Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 105752 7 pageshttpdxdoiorg1011552013105752

Research ArticleOperations on Soft Sets Revisited

Ping Zhu12 and Qiaoyan Wen2

1 School of Science Beijing University of Posts and Telecommunications Beijing 100876 China2 State Key Laboratory of Networking and Switching Beijing University of Posts and Telecommunications Beijing 100876 China

Correspondence should be addressed to Ping Zhu pzhubuptgmailcom

Received 14 November 2012 Accepted 7 January 2013

Academic Editor Han H Choi

Copyright copy 2013 P Zhu and Q Wen This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The concept of soft sets introduced byMolodtsov is a generalmathematical tool for dealingwith uncertainty Just as the conventionalset-theoretic operations of intersection union complement and difference some corresponding operations on soft sets have beenproposed Unfortunately such operations cannot keep all classical set-theoretic laws true for soft sets In this paper we redefinethe intersection complement and difference of soft sets and investigate the algebraic properties of these operations along with aknown union operation We find that the new operation system on soft sets inherits all basic properties of operations on classicalsets which justifies our definitions

1 Introduction

As a necessary supplement to some existing mathematicaltools for handling uncertainty Molodtsov [1] initiated theconcept of soft sets via a set-valued mapping A distinguish-ing feature of soft sets which is different from probabilitytheory fuzzy sets and interval mathematics is that precisequantity such as probability and membership grade is notessential This feature facilitates some applications becausein most realistic settings the underlying probabilities andmembership grades are not knownwith sufficient precision tojustify the use of numerical valuations Since its introductionthe concept of soft sets has gained considerable attention(see eg [2ndash31]) including some successful applications ininformation processing [32ndash36] decision [37ndash41] demandanalysis [42] clustering [43] and forecasting [44]

In [22] Maji et al made a theoretical study of the softset theory in more detail Especially they introduced theconcepts of subset intersection union and complement ofsoft sets and discussed their properties These operationsmake it possible to construct new soft sets from given softsets Unfortunately several basic properties presented in [22]are not true in general these have been pointed out andimproved by Yang [45] Ali et al [46] and Sezgin and Atagun[47] In particular to keep some classical set-theoretic lawstrue for soft sets Ali et al defined some restricted operations

on soft sets such as the restricted intersection the restrictedunion and the restricted difference and improved the notionof complement of a soft set Based upon these newly definedoperations they proved that certain De Morganrsquos laws holdfor soft sets It is worth noting that the concept of complement[22 46] (it should be stressed that there are two types ofcomplements defined in [46] one is defined with the NOTset of parameters and the other is defined without the NOTset of parameters) which is fundamental to DeMorganrsquos lawsis based on the so-called NOT set of a parameter set It meansthat the logic conjunction NOT is a prerequisite for definingthe complement of a soft set this is considerably beyond thedefinition of soft sets Moreover the union of a soft set andits complement is not exactly the whole universal soft set ingeneral which is considered less desirable

The purpose of this paper is to develop the theory of softsets by introducing new operations on soft sets that inheritall basic properties of classical set operations To this endwe redefine the intersection complement and difference ofsoft sets and then examine the algebraic properties of theseoperations along with a known union operation It turnsout that all basic properties of operations on classical setsincluding identity laws domination laws idempotent lawscommutative laws associative laws distributive laws and DeMorganrsquos laws hold for soft sets with respect to the newlydefined operations

2 Journal of Applied Mathematics

The remainder of this paper is structured as follows InSection 2 we briefly recall the notion of soft sets Section 3is devoted to the definitions of new operations on soft setsAn example is also provided to illustrate the newly definedoperations in this section We address the basic propertiesof the operations on soft sets in Section 4 and conclude thepaper in Section 5

2 Soft Sets

For subsequent need let us review the notion of soft sets Fora detailed introduction to the soft set theory the reader mayrefer to [1 22]

