An Introduction to the Theory of Imprecise Soft Sets

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1 IJ Itelliget Systems ad Appliatios Published Olie Otober 22 i MECS ( DOI: 585/isa229 A Itrodutio to the Theory of Impreise Soft Sets Tridiv Jyoti eog Dept of Mathematis CMJ Uiversity Shillog Meghalaya Idia tridivjyoti@gmailom Dusmata Kumar Sut Dept of Mathematis Saikia College Titabor Jorhat Idia sutdk@yahooom Abstrat This paper aims to itrodue the theory of impreise soft sets whih is a hybrid model of soft sets ad impreise sets It has bee established that two idepedet laws of radomess are eessary ad suffiiet to defie a law of fuzziess Further i ase of fuzzy sets the set theoreti axioms of exlusio ad otraditio are ot satisfied Aordigly the theory of impreise sets has bee developed where these mistakes arisig i the literature of fuzzy sets are abset Our work is a edeavor to ombie impreise sets with soft sets resultig i impreise soft sets We have put forward a matrix represetatio of impreise soft sets Fially we have studied the otio of similarity of two impreise soft sets ad put forward a appliatio of similarity i a deisio problem Idex Terms Impreise Sets Partial Presee Soft Sets Impreise Soft Sets Presee Level Matrix Similarity of Impreise Soft Sets I Itrodutio The disovery of fuzzy sets by Zadeh i 965 was a paradigm hage i the history of mathematis I the theory of fuzzy sets it has bee observed that the operatio of omplemetatio of a ormal fuzzy set does ot explai the priiples of exlusio ad otraditio followed by the lassial sets This is due to the fat that i the Zadehia defiitio of omplemetatio membership value ad membership futio had bee take to be of the same meaig [] I order to lik fuzziess with probability Zadeh the forwarded Probability-Possibility Cosistey Priiple But o osistey betwee probability ad fuzziess was refleted by that priiple ad two more Probability-Possibility Cosistey Priiples were forwarded thereafter by others Reetly Baruah [2] has itrodued the theory of impreise sets where these two mistakes i the literature of fuzzy sets are abset Most of the oepts we meet i our day to day life are vague i ature Mathematial modelig of suh day to day problems ivolvig uertaities is of great importae ow a days There are theories eg Probability Theory Fuzzy Set Theory Ituitioisti Fuzzy Set Theory Rough Set Theory et to deal with uertaities However these theories have their ow diffiulties Ifat the iadequay of the parameterizatio tool do ot allow these theories to hadle vagueess properly I 999 Molodtsov [3] itrodued the ovel oept of Soft Sets ad established the fudametal results of the ew theory Soft Set Theory is free from parameterizatio iadequay sydrome of Fuzzy Set Theory Rough Set Theory Probability Theory et Maji [4] studied the theory of soft sets ad iitiated some ew results I 2 eog ad Sut [5] put forward a ew defiitio of omplemet of a soft set ad showed that the axioms of exlusio ad otraditio are satisfied by the soft sets also I reet times researhes have otributed a lot towards fuzzifiatio of Soft Set Theory Combiig fuzzy sets ad soft sets Maji et al [6] put forward a ew model kow as fuzzy soft set He itrodued some properties regardig fuzzy soft uio itersetio omplemet of a fuzzy soft set De Morga Law et These results were further revised ad improved by Ahmad ad Kharal [7] They defied arbitrary fuzzy soft uio ad itersetio ad proved De Morga Ilusios ad De Morga Laws i Fuzzy Soft Set Theory eog ad Sut [89] have studied the theory of fuzzy soft sets i a ew perspetive ad iitiated several