International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 STUDY ON SOME OPERTIONS OF FUZZY SOFT SETS Manoj orah 1 Tridiv Jyoti Neog 2 Dusmanta Kumar Sut 3 1 Deptt. of Mathematis Jorhat Institute of Siene & Tehnology Jorhat ssam India 2 Deptt. of Mathematis D.K. High Shool Jorhat 3 Deptt. of Mathematis N.N.Saikia College Titabor ssam India STRCT The purpose of this paper is to study some operations results available in the literature of fuzzy soft sets. Instead of taking the notion of omplement of a fuzzy soft set put forward by Maji throughout our work we have taken the notion of omplement of a fuzzy soft set put forward by Neog Sut. Keywords Fuzzy Set Soft Set Fuzzy Soft Set. 1. INTRODUCTION In many ompliated problems arising in the fields of engineering soial siene eonomis medial siene et involving unertainties lassial methods are found to be inadequate in reent times. Molodtsov [2] pointed out that the important existing theories viz. Probability Theory Fuzzy Set Theory Intuitionisti Fuzzy Set Theory Rough Set Theory et. whih an be onsidered as mathematial tools for dealing with unertainties have their own diffiulties. He further pointed out that the reason for these diffiulties is possibly the inadequay of the parameterization tool of the theory. In 1999 he initiated the novel onept of Soft Set as a new mathematial tool for dealing with unertainties. Soft Set Theory initiated by Molodtsov [2] is free of the diffiulties present in these theories. In 2011 Neog Sut [9] put forward a new notion of omplement of a soft set aordingly some important results have been studied in their work. In reent times researhes have ontributed a lot towards fuzzifiation of Soft Set Theory. Maji et al. [6] introdued the onept of Fuzzy Soft Set some properties regarding fuzzy soft union intersetion omplement of a fuzzy soft set De Morgan Law et. These results were further revised improved by hmad Kharal [1]. Reently Neog Sut [8] have studied the notions of fuzzy soft union fuzzy soft intersetion omplement of a fuzzy soft set several other properties of fuzzy soft sets along with examples proofs of ertain results. In this paper we have studied some operations results available in the literature of fuzzy soft sets. Instead of taking the notion of omplement of a fuzzy soft set put forward by Maji et al. [6] throughout our work we have taken the notion of omplement of a fuzzy soft set put forward by Neog Sut [7]. 2. PRELIMINRIES In this setion we first reall the basi definitions related to soft sets fuzzy soft sets whih would be used in the sequel. 2.1. Soft Set [2] pair (F E) is alled a soft set (over U) if only if F is a mapping of E into the set of all subsets of the set U. In other words the soft set is a parameterized family of subsets of the set U. Every set ) E from this family may be onsidered as the set of - elements of the soft set (F E) or as the set of - approximate elements of the soft set. 2.2. Soft Null Set [5] soft set (F ) over U is said to be null soft set denoted by if ) = (Null set) 2.3. Soft bsolute Set [5] soft set (F ) over U is said to be absolute soft set denoted by if ) = U. 2.4. Soft Subset [5] For two soft sets (F ) ( ) over the universe U we say that (F ) is a soft subset of ( ) if (i) (ii) F G are idential approximations is written as (F ) ( ). Pei Miao [4] modified this definition of soft subset in the following way 2.5. Soft Subset Redefined [4] For two soft sets (F ) ( ) over the universe U we say that (F ) is a soft subset of ( ) if (i) (ii) F G is written as (F ) ( ). (F ) is said to be soft superset of ( ) if ( ) is a soft subset of (F ) we write (F ) ( ). 2.6. Union of Soft Sets [5] Union of two soft sets (F ) ( ) over a ommon universe U is the soft set (H where C C ) if H ( ) G( ) if ) G( ) if F H is written as www.ijmer.om 219 P a g e
