A STUDY OF PROPERTIES OF SOFT SET AND ITS APPLICATIONS

Transcription

1 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: STDY O POPETIES O SOT SET ND ITS PPLITIONS Shamshad usain 1 Km Shivani 2 1MPhil Studnt Pu mathmatics Shobhit nivsity Mut tta Padsh India 2Psot aduat Mathmatics S nivsity Mut tta Padsh India *** bstact - In this pap th authos study th thoy of soft In oth wods:- soft st ov is paamtizd sts initiatd by Molodtsov Equality of two soft sts subst family of substs fo may b consid as th supst of a soft st omplmnt of soft st null soft st and absolut of soft st and xampls Soft st opation lik O soft st of lmnts o appoximat Elmnts of ND union intsction lativ intsction lativ union th soft st thus is dfind as symmtic diffnc lativ symmtic diffnc and xampls Poptis of soft sts D Mogan s law with poof { P : E if } by paticula xampl associativ commutativ distibutiv tc Exampl 21- ssum that { c1 c2 c3 c c5 b univsal Ky Wods: Soft st subst and supst of soft st appoximation complmnt lativ omplmnt NOT st Null st Soft st opations 1 INTODTION Most of th taditional tools fo fomal modlling and computing a cisp dtministic and pcis in chaact owv th a many complicatd poblms in conomics ngining nvionmnt social scinc mdical scinc tc that involv data which a not always cisp W cannot succssfully us classical mthods bcaus of vaious typs of unctaintis psnt in ths poblms Th a thois thoy of pobability thoy fuzzy st thoy intuitionistic fuzzy sts thoy vagu st thoy intval mathmatics thoy ough st thoy which can considd mathmatical tools fo daling with unctaintis ut all ths thois hav thi inhnt difficultis as pointd out th ason fo ths difficultis is possibly th inadquacy of th paamtization tools of thois onsquntly Molodtsov initiatd th concpt of soft st thoy as a mathmatical tool fo daling with unctaintis which is f fom th abov difficultis W a awa of th soft sts dfind by Pawlak which is diffnt concpt and usful to solv som oth typ of poblms Soft st thoy has a ich potntial fo application in sval dictions fw of which had bn shown by Molodtsov in his pion wok in th psnt pap w mak a thotical study of th Soft St Thoy in mo dtail 2 SOT SET Lt b an initial univs st E is th st of paamts P Dnot th pow E thn pai is calld a soft st o attibuts with spct to Lt st of and ov wh is a mapping givn by : P st consticting of a st of Six as und sal Now E { } b th st of paamts with spct to Wh ach paamts i i stands fo {Expnsiv ood dsign Milag Modn Spac capacity} spctivly and { } E suppos a soft st Dscib th attaction by costum fo cas { c } 2 { c1 c3 c5} 3 { c3 c c5} 1 1 c { c2 c3} 5 { c2 c Thn th soft st is a paamts family i i 1235 of a subst dfind as { } i { c1 c2 c1 c3 c c3 c c5 c2 c3 c2 c Th soft st can also psntd as a st odd pai as follows { } { 1 c1 c2 2 c1 c3 c 3 c3 c c5 c2 c3 5 c2 c Notation- o and E 2018 IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 363 y Tabula fom Tabl Expnsiv ood Dsign Milag Modn Spac c c c c c 5 c