results related to fuzzy soft sets I their work Zadehia fuzzy sets have bee replaed with the exteded fuzzy sets iitiated by Baruah The otet of the paper is orgaized as follows: I setio II some prelimiary oepts related to exteded fuzzy sets impreise sets ad soft sets have bee put forward Setio III deals with the otio of impreise soft set theory ad some fudametal results I setio IV a matrix represetatio for a total impreise soft set has bee developed ad fially i setio V the otio of similarity of two total impreise soft sets has bee put forward ad the proposed otio has bee applied i a medial diagosis problem

2 76 A Itrodutio to the Theory of Impreise Soft Sets II Prelimiaries I this setio we first reall some oepts ad defiitios whih would be eeded i the sequel The followig otios regardig exteded defiitio of fuzzy sets ad impreise sets are due to Baruah ([][2][][][2]) 2 Exteded Defiitio of Uio ad Itersetio of Fuzzy Sets Let A x ; B 2 2 ad x ; be two fuzzy sets defied over the same uiverse U The the operatios itersetio ad uio are defied as A 2 B3 4 x mi max ; x U ad A B3 4 x max mi ; x U Complemet of a Fuzzy Set Usig Exteded Defiitio For usual fuzzy sets A x ( xu x ( ; xu ad B defied over the same uiverse U we have B x mi max ( x ( x xu A ; xu ); set ad B x max ( mi ( x xu A ; xu whih is othig but the ull ; whih is othig but the uiversal set U This meas if we defie a fuzzy set A x ( ; omplemet of A x ( xu it is othig but the 23 Impreise umber A impreise umber is a iterval aroud the real umber with the elemets i the iterval beig partially preset 24 Partial Presee Partial presee of a elemet i a impreise real umber is desribed by the presee level idiator futio p( whih is outed from the referee futio r( suh that the presee level for ay x x is (p( - r() where r( p( 25 ormal Impreise umber A ormal impreise umber is assoiated with a presee level idiator futio (x ) where ( if x ( 2( if x otherwise with a ostat referee futio i the etire real lie Here ( x ) is otiuous ad o-dereasig i the iterval ad 2( x ) is otiuous ad oireasig i the iterval with ( ) 2 ( ) ( ) 2 ( ) Here the impreise umber would be haraterized byx : xr R beig the real lie 26 Distributio Futio ad Complemetary Distributio Futio For a ormal impreise umber with a presee level idiator futio (x ) where ( if x ( 2( if x otherwise suh that ( ) 2 ( ) ( ) 2 ( ) with ostat referee futio equal to ( x ) is the distributio futio of a radom variable defied i the iterval ad 2( x ) is the omplemetary distributio futio of aother radom variable defied i the iterval We are usig the term radom variable here i the broader measure theoreti sese whih does ot require that the otio of probability eed to appear i defiig radomess 27 Complemet of a ormal Impreise umber For a ormal impreise umber x : xr the omplemet x : x R will have ostat presee

3 A Itrodutio to the Theory of Impreise Soft Sets 77 level idiator futio equal to the referee futio beig (x ) for x 28 Complemet of a Impreise Set If a ormal impreise umber is defied with a presee level idiator futio (x ) where ( if x ( 2( if x otherwise With ( ) 2 ( ) ( ) 2 ( ) the omplemet C should have the presee level idiator futio ( with ( x where ( is to be outed from ( x ) if x from 2 if x ad from otherwise Molodtsov [3]defied soft set i the followig way- 29 Soft Set A pair (F is alled a soft set (over U) if ad oly if F is a mappig of E ito the set of all subsets of the set U I other words the soft set is a parameterized family of subsets of the set U Every set F( ) E from this family may be osidered as the set of - elemets of the