We begin with some notations For classical set theorythe symbols 0 119860

119888 119860 cup 119861 119860 cap 119861 119860 119861 denote respectivelythe empty set the complement of 119860 with respect to someuniversal set 119880 the union of sets 119860 and 119861 the intersectionof sets 119860 and 119861 and the difference of sets 119860 and 119861 whoseelements belong to 119860 but not to 119861 In what follows we writeP(119880) for the power set of a universal set 119880 and denoteP(119880) 0 byPlowast(119880)

Throughout this paper let 119880 be a universal set and 119864 bethe set of all possible parameters under consideration withrespect to 119880 Usually parameters are attributes characteris-tics or properties of objects in 119880 We now recall the notionof soft sets due to Molodtsov [1]

Definition 1 (seeMolodtsov [1]) Let119880 be a universe and119864 theset of parameters A soft set over 119880 is a pair (119865 119860) consistingof a subset 119860 of 119864 and a mapping 119865 119860 rarr Plowast(119880)

Note that the above definition is slightly different fromthe original one in [1] where 119865 has P(119880) as its codomainIn other words we remove the parameters having the emptyset as images under 119865 It seems rational since this meansthat if there exists a parameter 119890 isin 119860 which is not theattribute characteristic or property of any object in 119880 thenthis parameter has no interest with respect to the knowledgestored in the soft set As a result a soft set of 119880 in the sense ofDefinition 1 is a parameterized family of nonempty subsets of119880

Clearly any soft set (119865 119860) over 119880 gives a partial function1198651015840

119864 rarr Plowast(119880) defined by

1198651015840

(119890) = 119865 (119890) if 119890 isin 119860

undefined otherwise(1)

for all 119890 isin 119864 Conversely any partial function 119891 from 119864 toPlowast(119880) gives rise to a soft set (119865

119891 119860119891) where 119860

119891= 119890 isin 119864 |

119891(119890) is defined and 119865119891is the restriction of 119891 on 119860

119891

To illustrate the above definition Molodtsov consideredseveral examples in [1] one of which we present blow

Example 2 Suppose that 119880 is the set of houses underconsideration say 119880 = ℎ

1 ℎ2 ℎ

5 Let 119864 be the set of

some attributes of such houses say 119864 = 1198901 1198902 119890

8 where

1198901 1198902 119890

8stand for the attributes ldquoexpensiverdquo ldquobeautifulrdquo

ldquowoodenrdquo ldquocheaprdquo ldquoin the green surroundingsrdquo ldquomodernrdquoldquoin good repairrdquo and ldquoin bad repairrdquo respectively

In this case to define a soft set means to point outexpensive houses beautiful houses and so on For examplethe soft set (119865 119860) that describes the ldquoattractiveness of thehousesrdquo in the opinion of a buyer say Alice may be definedas

119860 = 1198902 1198903 1198904 1198905 1198907

119865 (1198902) = ℎ

2 ℎ3 ℎ5 119865 (119890

3) = ℎ

2 ℎ4

119865 (1198904) = ℎ

1 119865 (119890

5) = 119880 119865 (119890

7) = ℎ

3 ℎ5

(2)

3 Operations on Soft Sets

In this section we generalize the basic operations on classicalsets to soft sets The examination of more properties of theseoperations is deferred to the next section

Let us start with the notions of empty and universal softsets Recall that in [22] a soft set (119865 119860) is called a null soft setif 119865(119890) = 0 for all 119890 isin 119860 Because 0 does not belong to thecodomain of 119865 in our framework we redefine the concept ofempty soft set as follows

Definition 3 A soft set (119865 119860) over 119880 is said to be emptywhenever 119860 = 0 Symbolically we write (0 0) for the emptysoft set over 119880

Definition 4 A soft set (119865 119860) over 119880 is called a universal softset if 119860 = 119864 and 119865(119890) = 119880 for all 119890 isin 119860 Symbolically we write(119880 119864) for the universal soft set over 119880

