Vol. 5, No. 5 May 2014 ISSN Journal of Emerging Trends in Computing and Information Sciences CIS Journal. All rights reserved.

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1 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. Notes on lt Soft trces D.Sngh, Onyeozl, I.., 3 lkl..j., 3 Deprtment of themtcs, hmd Bello Unversty Zr Nger Deprtment of themtcs, Unversty of b Nger BSTRCT Ths pper brefly descrbes mlt soft set, mlt soft mtrx nd ts mtrx representton, nd ponts ot certn msnderstndngs of the concept tht hve ppered n the ltertre. lso, t s shown tht ND nd OR opertons on mlt soft sets nd tht on ther correspondng soft mtrces re eqvlent. Keywords: mlt soft set, mlt soft mtrx, ND nd OR opertons. INTRODUCTION In the pst ten yers or so, reserches on soft set theory, orgnlly proposed by olodtsov[] n 999, hve gthered ttenton both on ts theoretcl[-9] nd pplcton[0-3] spects. The stndrd soft set dels wth bnry-vled nformton system. Recently Herwn et l.[4] ntrodced the noton of mlt soft set, representng mlt-vled nformton system. The de ws bsed on the decomposton of mlt-vled nformton system nto bnry-vled nformton system. Herwn et l.[5] defned ND nd OR opertons on mlt soft sets nd ppled ND-operton n ttrbte redcton n mlt-vled nformton system. Cgmn nd Engnogl[] ntted the concept of soft mtrx s mtrx representton of soft set nd sccessflly ppled t to decson mkng problem. Bsed on the work n [], Herwn et l.[6] presented mlt-soft mtrces, defned ND nd OR opertons on mlt soft mtrces nd ppled them n fndng redctons nd core ttrbtes n mlt-vled nformton system. Ths pper, besdes otlnng the notons of mltsoft sets nd mlt soft mtrces, ponts ot some msnderstndngs tht hve ppered n ths regrd n the ltertre nd demonstrtes tht ND nd OR opertons on mlt soft sets nd tht on ther correspondng soft mtrces re eqvlent.. ULTI SOFT SETS Defnton.: lt Soft Sets Let U,,,, 3 be fnte set of obects whch my be chrcterzed by fnte fmly of where ech,,,, prmeter sets 3 prmeter set,,,, represents the th clss of prmeter nd the elements of represent specfc property set. pr (F,) over U, whch conssts of set of soft sets F,,, mlt-soft set over U. In other words, ( F, ) F,, F,,, F, s clled Exmple. Let s consder mlt-soft set (F,) whch descrbes the ttrctveness of hoses n n Estte whch r. X (sy) s consderng to by. Let nd,, U h, h, h,, h be the set of ten hoses be fmly of three decsonprmeters nder consderton. Let = {expensve, chep, very expensve, very chep} represent cost prmeter, low densty, hgh densty} represent { locton prmeter, nd 3 { green, ble, red} represent colorprmeter. Let the correspondng soft sets be F, expensve = h, h, h, chep h, h nd h5 h8 h6 h7 h9 very expensve =,, very chep =,, ; F h h h h, low densty =,,,, F, green = h, h, h, h, ble h, h, h, h red = h, h. Then we cn vew the mlt-soft set (F,) s consstng of the followng: hgh densty = h, h, h, h, h, h, 4

2 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. ( F, ) ( F, ), F,, F, 3 h h h h h expensve =,,, chep, h5 h8 h6 h7 h9 very expensve =,, very chep =,,, low densty = h, h4, h6, h8, hgh densty = h, h, h, h, h, h, nd Exmple. Consder the nformton system n tble below: Tble U sze( ) Textre ( ) 3 Smll Corse Blck Bg Smooth Red 3 edm Smooth Green 4 edm Corse Green 5 Smll Corse Red Color ( ) green =,,,, ble,,, red = h, h. Defnton.[6]: lt-vled nformton system mlt-vled nformton system s qdrple S ( U,, V, f ) where U,,, U obects,,,, V s non-empty fnte set of sets of s non-empty fnte set of ttrbtes, s the domn (vle V wherev set) of ttrbte, nd f : U V s n nformton fncton sch tht f (, ) V f (, ) U., for every n Informton System cn be nttvely expressed n terms of n nformton tble (see Tble ), where the rows re lbeled by the obects (enttes), colmns re lbeled by the ttrbtes, nd the entry n row nd colmn hs the vle f(,), clled the ttrbte vle. Tble :, f,, h h h h h h h h f, f f, f f, f, f, f,, We hve: = { sze =, textre =, color = 3 } U = {,, 3, 4, 5 }V = { smll, bg, medm } V = { corse, smooth} V 3 = { blck, red, green}v = V, f : U V sch tht f(, ) = smll V, f(, ) = corse V, etc. Defnton.3 [6]: Decomposton of n Informton System Let S = (U,, V, f) be n nformton system where,,, sch tht for every, V f ( U, ) s fnte non-empty set nd for every U, f (, ). For every nder th - ttrbte consderton, nd v V, we defne the mp : 0, V U sch tht V ( ) f f (, ) v, otherwse V ( ) 0. Then the bnry-vled nformton S U,, V, f, s referred to system {0,} s decomposton of S = (U,, V, f) nto bnryvled nformton systems. Defnton.4 [6]: lt soft set n n Informton System Let S ( U,, V, f ) be mlt-vled nformton system nd let {0,} S U,, V, f, be the decomposton of S nto bnry-vled nformton systems defned by 4

