A New Approach to Multi-spaces Through the Application

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1 Neutrosophc Sets ad Systems Vol A New Approach to Mult-spaces Through the Applcato Mumtaz Al 1 Floret Smaradache Sad Broum 3 ad Muhammad Shabr 4 14 Departmet of Mathematcs Quad--Azam Uversty Islamabad 44000Paksta E-mal: mumtazal770@yahoocom mshbrbhatt@yahoocouk Uversty of New Mexco 705 Gurley Ave Gallup New Mexco USA E-mal: fsmaradache@gmalcom 3 Faculty of letters ad Humates Hay El Baraka Be M'sk Casablaca BP 7951 Uversty of Hassa II -Casablaca Morocco E-mal: broumsad78@gmalcom Abstract Mult-space s the oto combg dfferet felds to a ufyg feld whch s more applcable our daly lfe I ths paper we troduced the oto of mult-soft space whch s the approxmated collecto of the mult-subspaces of a mult-space Further we defed some basc operatos such as uo tersecto AND OR etc We also vestgated some propertes of mult-soft spaces Keywords: Mult-space soft set mult-soft space 1 Itroducto Mult-spaces [4] were troduced by Smaradache 1969 uder the dea of hybrd structures: combg dfferet felds to a ufyg feld [3] that are very effectve our real lfe Ths dea has a wde rage of acceptace the world of sceces I ay doma of kowledge a Smaradache multspace s the uo of dfferet spaces wth some dfferet for a teger Smaradache mult-space s a qualtatve oto as t s too huge whch clude both metrc ad o-metrc spaces Ths mult-space ca be used for both dscrete or coected spaces specally spacetmes ad geometres theoretcal physcs Mult-space theory has appled physcs successfully the Ufed Feld Theory whch ute the gravtatoal electromagetc weak ad strog teractos or the parallel quatum computg or the mu-bt theory etc Several mult-algebrac structures have bee troduced such as mult-groups mult-rgs mult-vector spaces mult-metrc spaces etc Lterature o mult-algebrac structures ca be foud [17] Molodtsov [0] proposed the theory of soft sets Ths mathematcal framework s free from parameterzato adequacy sydrome of fuzzy set theory rough set theory probablty theory ad so o Soft set theory has bee appled successfully may areas such as smoothess of fuctos game theory operato research Rema tegrato Perro tegrato ad probablty thoery Soft sets gaed much atteto of the researchers recetly from ts appearace ad some lterature o soft sets ca be see [1]- [16] Some other propertes ad algebras may be foud [18190] Some other cocepts together wth fuzzy set ad rough set were show [13] I secto we revew some basc cocepts ad otos o mult-spaces ad soft sets I secto 3 we defe mult-subspac The mult-soft spaces has bee troduced the curret secto Mult-soft space s a parameterzed collecto of mult-subspaces We also vestgated some propertes ad other otos of mult-soft spaces Basc Cocepts I ths secto we revew some basc materal of multspaces ad soft sets M Al F Smaradache S Broum ad M Shabr A New Approach to Mult-spaces Through the Applcato

2 Neutrosophc Sets ad Systems Vol Defto 1 [4] For ay teger 1 let M be a set wth esemble of law L ad the tersecto of k sets M M 1 M of them costras the k law I M M 1 M The the uo of M k 1 s called a mult-space M Let U be a tal uverse E s a set of parameters PU ( ) s the power set of U ad A B E Molodtsov defed the soft set the followg maer: Defto [0] A par ( FA ) s called a soft set over U where F s a mappg gve by F : A P( U ) I other words a soft set over U s a parameterzed famly of subsets of the uverse U For a A F a may be cosdered as the set of a -elemets of the soft set ( FA ) or as the set of a -approxmate elemets of the soft set Example 3 Suppose that U s the set of shops E s the set of parameters ad each parameter s a word or setece Let E 1 M hgh retormal ret good codto bad codto Let us cosder a soft set ( FA ) whch descrbes the attractveess of shops that MrZ s takg o ret Suppose that there are fve houses the uverse U { s1 s s3 s4 s 5} uder cosderato ad that A { a1 a a 3} be the set of parameters where a 1 stads for the parameter 'hgh ret a stads for the parameter 'ormal ret a 3 stads for the parameter ' good codto Suppose that F( a1) { s1 s 4} F( a ) { s s } 5 F( a ) { s } 3 3 The soft set ( FA ) s a approxmated famly { F( a ) 13} of subsets of the set U whch gves us a collecto of approxmate descrpto of a object The ( FA ) s a soft set as a collecto of approxmatos over U where F( a ) hgh ret { s s } 1 1 F( a ) ormal ret { s s } 5 F( a ) good codto { s } 3 3 Defto 4 [19] For two soft sets ( FA ) ad ( H B) over U ( FA ) s called a soft subset of ( H B) f 1 A B ad F( a) H( a ) for all x A Ths relatoshp s deoted by ( F A) ( H B) Smlarly ( FA ) s called a soft superset of ( H B) f ( H B) s a soft subset of ( FA ) whch s deoted by ( F A) ( H B) Defto 5 [19] Two soft sets ( FA ) ad ( HB) over U are called soft equal f ( FA ) s a soft subset of ( H B) ad ( H B) s a soft subset of ( FA ) Defto 6 [19] Let ( FA ) ad (GB) be two soft sets over a commo uverse U such that A B The ther restrcted tersecto s deoted by ( F A) R (GB) ( HC) where ( H C) s defed as H( c) F( c) c) for all c C A Defto 7 [1] The exteded tersecto of two soft sets ( FA ) ad (GB) over a commo uverse U s the soft set ( H C) where C A B ad for all c C Hc () s defed as M Al F Smaradache S Broum ad M Shabr A New Approach to Mult-spaces Through the Applcato