Let us now define the subsets of a soft set

Definition 5 Let (119865 119860) and (119866 119861) be two soft sets over 119880 Wesay that (119865 119860) is a subset of (119866 119861) denoted (119865 119860) sube (119866 119861)

if either (119865 119860) = (0 0) or 119860 sube 119861 and 119865(119890) sube 119866(119890) for every119890 isin 119860 Two soft sets (119865 119860) and (119866 119861) are said to be equaldenoted (119865 119860) = (119866 119861) if (119865 119860) sube (119866 119861) and (119866 119861) sube (119865 119860)

By definition two soft sets (119865 119860) and (119866 119861) are equal ifand only if 119860 = 119861 and 119865(119890) = 119866(119890) for all 119890 isin 119860 In [22]a similar notion called soft subset was defined by requiringthat 119860 sube 119861 and 119865(119890) = 119866(119890) for every 119890 isin 119860 By Definition 5the empty soft set (0 0) is a subset of any soft set It also followsfrom Definition 5 that any soft set is a subset of the universalsoft set (119880 119864) Formally we have the following proposition

Proposition 6 For any soft set (119865 119860) over 119880

(0 0) sube (119865 119860) sube (119880 119864) (3)

We are now in the position to introduce some operationson soft sets

Definition 7 Let (119865 119860) and (119866 119861) be two soft sets over119880Theintersection of (119865 119860) and (119866 119861) denoted by (119865 119860) cap (119866 119861) isdefined as (119865 cap 119866 119862) where

119862 = 119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 0 forall119890 isin 119862

(119865 cap 119866) (119890) = 119865 (119890) cap 119866 (119890)

(4)

Journal of Applied Mathematics 3

In particular if 119860 cap 119861 = 0 or 119865(119890) cap 119866(119890) = 0 for every119890 isin 119860 cap 119861 then we see that (119865 119860) cap (119866 119861) = (0 0)

The following definition of union of soft sets is the sameas in [22]

Definition 8 (see [22] Definition 211) Let (119865 119860) and (119866 119861)

be two soft sets over 119880 The union of (119865 119860) and (119866 119861)denoted by (119865 119860) cup (119866 119861) is defined as (119865 cup 119866 119862) where

119862 = 119860 cup 119861 forall119890 isin 119862

(119865 cup 119866) (119890) =

119865 (119890) if 119890 isin 119860 119861

119866 (119890) if 119890 isin 119861 119860

119865 (119890) cup 119866 (119890) otherwise

(5)

We now define the notion of complement in soft set the-ory It is worth noting that this is rather different from thosein the existing literature [22 46] where the complement isusually based on the so-called NOT set of a parameter set andthe union of a soft set and its complement is not exactly thewhole universal soft set in general

Definition 9 Let (119865 119860) be a soft set over 119880 The complementof (119865 119860) with respect to the universal soft set (119880 119864) denotedby (119865 119860)

119888 is defined as (119865119888

119862) where

119862 = 119864 119890 isin 119860 | 119865 (119890) = 119880

= 119890 isin 119860 | 119865 (119890) = 119880119888

forall119890 isin 119862

119865119888

(119890) = 119880 119865 (119890) if 119890 isin 119860

119880 otherwise

(6)

In certain settings the difference of two soft sets (119865 119860)

and (119866 119861) is useful

Definition 10 Let (119865 119860) and (119866 119861) be two soft sets over 119880The difference of (119865 119860) and (119866 119861) denoted by (119865 119860) (119866 119861)is defined as (119865 119866 119862) where

119862 = 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890) forall119890 isin 119862

(119865 119866) (119890) = 119865 (119890) 119866 (119890) if 119890 isin 119860 cap 119861

119865 (119890) otherwise

(7)

By Definitions 9 and 10 we find that (119865 119860)119888

= (119880 119864)

(119865 119860) holds for any soft set (119865 119860) That is the complement of(119865 119860)with respect to the universal soft set (119880 119864) is exactly thedifference of (119880 119864) and (119865 119860) In light of this (119865 119860)(119866 119861) isalso called the relative complement of (119866 119861) in (119865 119860) while(119865 119860)

