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Volume 1 No. 6, September 2011 ISSN 2222-9833 ARPN Journal of Systems and Software ©2010-11 AJSS Journal. All rights reserved http://www.scientific-journals.org Notes on the soft operations Fu Li Department of Mathematics, Qinghai Nationalities College Xining, Qinghai 810000, P.R. China fl0971@163.com ABSTRACT Soft set theory can be seen as a new mathematical approach to vagueness, authors P.K.Maji et al. introduced some operations and gave their some properties on soft sets, and in inference [12], authors defined some new operations and discussed their properties on soft sets. In this paper, We first point a small error in inference [4] and correct it; furthermore, based on some results of soft operations, using the DeMolan's laws , we give the distributive laws of the restricted union and the restricted intersection and the distributive laws of the union and the extended intersection. Keywords: soft set, soft operation, extended intersection, distributive law. 1. INTRODUCTION That is, the soft set is a parameterized family of subsets of the set U . Every set F (e), e E , from this family In 1999, Molodsov [1] initiated a novel concept may be considered as the set of e -elements of the soft set of soft set theory, which is a completely new approach ( F , E ) , or considered as the set of $e$-approximate for modeling vagueness and uncertainty. Soft set theory elements of the soft set. According to this manner, we can has a rich potential for applications in several directions, few of which had been shown by Molodsov in [1]. After view a soft set ( F , E ) as consisting of collection of Molodsov's work, some different applications of soft sets approximations: ( F , E ) {F (e) e E} . were studied in [2,3]. Furthermore, Maji, Biswas and Roy worked on soft set theory in [4]. Also Maji et al. [5] presented the definition of fuzzy soft set and Roy et al. Definition Let E {e1 , e2 , en } be a set of presented some applications of this notion to decision parameters .The NOT set of E denoted E is defined making problems in [6]. Recently, the many authors discuss the soft set, by E {e1 , e2 , , en } research on the soft set theory is progressing rapidly, for example, the concepts of soft semi-ring, soft group, soft Definition The complement of soft set ( F , A) BCK/BCI-algebra, soft BL-algebra, and fuzzy soft group c etc. have been proposed and investigated (see [7-11] is denoted by ( F , A) and is defined by respectively). M.Irfan Ali et.in [12] discussed new ( F , A)c ( F c , A) ,where F c : A P(U ) is a operations in soft set theory which the authors gave the mapping given by F ( ) U F ( ), A . c definition of the restricted intersection , the restricted difference and extended intersection of soft sets, and gave the DeMorgan's law in soft set theory. Proposition 1.4[4] Let E be a set of parameters and In this paper, we first point the some small errors in inference [4] and correct it, and using the DeMorgan's A, B E . Then law ,give their distributive laws. The rest of the paper is organized as following, in section 2, we give some (i) (A) A ; (ii) definitions and some results of soft sets which we will use ( A B) A B ; (iii) in this paper, and point an small error in the inference [4]. In section 3, we discuss the distributive laws of soft ( A B) A B . operations, conclusions are given in section 4 . The results (ii) and (iii) are not right. 2. PREMERILARY For example: Let X {a, b, c, d}, A {a, b, c}, B {b, d} , By Definition U be an initial universe set and E Let definition 1.2, A {d}, be a set of parameters . Let P(U ) denotes the power set B {a, c}, A B {a, b, c, d}, A B {b}, but, of U and A E . Then a pair ( F , A) is called a soft A B A B . set over U , where F P(U ) is a mapping. 205 Volume 1 No. 6, September 2011 ISSN 2222-9833 ARPN Journal of Systems and Software ©2010-11 AJSS Journal. All rights reserved http://www.scientific-journals.org We can revise it as: OPERATIONS Proposition 1.5 Let E be a set of parameters and Theorem 2.1 [12] Let ( F , A) , (G, B) be two soft sets A, B E . Then over the common universe U ，such that A B ， then (i) ( A B) A B ; (ii) ( A B) A B (i) (( F , A) R ((G, B))r ( F , A) (G, B)) r Proof: x ( A B) iff x A B iff x A (ii) (( F , A) ((G, B))r ( F , A) R ((G, B)r and x B iff x A and x B iff For the operation R and ω， by theorem 2.1， we x (A B) . The other is similar. know that they are duality operator with respect to the relative complement r. So， there are distributive law [12] Definition 1.6 The union of two soft sets ( F , A) about them: and (G, B) over a common universe U is the soft set Proposition 2.2 Let ( F , A) , (G, B) , ( H , C ) be soft ( H , C ) ， where C A B , e C ， denoted as sets in the common universe U ， then ( F , A) (G, B) ( H , C) ( H , A B) , where (i) ( F , A) R ((G, B) ( H , C )) = F (e), if e A B (( F , A) R (G, B)) (( F , A) R ( H , C )) H (e) G (e), if e B A . (ii) (( F , A) ((G, B) R ( H , C )) = F (e) G (e), if e A B (( F , A) (G, B)) R (( F , A) [12] ( H , C )) . Definition 1.7 The restricted intersection of two soft sets ( F , A) and (G, B) over a common universe U and called the distributive law of R r is the soft set is denoted as ( F , A) (G, B) and is and with respect to . defined as ( F , A) (G, B) ( H , C ) ,where C Proof: A B , and c C , H (c) F (c) G(c) Let (G, B) R (H , C) ( L, D) ( L, B C ) ， [4] d D Definition 1.8 The extended intersection of two L(d ) B C (d ) G(d ) H (d ) ,and let soft sets ( F , A) and (G, B) over a common universe ( F , A) R ( L, D) (M , K ) (M , A D) U is the soft set ( H , C ) , where C A B , and e C ，denoted as ( F , A) ó (G, B) ( H , C ) ( H , A B) , where then k M (k ) if and only if F (e), if e A B k ( F (k ) G(k )) ( F (k ) H (k )) if and only if H (e) G (e), if e B A k F (k ) G(k ) , and k F (k ) H (k ) if and F (e) G (e), if e A B only if k ( F , A) R (G, B) and k ( F , A) R ( H , C ) if and only if [12] Definition 1.9 Let ( F , A) and (G, B) be two soft k (( F , A) R (G, B)) (( F , A) R ( H , C )) . sets over common universe U such that a A B . The restricted union of ( F , A) and Similarly, we can proof the other equation. (G, B) is denoted as ( F , A) R (G, B) ， and is Proposition 2.3 Let A, B, C be sets, then ( F , A) R (G, B) ( H , C ) and， where defined as (i) ( A B) C ( A C ) ( B C ) (ii) C A B ， and' c C ， H (c) F (c) G(c) . A ( B C) ( A B) ( A C) 3. THE NOTE OF THE SOFT 206 Volume 1 No. 6, September 2011 ISSN 2222-9833 ARPN Journal of Systems and Software ©2010-11 AJSS Journal. All rights reserved http://www.scientific-journals.org Proof (i) x ( A B) C if and only if x A B , and x C if and only if x A or x B and x C if and only if ( x A and x C ) or ( x B and x C ) if and only if x A B or x B C if and only if x ( A C ) ( B C ) . Let (i) x A ( B C ) if and only if x A and x ( B C ) if and only if x A and x B and x C if and only if ( x A and x B ) and and ( x A and x C )if and only if x A B and x A C if and only if x ( A B) ( A C ) . and k A B , having Theorem 2.4[12] Let ( F , A),(G, B) be two soft sets F (k ), if k A B over the common universe U , then H1 (k ) G (k ), if k B A F (k ) G (k ), if k A B (i) (( F , A) (G, B))c ( F , A)c ó (G, B)c (ii) (( F , A) ó (G, B))c ( F , A)c (G, B)c And k B C , having By theorem 2.4, the operation and ó satisfied with if k B C G (k ), the De Morgan's laws, furthermore, we can prove they also have the distributive laws, that is : H 2 (k ) H (k ), if k C B G (k ) H (k ), if k B C Proposition 2.5 Let ( F , A),(G, B),( H , C ) be soft sets in the common universe U , then: k A B C , i. H1 (k ), if k ( A B) ( A C ) ii. H 3 (k ) H 2 (k ), if k ( A C ) ( A B) H (k ) H (k ), if k A ( B C ) 1 2 and called the distributive law with respect to Next, we need discuss and ó . Proof k A ( B C ), k ( B C ) A and k A ( B C ) respectively, by the definition of (G, B) ó ( H , C ) ( L, D) ( L, B C ), d D , and ó again, in any case, we all can prove and G (d ), if d B C holds. That is, we prove that the equation (i) holds. L(d ) H (d ), if d C B Similarly, we can prove the other equation holds , too. G (e) H (d ), if d B C 4. CONCLUSION Let ( F , A) ((G, B) ó ( H , C)) ( M , K ) ( M , A ( B C)) this paper ,we first point the some small errors in In inference [4] and correct it, gave the right relation of , m K .Using proposition 2.4 and the definitions of , , on soft sets, and, by means of the and ó , having DeMorgan's laws which introduced in [12] , we discuss 207 Volume 1 No. 6, September 2011 ISSN 2222-9833 ARPN Journal of Systems and Software ©2010-11 AJSS Journal. All rights reserved http://www.scientific-journals.org the distributive laws of R and r with respect to and [6] P.K.Maji,R.Biswas and A.R.Roy, Fuzzy soft sets, J. Fuzzy Mathematics, 9(3)(2001)589-602. the distributive laws of and ó . [7] A.R. Roy, P.K. Maji, A fuzzy soft set theoretic REFERENCES approach to decision making problems, Journal of Computational and Applied Mathematics 203 (2007) [1] D. Molodtsov, Soft set theory---first results, 412-418. Computers Math. Appl. 37(4/5),19-31(1999). [8] F.Feng,Y.B.Jun, X.Z.Zhao, Soft semi-rings, [2] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The Computers Math. Appl.56(2008)2621-2628. parameterization reduction of soft set and its applications, Computers Math. Appl. 49 (2005) 757- [9] H.Aktas,N.Cagman, Soft sets and soft groups, 763. Information Sciences,177(2007)2726-2735. [3] P.K. Maji, A.R. Roy, An application of Soft set in [10] Y.B.Jun , Soft BCK/BCI-algebras, Computers Math. decision making problem, Computers Math. Appl. 44 Appl.56 (2008)1408-1413 (2002) 1077-1083. [11] J.M.Zhan,Y.B.Jun, Soft BL-algebras, Computers [4] P.K.Maji, R.Biswas and A.R.Roy, Soft set theory, Math. Appl.(In press ) Computers Math. Appl.45(2003)555-562. [12] M.Irfan Ali, F.Feng, Xiaoyan Liu,Won Keun Min, [5] P.K. Maji, R. Biswas, A.R. Roy, Fuzzy soft sets, M.Shabir, On some new operations in soft sets Journal of Fuzzy Mathematics 9 (3) (2001) 589-602 theory, Computers and mathematics with Applications, 57(2009)1547-1553 208

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