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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
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