119888 is also called the absolute complement of (119865 119860)

Let us illustrate the previous operations on soft sets by asimple example

Example 11 Let us revisit Example 2 Recall that the soft set(119865 119860) describing the ldquoattractiveness of the housesrdquo in Alicersquosopinion was defined by

119860 = 1198902 1198903 1198904 1198905 1198907

119865 (1198902) = ℎ

2 ℎ3 ℎ5 119865 (119890

3) = ℎ

2 ℎ4

119865 (1198904) = ℎ

1 119865 (119890

5) = 119880 119865 (119890

7) = ℎ

3 ℎ5

(8)

In addition we assume that the ldquoattractiveness of the housesrdquoin the opinion of another buyer say Bob is described by thesoft set (119866 119861) where

119861 = 1198901 1198902 119890

7

119866 (1198901) = ℎ

3 ℎ5 119866 (119890

2) = ℎ

4

119866 (1198903) = ℎ

2 ℎ4 119866 (119890

4) = ℎ

1

119866 (1198905) = ℎ

2 ℎ3 ℎ4 ℎ5 119866 (119890

6) = 119866 (119890

7) = ℎ

3

(9)

Then by a direct computation we can readily obtain(119865 119860) cap (119866 119861) (119865 119860) cup (119866 119861) (119865 119860)

119888 and (119865 119860) (119866 119861) asfollows

(i) (119865 119860) cap (119866 119861) = (119865 cap 119866 1198903 1198904 1198905 1198907) where (119865 cap

119866)(1198903) = ℎ

2 ℎ4 (119865 cap 119866)(119890

4) = ℎ

1 (119865 cap 119866)(119890

5) =

ℎ2 ℎ3 ℎ4 ℎ5 and (119865 cap 119866)(119890

7) = ℎ

3 This means

that both Alice and Bob think that ℎ2and ℎ

4are

wooden ℎ1is cheap ℎ

2 ℎ3 ℎ4 ℎ5are in the green

surroundings and ℎ3is in the good repair

(ii) (119865 119860) cup (119866 119861) = (119865 cup 119866 1198901 1198902 119890

7) where (119865 cup

119866)(1198901) = ℎ

3 ℎ5 (119865 cup 119866)(119890

2) = ℎ

2 ℎ3 ℎ4 ℎ5 (119865 cup

119866)(1198903) = ℎ

2 ℎ4 (119865 cup 119866)(119890

4) = ℎ

1 (119865 cup 119866)(119890

5) =

119880 (119865 cup 119866)(1198906) = ℎ

3 and (119865 cup 119866)(119890

7) = ℎ

3 ℎ5 This

means that either Alice or Bob thinks that ℎ3is expen-

sive either Alice or Bob thinks that ℎ5is expensive

either Alice or Bob thinks that ℎ2is beautiful either

Alice or Bob thinks that ℎ3is beautiful and so on

(iii) (119865 119860)119888

= (119865119888

1198901 1198902 1198903 1198904 1198906 1198907 1198908) where 119865

119888

(1198901) =

119880 119865119888

(1198902) = ℎ

1 ℎ4 119865119888

(1198903) = ℎ

1 ℎ3 ℎ5 119865119888

(1198904) =

ℎ2 ℎ3 ℎ4 ℎ5 119865119888

(1198906) = 119880 119865

119888

(1198907) = ℎ

1 ℎ2 ℎ4 and

119865119888

(1198908) = 119880 This means that Alice thinks that none

of these houses is expensive neither ℎ1nor ℎ

4is

beautiful ℎ1 ℎ3 ℎ5are not wooden and so on

(iv) (119865 119860)(119866 119861) = (119865119866 1198902 1198905 1198907) where (119865119866)(119890

2) =

ℎ2 ℎ3 ℎ5 (119865 119866)(119890

5) = ℎ

1 and (119865 119866)(119890

7) =

ℎ5 This means that Alice thinks of ℎ

2 ℎ3 and ℎ

5

as beautiful but Bob does not think that these arebeautiful and so on

4 Algebraic Properties of Soft Set Operations

This section is devoted to some algebraic properties of soft setoperations defined in the last section