soft set (F or as the set of - approximate elemets of the soft set III A Itrodutio To The Theory of Impreise Soft Sets 3 Impreise Soft Set Let U be the iitial uiverse ad E be the set of parameters Let A E ad P ( U ) deote the set of all impreise subsets over the uiverse U The a pair (F is alled a impreise soft set over U where F : A P( U) is a mappig from A ito P ( U ) 32 Example U be the set of four ars uder osideratio ad Let e (ostly) (Beautiful) (Fuel Effiiet) E (ModerTeh ology) (Luxurious) be the set of parameters ad A e e e E F A 2 3 The F e ) ( F( ) F( ) is the impreise soft set represetig the attrativeess of the ar whih Mr X is goig to buy 33 Total Impreise Soft Set Let U be the iitial uiverse E be the set of parameters ad P ( U ) deote the set of all impreise subsets over the uiverse U The the pair (F is alled a total impreise soft set over U where F : E P( U) is a mappig from E ito P ( U ) 34 ull Impreise Soft Set A impreise soft set (F over U is said to be ull impreise soft set (with respet to the parameter set deoted by A if A F( ) is the ull set The ull Impreise Soft Set is ot uique it depeds upo the set of parameters uder osideratio 35 Absolute Impreise Soft Set A impreise soft set (F over U is said to be absolute impreise soft set (with respet to the parameter set deoted by U A if A F( ) is the absolute set U The Absolute Impreise Soft Set is ot uique it depeds upo the set of parameters uder osideratio 36 Uio of Impreise Soft Sets Uio of two impreise soft sets (F ad (G over (U is a impreise soft set (H where C A B ad C F( )if A B H ( ) G( ) if B A F( ) G( )if A B ad is writte as F A G B H C 37 Itersetio of Impreise Soft Sets Let (F ad (G be two impreise soft sets over (U with A B The itersetio of the impreise soft sets (F ad (G is a impreise soft set (H where C A B ad C H( ) F( ) G( ) We write F A G B H C 38 Impreise Soft Subset For two impreise soft sets (F ad (G over (U we say that (F is a impreise soft subset of (G if

4 78 A Itrodutio to the Theory of Impreise Soft Sets (i) A B (ii) For all A (F (G G F ad is writte as 39 Equality of Two Impreise Soft Sets For two impreise soft sets (F ad (G over (U we say that (F is equal to (G if (F (G ad (G (F 3 Complemet of a Impreise Soft Set The omplemet of a impreise soft set (F is deoted by (F ad is defied by (F = (F where F : A P ( U ) is a mappig give by F ( ) F( ) A I other words A if F( ) x F( )( xu the F ( ) ) x F( ; x U 3 AD Operatio of Two Impreise Soft Sets If (F ad (G be two impreise soft sets the (F AD (G is a impreise soft set deoted F A G B ad is defied by by F A G B H A B where F G A H ad B where is the operatio itersetio of two impreise sets 32 OR Operatio of Two Impreise Soft Sets If (F ad (G be two impreise soft sets the (F OR (G is a impreise soft set deoted by F A G B ad is defied by F A G B O A B where F G A O ad B where is the operatio uio of two impreise sets 33 Propositio For a impreise soft set (F over U we have F A F A U A 2 F A F A A Proof Let F A F A where A H () F( ) F ( ) F A F A H A F ( ) F( ) x F( )( xu x F( )( x xu x max mi xu ); ; F( ) F( ) x ; xu U F A F A U A Thus F A F A H A 2 Let F A F A where A H () F( ) F ( ) F ( ) F( ) x F( )( xu x F( )( x x mi F( )( max F( )( xu x ( x ); ; F( ) F( ) ); F A F A A Thus 34 Propositio For impreise soft sets (F (G ad (H over (U the followig results are valid A U A U A A 2 3 F A A F A 4 F A U A U A 5 F A A A 6 F A U A F A 7 F A B F A if ad oly if B A 8 F A U B U B if ad oly if A B 9 F A B A B F A U B F A B ( i) ( F ( G ( G ( F ( ii)( F ( G ( G ( F 2( i) ( F ( G ( H ( F ( G ( H ( ii)( F ( G ( H ( F ( G ( H 3( i) ( F ( G ( H ( F ( G ( F ( H ( ii)( F ( G ( H ( F ( G ( F ( H 4( i) ( F ( F ( F ( ii)( F ( F ( F ( F 5 ( F Proof: The result immediately follows from defiitio