2 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: o th pupos of stoing a soft st in comput w could psnt a soft st in th fom of tabl 1 cosponding to th soft st in th abov xampl 21 Soft Sub-st Dfinition- o two soft st and ov a common univs w say that is a soft subst of if a and b and a idntical appoximation w wit If is said to b a soft supp st of if is a soft subst of W Dnot if 22 Equality of two Sts Dfinition Two soft st and ov a common univs a said to b soft qual if is soft subst of and is soft subst of Exampl22 - Pvious xampl 21 E { } b th st of paamts with spct to and { 1 3 5} E { } E claly Lt and b two soft sts ov th sam univs { c1 c2 c3 c c5 such that 1 { c2 c} 2 { c1 c3} 3 { c3 c c5} { c2 c and { c } 3 { c3 c c5} 5 { c2 c 1 2 c Thfo and but i mak 21 lt soft st and ov a common univs Dos not imply that vy lmnt of is an lmnt of thfo dfinition of classical Subst dos not hold fo soft subst Exampl 23 lt { c1 c2 c3 c c5 b univsal and E { 1 2 3} b th st of paamts with spct to If { 1 } and { 1 3} fo { c } 1 c2 c 1 2 c 1 c2 c3 c 3 c1 c5 fo { 1 c2 c3 c 3 c1 c5} Thn and nc laly and Not st of a st of paamts Lt E { n} b th st of Paamts th Not st of E dnotd by E is dfind by E { n} wh i Not i i 123 n Poposition 21 a b c c mak 22 it has bn povd that and th E and so poposition hold but nw assumption that E and cam up with th following poposition a b D Mogan s law Exampl 2 onsid th abov xampl h E {Expnsiv ood dsign Milag Modn Spac capacity} thn th Not st of this is E {Not Expnsiv Not ood dsign No Milag No Modn No Spac capacity} i E { } thn Not st is E { } 2 omplimnt of soft st Th complmnt of a soft st is dnotd by and dfind as wh : P is a mapping givn by Exampl- 25 by Exampl 21 Thn th soft st is a paamts family i i 1235 of a subst dfind c c6 as { c c c } { c c } 3 3 { c1 c2 { c1 c c IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 36

3 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: { c1 c3 c5} { } { c1 c3 c5 c6 c2 c c6 c1 c2 c6 c1 c c5 c6 c1 c3 c5} Th soft st can also psntd as a st odd pai as follows { } { 1 c1 c3 c5 c6 2 c2 c c6 3 c1 c2 c6 c1 c c5 c6 5 c1 c3 c5} 25 lativ complmnt Th lativ complmnt of a soft st dnotd by and dfind as wh : P is a mapping givn by Exampl 26 by xampl 21 Thn th soft st is a paamts family i i 1235 of a subst dfind as { c c c } { c c } { c c } c6 2 2 c6 { c1 c c5 5 { c1 c3 c5 } { } c6 { c1 c3 c5 c6 c2 c c6 c1 c2 c6 c1 c c5 c6 c1 c3 c5} Th soft st can also psntd as a st odd pai as follows { } { 1 c1 c3 c5 c6 2 c2 c c6 3 c1 c2 c6 c1 c c5 c6 5 c1 c3 c5} Poposition 21- lt b th univs is soft st ov Thn a b c d 26 Null st soft st ov univsal st is said to b a null soft st dfind by if Exampl 27 suppos that is th st of woodn houss und th considation is th st of paamts Lt th b fiv houss in th univs givn by { h1 h2 h3 h and ={bick muddy stl ston} Th soft st dscib th constuction of th houss Th soft st is Dfind as bick =mans th bick built houss muddy = mans th muddy houss stl = mans th stl built houss ston = mans th ston built houss Th soft st is th collction of appoximations as blow: = {bick built houss = muddy houss= stl built houss= ston built houss= } 27 bsolut soft st soft st ov univsal st is said to b a absolut soft st dnotd by if laly and Exampl 28 Suppos that is th st of woodn houss und th considation is th st of paamts Lt th b fiv houss in th univs givn by { h1 h2 h3 h nd = {not bick not muddy not stl not ston} Th soft st dscib th constuction of th houss Th soft st is dfind as 2018 IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 365