Let us begin with some properties involving intersectionsand unions The first four laws are obvious We omit theirproofs here since the proofs follow directly from the defini-tions of intersection and union of soft sets

4 Journal of Applied Mathematics

Proposition 12 (Identity laws) For any soft set (119865 119860) over 119880we have that

(1) (119865 119860) cap (119880 119864) = (119865 119860)

(2) (119865 119860) cup (0 0) = (119865 119860)

Proposition 13 (Domination laws) For any soft set (119865 119860)

over 119880 we have that

(1) (119865 119860) cap (0 0) = (0 0)

(2) (119865 119860) cup (119880 119864) = (119880 119864)

Proposition 14 (Idempotent laws) For any soft set (119865 119860) over119880 we have that

(1) (119865 119860) cap (119865 119860) = (119865 119860)

(2) (119865 119860) cup (119865 119860) = (119865 119860)

Proposition 15 (Commutative laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)

(2) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)

Now we turn our attention to the associative laws

Proposition 16 (Associative laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) ((119865 119860) cap (119866B)) cap (119867 119862) = (119865 119860) cap ((119866 119861) cap (119867 119862))

(2) ((119865 119860) cup (119866 119861)) cup (119867 119862) = (119865 119860) cup ((119866 119861) cup (119867 119862))

Proof We only prove the first assertion since the secondone is the same as Proposition 25(i) in [22] For simplicitywe write (119871 119860

1015840

) (119877 1198611015840

) and (119865 cap 119866 1198601) for ((119865 119860) cap

(119866 119861))cap(119867 119862) (119865 119860)cap((119866 119861)cap(119867 119862)) and (119865 119860)cap(119866 119861)respectively We thus get by definition that

1198601015840

= 119890 isin 1198601

cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 1198601

| (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) = 0 (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) cap 119867 (119890) = 0

(10)

By the same token we have that 1198611015840

= 119890 isin 119860 cap 119861 cap 119862 | 119865(119890) cap

119866(119890) cap 119867(119890) = 0 and thus 1198601015840

= 1198611015840 Moreover for any 119890 isin 119860

1015840we have that

119871 (119890) = (119865 cap 119866) (119890) cap 119867 (119890)

= 119865 (119890) cap 119866 (119890) cap 119867 (119890)

= 119865 (119890) cap (119866 (119890) cap 119867 (119890))

= 119865 (119890) cap (119866 cap 119867) (119890)

= 119877 (119890)

(11)

namely 119871(119890) = 119877(119890) Therefore the assertion (1) holds

Proposition 17 (Distributive laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) (119865 119860)cap((119866 119861)cup(119867 119862)) = ((119865 119860)cap(119866 119861))cup((119865 119860)cap

(119867 119862))(2) (119865 119860)cup((119866 119861)cap(119867 119862)) = ((119865 119860)cup(119866 119861))cap((119865 119860)cup

(119867 119862))

Proof We only verify the first assertion the second one canbe verified similarly For simplicity we write (119871 119860

1015840

) and(119877 1198611015840

) for (119865 119860) cap ((119866 119861) cup (119867 119862)) and ((119865 119860) cap (119866 119861)) cup

((119865 119860) cap (119867 119862)) respectively We thus see that

1198601015840

= 119890 isin 119860 cap (119861 cup 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin (119860 cap 119861) cup (119860 cap 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap (119866 (119890) cup 119867 (119890)) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 1198611015840

(12)

namely 1198601015840 = 1198611015840 Furthermore for any 119890 isin 119860

1015840 one can checkthat 119871(119890) = 119865(119890) cap (119866 cup 119867)(119890) = ((119865 cap 119866) cup (119865 cap 119867))(119890) = 119877(119890)

by a routine computation We do not go into the details hereHence the assertion (1) holds