5 A Itrodutio to the Theory of Impreise Soft Sets 79 De Morga Laws are ot valid for impreise soft sets with differet sets of parameters Istead we have the followig ilusios De Morga Laws are valid for impreise soft sets with the same set of parameter 35 Propositio For impreise soft sets (F ad (G over (U oe has the followig: F A G B F A G B 2 F A G B F A G B Proof Let F A G B HC ad C H () where C A B x F ( )( : if A-B x G( )( : if B-A xmax F ( )( G( )( : Thus F A G B HC C A B ad C H ( ) H ( ) if A B H C x F ( )( : if A-B x G( )( : if B-A x max F ( )( G( )( : where if A B F A G B say where J A B ad J Agai F A G B = J I () F ( ) G ( ) x F( )( : xu x G( )( x max ( x ) ( x ) x U : xu : F( ) G( ) We see that J C ad J I( ) H ( ) Thus F A G B F A G B It follows immediately that F A G B F A G B 2 Let F A G B HC ad C H () F( ) G( ) I where C A B x F( )( : xu x G( )( x mi xu : xu : F( ) G( ) F A G B A B C Thus HC C ad H ( ) F ( ) G( ) H C ) x mi F( )( G( )( : x U x mi F( )( x ) G( )( x : x U Agai F A G B = F A G B say where J A B ad J I () x F ( ) : if A-B x G( ) : if B-A x mi F ( ) G( ) : where = J I if A B We see that C J ad C H ( ) I( ) It follows that F A G B F A G B 36 Propositio (De Morga Laws) For impreise soft sets (F ad (G over (U oe has the followig: F A G A F A G A 2 F A G A F A G A Proof Let F A G A H A H () F( ) G( ) where A x max F( ) G( ) : x U Thus F A G A H A A H ( ) H ( ) H A x max ( x ) ( x ) F( ) G( ) : x U F A G A where Agai F A G A = A say where A I () F ( ) G ( ) x max F( ) G( ) : xu Thus F A G A F A G A 2 Let F A G A H A where A I

6 8 A Itrodutio to the Theory of Impreise Soft Sets H () F( ) G( ) x mi F( ) G( ) : xu F A G A H A H A where Thus A H ( ) H ( ) x mi ( x ) ( x ) F( ) G( ) : x U = F A G A I say where A Agai F A G A = A I () F ( ) G ( ) x mi F( ) G( ) : xu Thus F A G A F A G A 37 Propositio For impreise soft sets (F ad (G over (U oe has the followig: F A G B F A G B 2 F A G B F A G B Proof where A ad B Let F A G B H A B H F G x F( ) ( x ) : x U x G( ) ( x ) : x U mi ( x ) ( x ) x F( ) G( ) : x U Thus F A G B H A B where A B H H F G H A B mi F( ) G( ) ) x : x U x mi F( ) G( ) ( x : xu Let F A G B where F A G B A ad B O F G O A B x F( ) ( x ) : x U x G( ) ( x ) : x U mi ( x ) ( x ) x F( ) G( ) : x U Thus F A G B F A G B where A ad B 2 Let F A G B H A B H F G x F( ) : xu x G( ) x max ( x ) ( x ) x U : xu : F( ) G( ) Thus F A G B H A B where A B H H F G max F( ) G( ) H A B ) x : x U x max F( ) G( ) ( x : xu F A G B Let F A G B where A ad B O F G O A B x F( ) : xu x G( ) x max ( x ) ( x ) x U : xu : F( ) G( ) Thus F A G B F A G B IV Matrix Represetatio of Total Impreise Soft Set Let U { 2 3 m} be the uiversal set ad E be the set of parameters give by E { e e} The the total impreise soft set F E a be expressed i matrix form as A = [ a ] m or simply by [ a ] i = 23 m ; j = 23 ad a p ) r ( ) ; where p ) ad r j ( i ) j ( i j i j ( i represet the presee level idiator futio ad the referee futio respetively of i i the impreise set F e ) so that ) p ( ) r ( ) ( j ( i j i j i gives the presee level of i We shall idetify a total impreise soft set with its total impreise soft matrix ad use these two oepts iterhageable The set of all m total impreise soft matries over U will be deoted bytism m 4 Presee Level Matrix of a Total Impreise Soft Matrix We defie the presee level matrix orrespodig to A [ ] where the matrix A as A m