4 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: bick =mans th houss not built by bick muddy = mans th houss not by muddy stl = mans th houss not built by stl ston = mans th houss not built by ston Th soft st is th collction of appoximations as blow: = {not bick built houss = { h 1 h2 h3 h not muddy houss= { h 1 h2 h3 h not stl built houss= { h 1 h2 h3 h not ston built houss= { h 1 h2 h3 h } Th soft st is th absolut soft st 28 lativ null soft st b a univs E b a st of paamts Lt and E is calld a lativ Null soft st with spct to dnotd by 29 lativ whol soft st if Lt b a univs E b a st of paamts and E is calld lativ whol soft st o univsal with spct to dnotd by if 210 lativ bsolut soft st Th lativ whol soft st with spct to E dnotd b is calld th lativ absolut soft st ov Exampl 29 lt E { } if { 2 3 } such that { c } 3 2 2c thn soft st { c c } 2 2 if { 1 3} such that th soft st { 1 3 } thn th soft st is a lativ null soft st if { 1 2} such that 1 2 thn th soft st is a lativ whol soft st E c if D E such that 1 i E i 123 thn th soft st D E is an absolut soft st Poposition 22- lt b th univs E a st of paamts E if and a soft st ov Thn f g h and implis and implis 3 SOT SET OPETIONS 31 nion Lt and b two soft sts ov a common univs thn th union of and dnotd by is a soft st wh and Exampl 31 onsid th soft st which dscibs th cost of th houss and th soft st which dscibs th attactivnss of th houss Suppos that { h1 h2 h3 h h5 h6 h7 h8 h9 h10} {Vy costly ostly hap} nd {hap autiful in th gn suoundings} { 1 2 3} nd { 3 5} spctivly Lt 1 { h2 h h7 h8} 2 { h1 h3 3 { h6 h9 h10} nd 3 { h6 h9 h10} { h2 h3 h7} 5 { h5 h6 h8} thn = wh 1 { h2 h h7 h8} 2 { h1 h3 3 { h6 h9 h10} { h2 h3 h7} 5 { h5 h6 h8} 2018 IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 366

5 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: Intsction Lt and b two soft sts ov a common univs thn th intsction of nd dnotd by is a soft st wh o as both a sam st Exampl 32 by xampl 31 = Wh { h6 h9 h10} Poposition 31 -lt b th nivs is soft st ov Thn a b c d wh is a null st wh is a null st wh f Poposition 32 Poof- a b is absolut st a Suppos that wh if if if Thfo Now Thfo if if if gain K Wh Say K if if if and K a sam function nc povd b Suppos that Thfo gain K Say Wh w hav K o = o Wh = = nd K a sam function nc povd Poposition 33 if and a th soft sts ov Thn 33 ND a b c d Lt and b two soft sts ov a common univs thn th ND opation of and dnotd by ND o by is a soft st dfind by = wh a b a b a b Exampl 33 by xampl 31 = wh { h 2 h7} 1 5 { h 8 } IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 367

6 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: { h 3 } 2 5 { h 5 } 3 3 { h6 h9 h10} { h 6 } 3 O Lt and b two soft sts ov a common univs thn th ND opation of and dnotd by O thfo by is a soft st dfind by = wh a b a b a b Exampl 3 by xampl 31 Thn = wh 1 3 = { h 2 h h6 h7 h8 h9 h10} 1 { h2 h3 h h5 h7 h8} 1 5 { h2 h h5 h6 h7 h8} 2 3 { h1 h3 h5 h6 h9 h10} 2 { h1 h2 h3 h5 h7 h8} 2 5 { h1 h3 h5 h6 h8} 3 3 { h6 h9 h10} 3 { h2 h3 h6 h7 h9} 3 5 { h5 h6 h8 h9 h10} mak 31 W also us th tabula fom fo ND O solving th xampls Poposition 3 Poof- c d c Suppos that Thfo now K wh K x y x y K Now tak Thfo K [ K nd K a sam function nc povd d Suppos that Thfo Now K wh K x y x y K Now tak Thfo K [ [ ] [ ] K nd K a sam function nc povd 35 Extndd intsction Lt and b two soft sts ov a common univs thn th xtndd intsction of and dnotd by is a soft st wh and Exampl 35- onsid th soft st which dscibs th cost of th houss and th soft st which dscibs th attactivnss of th houss Suppos that { h1 h2 h3 h h5 h6} E { } spctivly E {Vy costly ostly hap bautiful in th gn suoundings} {Vy costly ostly hap} th gn suoundings} {chap bautiful in { 1 2 3} E { 3 5} E spctivly Lt { h } 2 { h1 h3 3 { h3 h and 1 2 h 3 { h1 h2 h3} { h2 h3 h6} 5 { h2 h3 h} thn 2018 IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 368