3 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. S U,, V, f {0,} Then, we defne,, {0,},,,, {0,},, S U V f F S U V f F {0,} S U V f F,,,,. ( F, ) F,, F,,, F, s mltsoft set over U representng mlt-vled nformton system S ( U,, V, f ). Exmple.3 Consder the nformton system n Exmple.(Tble ) whose decomposton nto = 3 bnryvled nformton systems re gven s follows: U Smll bg edm U Corse Smooth U 3 Blck Red Green s defned s F, ND F, H, where v, v H v, v F v F v,for,., ; (b) The OR operton between F, nd F, denoted by, OR, F F s defned s F, OR F, H, v, v ;, where H v, v F v F v, for,. Exmple.4 Consder the mlt soft set (F,) n Exmple.3 where (F,)= { (F, ), (F, ), (F, 3 ) } sch tht (F, ) = { smll = {, 5 }, bg = { }, medm = { 3, 4 }} (F, ) = { corse = {,, 5 }, smooth = { 3, 4 } } (F, 3 ) = {blck = { }, red = { 4, 5 }, green = {, 3 } } Then () (F, ) ND (F, ) = (H, ) (bg, corse) = {}, (bg, smooth) = {}, = { ( smll, corse) = {, 5 }, (smll, smooth) = {}, ( medm, corse) = {}, (medm, smooth) = { 3, 4 } () (F, ) OR (F, ) = (K, ) = { (smll, corse) = {,, 5 }, (smll, smooth) = {, 3, 4, 5 ), (bg, corse) = {,, 5 }, (bg, smooth) = {, 3, 4 }, (medm, corse) = {, 3, 4 ), (medm, smooth) = { 3, 4 ). 3. TRIX REPRESENTTION OF ULTI SOFT SET 3. lt Soft trx The correspondng soft sets re s follows: (F, ) = {smll = {, 5 }, bg = { }, medm = { 3, 4 }} F, ) = {corse = {,, 5 } smooth = { 3, 4 } (F, 3 ) = {blck = { }, red = { 4, 5 }, green = {, 3 } Ths the mlt soft set (F,) over U, representng tble, s gven by (F,) = {(F, ), (F, ), (F, 3 )} Defnton.5 [6]: ND nd OR Opertons on mlt soft sets ( F, ) F, :,,, be Let mlt soft set over nverse U, representng mltvled nformton system S = (U,, V, f). () The ND operton between, nd, F, ND F, F F denoted by Defnton 3. [6]: lt soft mtrx ( F, ) F, :,,, be Let mlt soft set over U representng mlt-vled nformton system S = (U,, V, f). The mtrx, re defned whose entres by, f f (, ),, V, U, V 0, f f(, ) = 0 nd whose dmenson Dm V, s clled the mtrx representton of the soft set F, n the mlt soft set (F,).The collecton of ll mtrces representng (F,), denoted by clled the mlt soft mtrx s defned by 43

4 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. :. Exmple 3. Consder the followng mlt soft set (F,) tken n Exmple.3 where, (F,) = { (F, ),(F, ), (F, 3 ) }. Then the mlt soft mtrx, representng (F,) s the collecton of ll mtrces gven by; , 0, Defnton 3. [6]: ND nd OR opertons n mlt soft mtrces Let [ ],, [ ] mn m, k, n v nd, l v, be two mtrces n mlt soft mtrx representng mlt soft set (F,) over U of mlt-vled nformton system. () ND operton of mtrces,denoted by kl ND s ND [ ] pq nd, s defned wth p, q v v where mn{, }, mn{, }, p m k p m k..., mn{, }. p v v m v k v () OR operton of,denoted by s OR [ b ] pq mtrces nd OR, s defned wth p, q v v where b mx{, }, b mx{, }, p m k p m k..., b mx{, }. p v v m v k v Exmple 3. From Exmple 3., we hve () ND () OR OR ND 44