7 A Itrodutio to the Theory of Impreise Soft Sets 8 A p j ( i ) r j ( i ) i 23 m ad j 23 where p ) ad r ) represet j ( i j ( i the presee level idiator futio ad the referee futio respetively of i i the impreise set F ( e j ) 42 Example Let U { 2 3 4} be the uiversal set ad E be the set of parameters give by E e e 2 e 3 We osider a total impreise soft set F E F( e ) F( ) F( ) We would represet this total impreise soft set i matrix form as 7 A = [ a ] The presee level matrix orrespodig to the matrix A is give by A V Similarity Betwee Two Total Impreise Soft Sets Ad Its Appliatio I Medial Diagosis 5 Similarity Betwee Two Total Impreise Soft Sets Let (F ad (G be two total impreise soft sets over (U where U { 2 3 m} ad E { e e} Let F E G E P E F E G E Q E ad We assume that A a ] ad B b ] are the total [ [ impreise soft matries orrespodig to (P ad (Q a p ) r ( ) ad respetively where / j / j b p ( ) r ( ) i i j ( i j i The Presee Level Matries orrespodig to the matries A ad B are respetively ( ] [ ( where p ( ) r ( ) ad ( ] ( j i j i / / ( j i j i where p ( ) r ( ) [ ( Let S(( F ( G deote the similarity betwee the total impreise soft sets (F ad (G Let S j (( F ( G represet the similarity betwee the e j approximatios F(e j ) ad G(e j ) The we defie m ( ( i S j (( F ( G m ( ( i S(( F ( G max 52 Propositio j ad (( F ( G j 23 S j e j Let (F (G ad (H be three total impreise soft sets over (U The the followig results are valid (i) S (( F ( F ) (ii) S(( F ( G S(( G ( F (iii) ( F ( G S (( F ( G (iv) ( F ( G S(( F ( G (v) ( F ( H ( G S(( F ( G S(( H ( G Proof: The proof is straight forward 53 Sigifiatly Similar Total Impreise Soft Sets Let F Ead G E be two total impreise soft sets over the same uiverse (U We all the two total impreise soft sets to be sigifiatly similar if S(( F ( G 2 54 Appliatio of Similarity i Medial Diagosis I this setio we will try to fid out whether a ill perso havig ertai symptoms is sufferig from peumoia or ot We first ostrut a model impreise soft set for peumoia ad the impreise soft set of symptoms for the ill persos ext we fid the similarity of these two sets If they are sigifiatly similar the we olude that the perso is possibly sufferig from peumoia Let our uiversal set otai oly two elemets yes (y) ad o () ie U y Here the set of parameters E is the set of ertai visible symptoms Let E = {e e 2 e 3 e 4 e 5 e 6 e 7 e 8 } where e = body temperature e 2 = ough with hest ogestio e 3 =

8 82 A Itrodutio to the Theory of Impreise Soft Sets ough with o hest ogestio e 4 = body ahe e 5 = headahe e 6 = loose motio e 7 = breathig trouble Our model fuzzy soft set for peumoia F E is give i Table ad this a be prepared with the help of a physiia I a similar fashio we ostrut the impreise soft sets orrespodig to the two ill persos uder osideratio as give i Table 2 3 respetively Table : Model Impreise Soft Set for Peumoia ( F e y () () () () () () () Table 2: () () () () Impreise Soft Set for first ill perso () (3) (2) () () (3) () (5) () ( F e y (2) () (6) (3) (5) () () Table 3: Impreise Soft Set for seod ill perso () () (2) () (3) (7) (6) ( F2 e y (7) (9) (2) (6) (5) () (8) Case I: Similarity betwee F E ad F E F E F E P E () (3) () () () () () ( P e y () () (6) () (5) () () F E F E Q E () () (2) () (3) (5) (6) ( Q e y (2) () () (3) () () () () The Impreise soft matries orrespodig to these two Impreise soft sets P E ad Q Eare give by A B The orrespodig presee level matries are give by A B 2 Calulatios give F E F E S F E F E F E F E S F E F E S 8 S F E F E S F E F E S 5 2 F E F E S 7 6 Thus we have S F E F E Case II: Similarity betwee F E ad F E F E F E P E 2 2 () () () 2 ( P2 e y () () (2) () (5) () () F E F E Q E 2 2 () () (2) () () () (3) () (7) () ( Q2 e y (7) (9) () (6) () () (8) () The impreise soft matries orrespodig to these two impreise soft sets P 2 E ad Q 2 E are give by A 2 B The orrespodig presee level matries are give by A 2 2 5