3 36 Neutrosophc Sets ad Systems Vol F( c) f c A B H( c) G( c) f c B A F( c) G( c) f c A We wrte ( F A) (GB) ( H C) Defto 8 [19] The restrcted uo of two soft sets ( FA ) ad (GB) over a commo uverse U s the soft set ( H B) where C A ad for all c C Hc () s defed as H( c) F( c) G( c) for all c C We wrte t as ( F A) R (GB) ( HC) Defto 9 [1] The exteded uo of two soft sets ( FA ) ad (GB) over a commo uverse U s the soft set ( H B) where C A ad for all c C Hc () s defed as F( c) f c A B H( c) G( c) f c B A F( c) G( c) f c A B We wrte ( F A) (GB) ( H C) I the ext secto we troduced mult-soft spaces 3 Mult-Soft Space ad Its Propertes I ths secto frst we troduced the defto of mult-subspace Further we troduced mult-soft spaces ad ther core propertes ' Defto 31 Let M be a mult-space ad M M ' ' The M s called a mult-subspace f M s a multspace uder the operatos ad costats of M Defto 3 Let A1 { a j : j J} A { ak : k K} A { a : L} be -set of parameters Let ( F1 A1 )( F A )( F A ) are soft set over the dstct uverses M1 M M respectvely The ( HC ) s called a mult-soft space over M M1 M where (HC) ( F1 A1 ) E ( F A ) E E( F A ) such that C A1 A A ad for all c C Hc () s defed by H( c) F ( c) F ( c) F ( c) 1 k f c ( A A A ) ( A A A ) 1 k k 1 k where ( 1 k k 1 ) are all possble permutatos of the dexes (1 ) k = 1 There are 1 peces of the pece-wse fucto ( HC ) Proposto 33 Let M be a uverse of dscourse ad ( FA ) s a soft set over M The ( FA ) s a mult-soft space over M f ad oly f M s a mult-space Proof: Suppose that M s a mult-space ad F : A P(M) be a mappg The clearly for each a A the Fa ( ) s a subset of M whch s a multsubspace Thus each Fa ( ) s a mult-subspace of M ad so the soft set ( FA ) s the parameterzed collecto of mult-subspaces of M Hece ( FA ) s a mult-soft space over M For coverse suppose that ( FA ) s a mult-soft space over M Ths mples that Fa ( ) s a mult-subspace of M for all a A Therefore M s a mutl-space Ths stuato ca be llustrated the followg Example Example 34 Let M { u1 u u3 u4 u5 u6 u7} be a tal uverse such that M s a mult-space Let A1 { a1 a a3 a8} A { a a4 a5 a6 a8} ad A { a a a } are set of parameters Let F1 A1 F A ad 3 3 ( )( ) sets over M as followg: ( F A ) respectvely be the soft F1 ( a1) {m 1m m 3} F1 ( a) {m 4m 5} M Al F Smaradache S Broum ad M Shabr A New Approach to Mult-spaces Through the Applcato ad F a {m m m m } F a {m m m m } F ( a ) {m m m m m } F ( a ) {m m m m } F a {m m m } 5 4 5