5 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. 3. Proof tht Propostons () nd (b) of [6] re Incorrect Proposton 3.: Propostons () nd (b) of [6] Let, be mtrces n the collecton of soft mtrces representng mlt soft set (F,). The followng hold: () (b) ND OR Idempotent Lw Idempotent Lw Proof: The bove propostons re ncorrect s cn be seen below: () Dm( ) V, Bt Dm( ND ) V V. Ths s not closed nder ND operton. Hence. ND (b) Smlr rgment holds. Hence OR. We llstrte these foresd rgments sng or exmple () ND 0 ND The sme exmple works for (b) s well. 4. COPRING (ND) ND (OR) OPERTIONS IN ULTI SOFT SETS WITH THEIR CORRESPONDING ULTI SOFT TRICES In ths secton, we show tht ND nd OR opertons n mlt soft sets nd tht on ther correspondng mlt soft mtrces re eqvlent. Consder the mlt soft set (F,)tken n exmple.3 where (F,) = { (F, ), (F, ), (F, 3 )} sch tht, (F, ) = {smll = {, 5 }, bg = { }, medm = { 3, 4 } }; (F, ) = {corse = {,, 5 }, smooth{ 3, 4 }} nd ther correspondng mlt soft mtrces (Exmple 3) gven by nd , respectvely. 0 0 Now from Exmple.5, we hve tht the mlt soft set: F, ND F, H, {( smll, corse), 5, bg, corse, medm, smooth {, }}. 3 4 Then the correspondng mlt soft mtrx, denoted, representng the mlt soft seth, s gven by ND lso from Exmple.5, F, OR F, K, {( smll, corse) {,, },( smll, smooth) {,,, }, ( bg, corse) {,, },( bg, smooth) {,, },( medm, corse) U, ( medm, smooth) {, }} nd whose correspondng mlt soft mtrx s 45

6 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. gven by OR 5. CONCLUSION The notons of mlt soft set nd mlt soft mtrx re explned nd llstrted n prtclr reference to mlt-vled nformton system. ND nd OR-opertons on mlt soft sets nd ther correspondng mlt soft mtrces re dscssed nd re shown to be eqvlent. Some msnderstndngs of the concept tht hve ppered n the ltertre were clrfed. REFERENCES [] olodtsov, D.(999). Soft set Theory-Frst reslts, Compters nd themtcs wth pplctons, 37(4-5), 9-3. [] l,.i., F. Feng, X. L, W.K n nd. Shbr(009) On some new opertons n soft set theory, Compters nd themtcs wth pplctons, 57(9), [3] l,. I., Shbr,. nd Nz,. (0). lgebrc strctres of soft sets ssocted wth new operton, Compters nd themtcs wth pplctons, 6(9), [4] tgn,. O. nd Sezgn,. (0). Soft sbstrctres of rngs, felds nd modles, Compters nd themtcs wth pplctons,6(4), [9] Sngh, D nd Onyeozl, I.(0). Notes on soft mtrces opertons, RPN Jornl of scence nd technology, Vol, No 9, p [0] Cgmn, N. nd Engnogl, S. (00).Soft set theory nd n- nt decson mkng, Eropen Jornl of opertonl reserch, 07(), [] Cgmn, N. nd Engnogl, S. (00). Soft mtrx theory nd ts decson mkng, Compters nd themtcs wth pplctons, 59(0), [] Chen, D., Tsng, E.C.C., Yeng, D. S. nd Wng,X. (005). The prmeterzed redcton of soft sets nd ts pplctons, Compters nd themtcs wth pplctons, 49(5-6), [3], P.K., Bsws, R. nd Roy,.R. (00). n pplcton of soft sets n decsons mkng problems, Compters nd themtcs wth pplctons, 44(8-9), [4] Herwn, T., nd stf,.d(009). On mlt soft sets constrcton n nformton systems, In : Hng O.S et l(eds) ICIC 009, LNCS(LNI) vol.5755 p.0-0, Sprnger Verlg, Hedelberg. [5] Herwn, T., Ghzl, R., nd stf,.d(00). Soft set theoretc pproch for Dmensonlty Redcton, Interntonl Jornl of Dtbse Theory nd pplcton. 3() [6] Herwn, T., stf,.d., nd Jeml, H.., (00). trces Representton of mlt soft sets nd ts pplcton In: D Tnr et l.(eds) ICCS(00) prt III, LNCS vol. 608, p.0-4, Sprnger Verlg, Berln. [5] tks, H. nd Cgmn, N. (007). Soft sets nd soft grops, Informton scences, (3), [6] Bbth, K. V. nd Snl, J. J. (00). Soft set relton nd fnctons, Compters nd themtcs wth pplctons, 60(7), [7] Jn, Y.B. (008). Soft BCK/BCI-lgebrs, Compters nd themtcs wth pplctons, 56(5), [8], P.K., Bsws, R. nd Roy,.R. (003). Softset theory, Compters nd themtcs wth pplctons,45(4-5), UTHORS INTRODUCTION Sngh, D. s former professor (IIT Bombey) nd crrently professor, Deprtment of themtcs, hmd Bello Unversty, Zr- Nger. res of speclzton re Sets, ltsets, Fzzy set, Fzzymltsets nd soft sets theory. Onyeozl, I.. receved her sc. degree n themtcs from Unversty of Nger Nskk n 99. She s Ph.D. reserch scholr n the deprtment of themtcs, hmd Bello Unversty, Zr- Nger. Crrently, she s senor lectrer n the deprtment of themtcs, Unversty of b- Nger. lkl, J. receved hs sc. degree n themtcs from hmd Bello Unversty, Zr- Nger n 00. He s crrently, Ph.D. reserch scholr 46

7 Vol. 5, No. 5 y 04 ISSN Jornl of Emergng Trends n Comptng nd Informton Scences CIS Jornl. ll rghts reserved. nd lectrer n the deprtment of themtcs, hmd Bello Unversty, Zr-Nger. 47