Beautful e uel Effent e ModernTehnology e Luxurous} E 5 be the set of parameters and { e e e } E { e e } E { e e e e } E. We onsder three fuzzy soft sets and as follows. { e { /0.7 /0. /0. /0.6} e { /0.8 /0.6 /0. /0.5} e { /0. /0. /0.7 /0.}} { e { /0. /0.6 /0. /0.6} e { /0.8 /0. /0. /0.5}} { e { /0. /0. /0.5 /0} e { /0. /0.9 /0.5 /0.5} e { /0. /0. /0.6 /0.} e { /0.8 /0. /0.5 /0.}} Thus where { e } and { e { /0. /0. /0. /0}} ollowng the defntons 9 and 0 we now propose the followng two propostons. Proposton. Let I { I} be a famly of fuzzy soft sets n a fuzzy soft lass U E.Then I Proof. Let where and α I α α where α f α α ϕ f α learly.e and α I α α.e. α α Thus I Example 9. or the three fuzzy soft sets and gven n Example 7 we see that where { e e e e } and { e { /0.7 /0.6 /0.5 /0.6} e { /0.8 /0.9 /0.5 /0.5} e { /0. /0. /0.7 /0.} e { /0.8 /0. /0.5 /0.5}} Thus and for IJT SPT-OT 0 valable onlne@ 7
14 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Proposton 5. Let I { I} be a famly of fuzzy soft sets n a fuzzy soft lass U E.Then I Proof. Suppose that where and α I α α Now and α I α α α Thus α α α and hene the result follows. Example 0. or the three fuzzy soft sets and gven n Example 8 we see that where { e } and { e { /0. /0. /0. /0}. Thus and for. hmad and Kharal [] proved DeMorgan Laws for soft sets and G n a soft lass U E. e further generalzed DeMorgan Laws for a famly of fuzzy soft sets n a fuzzy soft lass U E as follows- Theorem. [] Let I { I} be a famly of fuzzy soft sets n a fuzzy soft lass U E.Then one has the followng -.. ere we gve the proof of ths theorem. Proof..We have say Where α α α... gan suppose that I.Then I I where IJT SPT-OT 0 valable onlne@ 7
15 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 I α [ I α ] α α we have I α α α... rom and we get the desred result.. We have say where α α α gan suppose that I.Then I I where I α [ I α ] α α we have I α α α rom and we get the desred result. Example. Let U { } be the set of four ars under onsderaton and E { e ostly e Beautful e uel Effent e ModernTehnology e 5 Luxurous} be the set of parameters and { e e } E We onsder three fuzzy soft sets and as follows. { e { /0.7 /0. /0. /0.6} e { /0.8 /0.6 /0. /0.5}} { e { /0. /0.9 /0.8 /0.} e { /0. /0. /0.9 /0.5}} { e { /0. /0.6 /0. /0.6} e { /0.8 /0. /0. /0.5}} { e { /0.6 /0. /0.9 /0.} e { /0. /0.9 /0.6 /0.5}} IJT SPT-OT 0 valable onlne@ 7
16 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 { e { /0. /0. /0.5 /0} e { /0.8 /0. /0.5 /0.}} { e { /0.8 /0.7 /0.5 /} e { /0. /0.7 /0.5 /0.9}} Thus where { e { /0.7 /0.6 /0.5 /0.6} e { /0.8 /0.6 /0.5 /0.5}} { e { /0. /0. /0.5 /0.} e { /0. /0. /0.5 /0.5}} and where { e { /0. /0. /0. /0} e { /0.8 /0. /0. /0.}} { e { /0.8 /0.9 /0.9 /} e { /0. /0.9 /0.9 /0.9}} Now I { I e { /0.8 /0.9 /0.9 /} I e { /0. /0.9 /0.9 /0.9}} nd J {J e { /0. /0. /0.5 /0.} J e { /0. /0. /0.5 /0.5}} It s lear that. onluson. The Soft Set Theory of Molodstov [7] offers a general mathematal tool for dealng wth unertan and vague objets. t present work on the extenson of soft set theory s progressng rapdly. Maj et al. [5] proposed the onept of fuzzy soft set and developed some propertes of fuzzy soft sets and n reent years the researhers have ontrbuted a lot towards the fuzzfaton of Soft Set Theory. Ths paper ontrbutes some more propertes regardng fuzzy soft unon and nterseton and support these propostons wth proof and examples. We further gve the proof of some propostons ntrodued by hmad and Kharal [] and support them wth examples. We hope that our fndngs wll help enhanng ths study on fuzzy soft sets. and IJT SPT-OT 0 valable onlne@ 75
17 Dusmanta Kumar Sut et al Int. J. omp. Teh. ppl. Vol ISSN:9-609 Referenes. hmad B and Kharal thar On uzzy Soft Sets dvanes n uzzy Systems Volume l M.I eng Lu XY Mn WK Shabr M 009 On some new operatons n soft set theory. omputers and Mathemats wth pplatons 57: hen D Tsang E Yeung D S and Wang X The parameterzaton reduton of soft sets and ts applatons omputers & Mathemats wth pplatons vol. 9 no.5-6 pp Maj P K and Roy R n pplaton of Soft Sets n Deson Makng Problem omputers and Mathemats wth pplatons Maj P K Bswas R and Roy R uzzy Soft Sets Journal of uzzy Mathemats Vol 9 no.pp Maj P K and Roy R Soft Set Theory omputers and Mathemats wth pplatons Molodstov D Soft Set Theory - rst Result omputers and Mathemats wth pplatons Pe D and Mao D rom soft sets to nformaton systems n Proeedngs of the IEEE Internatonal onferene on Granular omputng vol. pp IJT SPT-OT 0 valable onlne@ 76
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