International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 2.7. Intersetion of Soft Sets [5] soft set (F ) over U is said to be absolute fuzzy soft set Intersetion of two soft sets (F ) ( ) over a denoted by if ) is the absolute fuzzy set 1 ommon universe U is the soft set (H where of U where 1( 1x U C C H( ) ) or G( ) (as both are same set) is written as F 2.16. Fuzzy Soft Subset [6] H C. For two fuzzy soft sets (F ) ( ) in a fuzzy soft lass (U E) we say that (F ) is a fuzzy soft subset of ( ) if Pei Miao [4] pointed out that generally ) or (i) G() may not be idential. Moreover in order to avoid the (ii) For all F G is written as degenerate ase hmad Kharal [1] proposed that must be non-empty thus revised the above (F ) ( ). definition as follows. 2.17. Union of Fuzzy Soft Sets [6] Union of two fuzzy soft sets (F ) ( ) in a soft lass 2.8. Intersetion of Soft Sets Redefined [1] (U E) is a fuzzy soft set (H where C Let (F ) ( ) be two soft sets over a ommon C universe U with. Intersetion of two soft ) if sets (F ) ( ) is a soft set (H where H( ) C C H( ) ) G( ). G( ) if F H ) G( ) if We write 2.9. ND Operation of Soft Sets [5] If (F ) ( ) be two soft sets then (F ) ND ( F is defined ) is a soft set denoted by by F H where H F G where is the operation intersetion of two sets. 2.10. OR Operation of Soft Sets [5] If (F ) ( ) be two soft sets then (F ) OR ( ) F is defined by is a soft set denoted by F K where K F G where is the operation union of two sets. 2.11. Complement of a Soft Set [9] The omplement of a soft set (F ) is denoted by (F ) is defined by F = (F ) where F : P (U ) is a mapping given by F F for all.. nd is written as F H C 2.18. Intersetion of Fuzzy Soft Sets [6] Intersetion of two fuzzy soft sets (F ) ( ) in a soft lass (U E) is a fuzzy soft set (H where C C H( ) ) or G( ) (as both are same fuzzy F H set) is written as hmad Kharal [1] pointed out that generally ) or G() may not be idential. Moreover in order to avoid the degenerate ase he proposed that must be non-empty thus revised the above definition as follows - 2.19. Intersetion of Fuzzy Soft Sets Redefined [1] Let (F ) ( ) be two fuzzy soft sets in a soft lass (U E) with. Intersetion of two fuzzy soft sets (F ) ( ) in a soft lass (U E) is a fuzzy soft set (H where C C H( ) ) G( ) F H. We write 2.12. Fuzzy Soft Set [6] pair (F ) is alled a fuzzy soft set over U where F : P( U) is a mapping from into P ( U ). 2.13. Fuzzy Soft Class [1] Let U be a universe E a set of attributes. the pair (U E) denotes the olletion of all fuzzy soft sets on U with attributes from E is alled a fuzzy soft lass. 2.14. Fuzzy Soft Null Set [6] soft set (F ) over U is said to be null fuzzy soft set denoted by if ) is the null fuzzy set 0 of U where 0 ( 0x U. 2.15. Fuzzy Soft bsolute Set [6] 2.20. Complement of a Fuzzy Soft Set [7] The omplement of a fuzzy soft set (F ) is denoted by (F ) is defined by (F ) = (F ) where F : P ( U ) is a mapping given by F ( ) F ( ). 2.21. ND Operation of Fuzzy Soft Sets [6] If (F ) ( ) be two fuzzy soft sets then (F ) F ND ( ) is a fuzzy soft set denoted by is defined by F H where H F G where is the operation intersetion of two fuzzy sets. 2.22. OR Operation of Soft Sets [6] www.ijmer.om 220 P a g e