7 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: = wh { h } 2 { h1 h3 1 2 h 3 { h 3 } { h2 h3 h6} 5 { h2 h3 h} 36 stictd Intsction Lt and b two soft sts ov a common univs thn th stictd intsction of and dnotd by is a soft st wh Exampl 36 by xampl 35 Thn = wh stictd nion { h 3 } Lt and b two soft sts ov a common univs thn th stictd union of and dnotd by is a soft st wh Exampl37 by xampl 35 Thn = wh soft st wh Exampl 38 by xampl 35 Thn = wh { h 39 stictd symmtic diffnc Lt and b two soft sts ov a common univs thn th stictd symmtic diffnc of and dnotd by is a soft st wh In oth wods Lt and b two soft sts ov a common univs thn th stictd symmtic diffnc of and dnotd is a soft st dfind by Exampl 39 by xampl 35 Thn now = wh { h stictd diffnc { h1 h2 h3 h Lt and b two soft sts ov a common univs thn th stictd diffnc of and dnotd by is a 5 = K D wh D D K K 1 K 2 K 3 h 1 h } K { 2 K 5 now th valu of 2018 IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 369

8 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 370 = D K wh D Z Z K J 1 J 2 J 3 J } { h h h h J 5 J POPETIES O SOT SET OPETION 1 Idmpotnt poptis a b 2 Idntity poptis a b a b 3 Domination poptis a b omplmntation poptis a b 5 Doubl omplmntation involution popty a 6 Exclusion popty a 7 ontadiction popty a mak 1 Exclusion and contadiction poptis do not hold with spct to complmnt in dfinition 26 8 D Mogan s poptis a b c d f g h i j mak 2 D Mogan s popty dos not hold fo stictd union and stictd intsction with spct to complmnt in dfinition that mans a b 9 bsoption poptis a b c d mak 3 a nd do not absob ov ach oth b nd do not absob ov ach oth 10 ommutativ poptis a b c d

9 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 371 mak a nd do not commut ov ach oth 11 ssociativ poptis a b c d f 12 omplmntation poptis a b c d f g h i j k l mak 5 nd do not distibut ov ach oth nd do not distibut ov ach oth and do not distibut ov nd do not distibut ov and Distibut ov but convs is fals Distibut ov but th convs is fals 5 ONLSIONS In this pap w hav discussd in dtail th fundamntals of soft st thoy such as quality of soft st subst omplmnt with xampls Its poptis with solvd xampls It was obsv that som poptis dos not hold som classic sult and soft st opation In this pap w discussd th D Mogan s law nd all opation union intsction ND O and also lativ opations and its xampls Sam as w poof th all poptis in futu EEENES [1] hmad and Khaal 2009 On uzzy soft sts dvancs in fuzzy systms 1-6 [2] li MI ng LuiX MinWK and Shabi M2009 On som nw opations in soft st thoy omputs and mathmatics with applications [3] abitha KV and Sunil JJ 2010 Soft st lations and functions omputs and mathmatics with applications [] abitha KV and Sunil JJ2011 Tansitiv closu and oding on soft sts omputs and mathmatics with applications [5] agman N and Enginoglu S2010 Soft matix thoy and its dcision making omputs and mathmatics with applications [6] ng ngyoung a Jun and Xianzhong Zhao2008Soft smi ings omputs and mathmatic with applications [7] u Li2011 Nots on soft st opations PN Jounal of systms and softwas [8] X and Yang S2011 Invstigations on som opations of soft sts Wold acadmy of Scinc Engining and Tchnology [9] Maji PK iswas and oy 2003 Soft st thoy omputs and mathmatics with applications

10 Intnational sach Jounal of Engining and Tchnology IJET -ISSN: Volum: 05 Issu: 01 Jan-2018 wwwijtnt p-issn: [10] Molodtsov D1999 Soft st thoy - ist sults omputs and mathmatics with applications [11] Pi D and Miao D2005 om soft st to infomation systms In: Pocdings of anula computing IEEE [12] Qin K and ong Z 2010 On soft quality Jounal of computational and applid mathmatics [13] Szgin and tagun O2011 On opations of soft sts omputs and mathmatics with applications [1] Singh D and Onyozili I 2012 Som concptual misundstanding of th fundamntals of soft st thoy PN Jounal of systms and softwas [15] Singh D and Onyozili I 2012 Som sults on Distibutiv and absoption poptis on soft opations IOS Jounal of mathmatics IOSJM IJET Impact acto valu: 6171 ISO 9001:2008 tifid Jounal Pag 372