9 A Itrodutio to the Theory of Impreise Soft Sets 83 We have B F E F2 E S F E F2 E F E F E S F E F E S 64 S F E F E S F E F E S F E F E S Thus we have S F E F E I view of our otio of similarity of two impreise soft sets we a olude that the seod ill perso is possibly sufferig from peumoia VI Colusio Baruah has already established that two idepedet laws of radomess are eessary ad suffiiet to defie a law of fuzziess Seodly the existig defiitio of omplemet of a fuzzy set is logially iorret He has itrodued the theory of impreise sets i whih these two fuzzy set theoreti bluders are abset Keepig i view we have iitiated the otio of impreise soft sets whih is a hybrid model of Molodtsov s soft sets ad Baruah s impreise sets Impreise soft sets are differet from Maji s fuzzy soft sets i the sese that i our work we have replaed Zadeh s fuzzy sets with Baruah s impreise sets It is hoped that our work would help ehaig the study i fuzziess [4] P K Maji ad AR Roy Soft Set Theory Computers ad Mathematis with Appliatios 45 (23) [5] TJ eog ad D K Sut A ew Approah To The Theory of Soft Sets Iteratioal Joural of Computer Appliatios Vol 32 o 2Otober 2 pp -6 [6] PKMaji RBiswas ad ARRoy Fuzzy soft Sets Joural of Fuzzy Mathematis Vol 9 o3 pp [7] B Ahmad ad A Kharal O Fuzzy Soft Sets Advaes i Fuzzy Systems Volume 29 [8] TJ eog D K Sut O Fuzzy Soft Complemet Ad Related Properties Iteratioal Joural of Eergy Iformatio ad Commuiatios Volume 3 Issue February-22 pp [9] TJ eog D K Sut Theoty of Fuzzy Soft Sets from a ew perspetive Iteratioal Joural of Latest Treds i omputig Vol2 o 3 September 2 pp [] H K Baruah Towards Formig A Field of Fuzzy Sets Iteratioal Joural of Eergy Iformatio ad Commuiatios Vol 2 Issue pp 6-2 February 2 [] H K Baruah Theory of Fuzzy Sets: The Case of Subormality Iteratioal Joural of Eergy Iformatio ad Commuiatios Vol 2 Issue 3 pp -8 August 2 [2] H K Baruah I searh of the root of fuzziess: The measure theoreti meaig of partial presee Aals of Fuzzy Mathematis ad Iformatis 2 Akowledgmet The authors would like to thak the aoymous reviewers for their areful readig of this paper ad for their helpful ommets whih have improved this work Referees [] H K Baruah The Theory of Fuzzy Sets: Beliefs ad Realities Iteratioal Joural of Eergy Iformatio ad Commuiatios Vol 2 Issue 2 pp -22 May 2 [2] H K Baruah A Itrodutio to The Theory of Impreise Sets: The Mathematis of Partial Presee Joural of Mathematial ad Computatioal Siees Vol 2 o 2 pp [3] DA Molodtsov Soft Set theory First Result Computer ad Mathematis with Appliatios 37 (999) 9-3 Authors Profiles Tridiv Jyoti eog (98-) reeived his MS degree i Mathematis from Dibrugarh Uiversity Idia i 24 He is a researh sholar i the departmet of Mathematis Faulty of Siee CMJ Uiversity Shillog Meghalaya Idia He was awarded BIADI MEDHI MEMORIAL AWARD for beig the best graduate i BS examiatio 2 uder Dibrugarh Uiversity Dusmata Kumar Sut (975-) reeived his MS degree i Mathematis from Dibrugarh Uiversity Dibrugarh Idia i 22 ad his Phd i Mathematis from Dibrugarh Uiversity Idia i 27 He is a assistat professor i the departmet of Mathematis Saikia College Titabor Idia His researh iterests are i Fuzzy Mathematis Fluid dyamis ad Graph Theory