International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 If (F ) ( ) be two fuzzy soft sets then (F ) OR : xu ( ) is a fuzzy soft set denoted by F is 1 : xu defined by F K where K F G 11: xu where is the operation union of two fuzzy sets. 0: xu 3. STUDY ON THE OPERTIONS IN FUZZY SOFT SETS In this setion we shall endeavour to study the basi operations results available in the literature of fuzzy soft sets. hmad Kharal [1] has pointed out that the null (resp. absolute) fuzzy soft set as defined by Maji et al. [6] is not unique in a fuzzy soft lass (U E) rather it depends upon E. s suh whenever we would refer to a fuzzy soft null set or fuzzy soft absolute set we would refer to the set E of parameters under onsideration. In ase of soft sets this has already been pointed out by Ge Yang in [10] aordingly they have studied some basi results regarding soft sets. Here we put forward some results of fuzzy soft sets in our way. We are using the notation to represent the fuzzy soft null set with respet to the set of parameters the notation U to represent the fuzzy soft absolute set with respet to the set of parameters. U 3. F F We have F ( 0 F x ) : x U : xu max ( 0 : F ( : x U x F xu F ( ) F F 3.1. Proposition. U 1 Let F F () : xu F 0 : xu F ( ) where F F ( ) : xu 1 : xu 1 0 : xu 1 : xu U U. U 2 Let U F F () : xu U F 1 : xu F ( ) where F F ( ) www.ijmer.om 4. F U U We have F ( U 1 F U 1 U x ) : x U : xu max ( 1 F U U 5. F We have x ) : x U : xu F ( 0 F 0 x ) : x U : xu min ( 0 F 6. F U F We have x ) : xu : xu F ( U 1 x ) : x U : xu 221 P a g e
International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 F U min 1 : xu : xuif : x U G( : xuif F max G( : xu if F U F 7. F F if only if We have for F F () : xu lso let G () 0: xu Let F F ( H where C C H () : xuif G( : xuif max G( : xu ( : xuif ) 0: xuif max 0 : xu ( : xuif if ) 0: xuif : xu if Let H () F ( : x U ( : x ) F ( ) F F Conversely let 8 if if if. F U U if only if : xu if 1 : xuif if Let max 1 : xu x ( : xuif ) 1 : xu 1 : xu H () if if 1 : xu 1 : xu if if G() F U U Conversely let 9. F We have for F F () : xu lso let G () 0: xu Let F F ( H where C C H () min F G( : x ( U min 0 : xu 0: xu F 10. F U F We have for F F () : xu lso let U G () 1: xu Let F U F ( H where C C H () We have for F F () : xu lso let U G () 1: xu Let F U F ( H where C C H () min F G( : x ( U min 1 : xu www.ijmer.om 222 P a g e
International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 : x U F = F G = I J say F U F Where J J F ( )if - It is well known that De Morgan Laws inter-relate union I ( ) G ( )if - intersetion via omplements. Maji et al [6] gave the following proposition- F ( ) G ( )if 3.2. Proposition F F 1. 2. F F hmad Kharal [1] proved by ounter examples that these propositions are not valid. However the following inlusions are due to hmad Kharal [1]. They proved these results with the definition of omplement initiated by Maji et al. [6]. elow we are giving the proof in our way. 3.3. Proposition For fuzzy soft sets (F ) ( ) over the same universe U we have the following - F F 1. 2. F F where C 1. Let F H C C H () )if - G( ) if - ) G( )if x if x G( if x max ) G( ) - - F H C H C if where C C H ( ) H ( ) gain ) if ε G( ) if ) G( ) - - if x 1 if - x 1 G( if - 1 max ) G( ) if x 1 if - x 1 G( if - x min1 1 G( if www.ijmer.om x ( if - F ( ) x ( if - G ( ) x max ( ( F ( ) G ( ) x 1 if - x 1 G( if - x max1 ) 1 G( ) We see that C = J C H ( ) I( ) F F 2. Let F H C Where C C H () ) G( ) min G( F H C H C Where C C H ( ) F ( ) G( ) 1 min G( if if max 1 1 G( gain F = F G Where J J I ( ) F ( ) G ( ) min ( ( F ( ) G ( ) = J min 1 1 G( We see that C = J C I( ) H ( ) F F I say 3.4. Proposition (De Morgan Inlusions) For fuzzy soft sets (F ) ( ) over the same universe U we have the following - F F 1. 2. F F Proof where C 1. Let F H C C H () )if - G( ) if - ) G( )if 223 P a g e