4 Neutrosophc Sets ad Systems Vol Also F ( a6) {m 1m 7} F a {m m m m } F( a ) {m m m m m } F3 ( a7) {m 4m 5m 7} F a {m } 3 8 A A A A { a a a a a a a a } Let The the mult-soft space of ( F1 A1 )(F A ) ad ( F3 A 3) s ( FA ) where ( F A) ( F A ) ( F A ) ( F A ) such that 1 1 E E 3 3 F( a ) F ( a ) {m m m } as a A A A F( a ) F ( a ) F ( a ) {m m m m m m m } a A A A as 1 3 F( a ) F a {m m m m } as a3 A1 A A3 F( a ) F ( a ) {m m m m } as a A A A F( a ) F ( a ) F ( a ) {m m m m m } a A A A as F( a6) F ( a6) {m 1m 7} as a6 A A1 A3 F( a ) F ( a ) {m m m } as a A A A F( a ) F a F ( a ) F ( a ) {m m m m m m m } as a8 A1 A A3 Defto 35 Let ( FA ) ad ( HB ) be two multsoft spaces over M1 M The ( FA ) s called a mult-soft subspace of ( HB ) f 1 A B ad F( a) H( a ) for all a A Ths ca be deoted by ( F A) ( H B ) Smlarly ( FA ) s called a mult-soft superspace of ( FA ) f ( FA ) s a mult-soft subspace of ( FA ) whch s deoted by ( F A) ( H B ) Defto 36 Two mult-soft spaces ( FA ) ad ( HB ) over M1 M are called mult-soft mult-equal f ( FA ) s a mult-soft subspace of ( HB ) ad ( HB ) s a mult-soft subspace of ( FA ) Proposto 36 Let ( FA ) ad ( KB ) be two multsoft spaces over M1 M such that A B The ther restrcted tersecto ( F A) R ( K B) ( H C ) s also a mult-soft space over M1 M Proposto 37 The exteded tersecto of two multsoft mult-spaces ( FA ) ad ( KB ) over M1 M s aga a mult-soft mult-space M M M over 1 Proposto 38 Let ( FA ) ad ( KB ) be two multsoft mult-spaces over M1 M such that A B The ther restrcted uo ( F A) R ( K B) ( H C ) s also a mult-soft mutspace over M1 M Proposto 39 The exteded uo of two mult-soft mult-spaces ( FA ) ad ( KB ) over M1 M M s aga a mult-soft mult-space over M1 M M Al F Smaradache S Broum ad M Shabr A New Approach to Mult-spaces Through the Applcato

5 38 Neutrosophc Sets ad Systems Vol Proposto 310 The AND operato of two mult-soft mult-spaces ( FA ) ad ( KB ) over M1 M s aga a mult-soft mult-space M M M over 1 Proposto 311 The OR operato of two mult-soft mult-spaces ( FA ) ad ( KB ) over M1 M s aga a mult-soft mult-space M M M over 1 Proposto 31 The complemet of a mult-soft space over a mult-space M s aga a mult-soft space over M Prof Ths s straghtforward Defto 313 A mult-soft mult-space ( FA ) over M1 M s called absolute mult-soft mult-space f Fa () = M1 M for all a A Proposto 314 Let ( FA ) ( GB ) ad ( HC ) are three mult-soft mult-spaces over M1 M The 1 ( F A) E ( G B) E ( H C) ( F A) E ( G B) E ( H C ) ( F A) R ( G B) R ( H C) ( F A) R ( G B) R ( H C ) Proposto 315 Let ( FA ) ( GB ) ad ( HC ) are three mult-soft mult-spaces over M1 M The 1 ( F A) ( G B) ( H C ) ( F A) ( G B) ( H C ) ( F A) ( G B) ( H C ) ( F A) ( G B) ( H C ) Cocluso I ths paper we troduced mult-soft spaces whch s a frst attempt to study the mult-spaces the cotext of soft sets Mult-soft spaces are more rch structure tha the mult-spaces whch represet dfferet felds a approxmated ufyg feld We also studed some propertes of mult-soft spaces A lot of further research ca do the future ths area I the future oe ca defe the algebrac structures of mult-soft spaces Refereces [1] H Aktas N Cagma Soft sets ad soft groups If Sc 177 (007) [] K Ataassov Itutostc fuzzy sets Fuzzy Sets Syst 64 () (1986) [3] M Al F SmaradacheM Shabr M Naz Soft Neutrosophc Bgroup ad Soft Neutrosophc N-group Neutrosophc Sets ad Systems (014) [4] M Al F Smaradache M Shabr M Naz Soft Neutrosophc Rgad Soft Neutrosophc Feld Neutrosophc Sets ad Systems 3 (014) [5] M Al C Dyer M Shabr F Smaradache Soft Neutrosophc Loops ad Ther Geeralzato Neutrosophc Sets ad Systems 4 (014) [6] M Al F Smaradache ad M Shabr Soft Neutrosophc B-LA-Semgroup ad Soft Neutrosophc N-LA-Semgroup Neutrosophc Sets ad Systems 5 (014) [7] M Al F Smaradache L Vladareau ad M Shabr Geeralzato of Soft Neutrosophc Rg ad Soft Neutrosophc Felds Neutrosophc Sets ad Systems 6 (014) [8] M Al F SmaradacheM Shabr M Naz Soft Neutrosophc Bgroup ad Soft Neutrosophc N-group Neutrosophc Sets ad Systems (014) [9] M Al F Smaradache M Shabr ad M Naz Soft Neutrosophc Semgroups ad Ther Geeralzato Sceta Maga 10(1) (014) [10] M Al F Smaradache ad M Shabr Soft Neutrosophc Groupods ad Ther Geeralzato Neutrosophc Sets ad Systems 6 (014) 6-81 [11] M Al F Smaradache ad M Shabr Soft Neutrosophc Algebrac Structures ad Ther Geeralzato Vol EuropaNova ASBL 3E clos du Paraasse Brussels 1000 Belgum [1] M I Al F Feg X Lu W K M M Shabr O some ew operatos soft set theory Comp Math Appl 57(009) [13] M Aslam M Shabr A Mehmood Some studes soft LA-semgroup Joural of M Al F Smaradache S Broum ad M Shabr A New Approach to Mult-spaces Through the Applcato

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