International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 x if x G( if x max ) G( ) - - F H C H C if where C C H ( ) H ( ) ) if ε G( ) if ) G( ) - - if x 1 if - x 1 G( if - 1 max ) G( ) if x 1 if - x 1 G( if - x min1 1 G( if gain F = F G = I J say Where J J I ( ) F ( ) G ( ) min ( ( F ( ) G ( ) min 1 1 G( We see that J C J I( ) H ( ) F F Where C 2. Let F H C C H () ) G( ) min G( F H C H C Where C C H ( ) F ( ) G( ) 1 min G( max 1 1 G( gain F = F G = I J say Where J J I ( ) F ( )if - G ( )if - F ( ) G ( )if x ( if - F ( ) x ( if - G ( ) x max ( ( F ( ) G ( ) x 1 if - x 1 G( if - x max1 ) 1 G( ) We see that C J C H ( ) I( ) F F if if De Morgan Laws are valid for fuzzy soft sets with the same set of parameter. for fuzzy soft sets we have the following De Morgan Laws: 3.5. Proposition (De Morgan Laws) For fuzzy soft sets (F ) ( ) over the same universe U we have the following - F F 1. 2. F F where 1. Let F H H () ) G( ) max ) G( ) F H H where H ( ) H ( ) F ( ) G( ) 1 max ) G( ) min 1 1 G( F F G = gain Where I ( ) F ( ) G ( ) min ( ( F ( ) G ( ) = min 1 1 G( F F 2. Let F H H () ) G( ) where min ) G( ) F H H where H ( ) H ( ) F ( ) G( ) x 1 min ) G( ) I say www.ijmer.om 224 P a g e
International Journal of Modern Engineering Researh (IJMER) www.ijmer.om Vol.2 Issue.2 Mar-pr 2012 pp-219-225 ISSN: 2249-6645 x max1 1 G( x max F ( G ( gain F = F G = I say Where F H I ( ) F ( ) G ( ) H where max ( ( F ( ) G ( ) max 1 1 G( F F Maji et al [6] proved the following De Morgan Types of results for fuzzy soft sets (F ) ( ) over the same universe U. We an verify that these De Morgan types of results are valid in our way also. 3.6. Proposition For fuzzy soft sets (F ) ( ) over the same universe U we have the following - 1. F F 2. F F H H 1 max F ( G ( min1 F ( 1 G ( Let F F G O Where O F G where is the operation union of two fuzzy sets. min ( ( F ( ) G ( ) min 1 ) ( 1 G( It follows that F F 4. CONCLUSION We have made an investigation on existing basi notions results on fuzzy soft sets. Some new results have been stated in our work. Future work in this regard would be required to study whether the notions put forward in this paper yield a fruitful result F H 1. Let Where H F G where is the operation intersetion of two fuzzy sets. F G H F H H min F ( G ( H where H 1 min F ( G ( max1 F ( 1 G ( Let F F G O Where O F G where is the operation union of two fuzzy sets. max ( ( F ( ) G ( ) max 1 ) ( 1 G( It follows that F F 2. Let F H Where H F G where is the operation union of two sets. F G H 5. REFERENCES [1]. hmad. Kharal On Fuzzy Soft Sets dvanes in Fuzzy Systems Volume 2009. [2] D.. Molodtsov Soft Set Theory - First Result Computers Mathematis with ppliations Vol. 37 pp. 19-31 1999 [3] D. Chen E.C.C. Tsang D.S. Yeung X.Wang The parameterization redution of soft sets its appliations Computers & Mathematis with ppliations vol. 49 no.5-6 pp. 757 763 2005. [4] D. Pei D. Miao From soft sets to information systems in Proeedings of the IEEE International Conferene on Granular Computing vol. 2 pp. 617 621 2005. [5] P. K. Maji.R. Roy Soft Set Theory Computers Mathematis with ppliations 45 (2003) 555 562 [6] P. K. Maji R. iswas.r. Roy Fuzzy Soft Sets Journal of Fuzzy Mathematis Vol 9 no.3pp.-589-6022001 [7] T. J. Neog D. K. Sut On Fuzzy Soft Complement Related Properties epted for publiation in International Journal of Energy Information ommuniations (IJEI. [8] T. J. Neog D. K. Sut On Union Intersetion of Fuzzy Soft Sets Int.J. Comp. Teh. ppl. Vol 2 (5) 1160-1176 [9] T. J. Neog D. K. Sut New pproah To The Theory of Soft Sets International Journal of Computer ppliations Vol 32 No 2Otober 2011 pp 1-6 [10] Xun Ge Songlin Yang Investigations on some operations of soft sets World ademy of Siene Engineering Tehnology 75 2011 pp. 1113-1116 www.ijmer.om 225 P a g e