Like usual sets soft sets are monotonic with respect tointersection and union

Journal of Applied Mathematics 5

Proposition 18 Let (119865119894 119860119894) and (119866

119894 119861119894) 119894 = 1 2 be soft sets

over 119880 If (119865119894 119860119894) sube (119866

119894 119861119894) 119894 = 1 2 then we have that

(1) (1198651 1198601) cap (119865

2 1198602) sube (119866

1 1198611) cap (119866

2 1198612)

(2) (1198651 1198601) cup (119865

2 1198602) sube (119866

1 1198611) cup (119866

2 1198612)

Proof It is clear by the definitions of intersection union andsubset of soft sets

Recall that in classical set theory we have that 119883 sube 119884 ifand only if 119883 cap 119884 = 119883 which is also equivalent to 119883 cup 119884 = 119884For soft sets we have the following observation

Proposition 19 Let (119865 119860) and (119866 119861) be soft sets over119880Thenthe following are equivalent

(1) (119865 119860) sube (119866 119861)(2) (119865 119860) cap (119866 119861) = (119865 119860)(3) (119865 119860) cup (119866 119861) = (119866 119861)

Proof Again it is obvious by the definitions of intersectionunion and subset of soft sets

The following several properties are concerned with thecomplement of soft sets

Proposition 20 Let (119865 119860) and (119866 119861) be two soft sets over 119880Then (119866 119861) = (119865 119860)

119888 if and only if (119865 119860) cap (119866 119861) = (0 0) and(119865 119860) cup (119866 119861) = (119880 119864)

Proof If (119866 119861) = (119865 119860)119888 then we see by definition that

(119865 119860)cap(119865 119860)119888

= (0 0) and (119865 119860)cup(119865 119860)119888

= (119865 119860)cup(119865119888

119890 isin

119860 | 119865(119890) = 119880119888

) = (119880 119864) Whence the necessity is trueConversely assume that (119865 119860) cap (119866 119861) = (0 0) and

(119865 119860) cup (119866 119861) = (119880 119864) The latter means that 119860 cup 119861 = 119864Moreover we obtain that 119865(119890) = 119880 for all 119890 isin 119860 119861 and119866(119890) = 119880 for all 119890 isin 119861 119860 For any 119890 isin 119860 cap 119861 it followsfrom (119865 119860) cap (119866 119861) = (0 0) and (F 119860) cup (119866 119861) = (119880 119864) that119865(119890) cup 119866(119890) = 119880 and 119865(119890) cap 119866(119890) = 0 As neither 119865(119890) nor119866(119890) is empty this forces that 119861 = 119890 isin 119860 | 119865(119890) = 119880

119888For any 119890 isin 119861 if 119890 isin 119860 then 119866(119890) = 119865(119890)

119888

= 119865119888

(119890)if 119890 isin 119861 119860 then 119866(119890) = 119880 = 119865

119888

(119890) This implies that(119865 119860)

119888

= (119865119888

119890 isin 119860 | 119865(119890) = 119880119888

) = (119866 119861) finishing theproof

The following fact follows immediately fromProposition 20

Corollary 21 For any soft set (119865 119860) over 119880 we have that

((119865 119860)119888

)119888

= (119865 119860) (13)

Proof Note that (119865 119860)119888

cap(119865 119860) = (0 0) and (119865 119860)119888

cup(119865 119860) =

(119880 119864) It therefore follows from Proposition 20 that (119865 119860) =

((119865 119860)119888

)119888 as desired

With the above corollary we can prove the De Morganrsquoslaws of soft sets

Proposition 22 (De Morganrsquos laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) ((119865 119860) cap (119866 119861))119888

= (119865 119860)119888

cup (119866 119861)119888

(2) ((119865 119860) cup (119866 119861))119888

= (119865 119860)119888

cap (119866 119861)119888

Proof (1) For convenience let 1198600

= 119890 isin 119860 | 119865(119890) = 1198801198610

= 119890 isin 119861 | 119866(119890) = 119880 1198620

= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 119880and 119862

1= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 0 Then we have that

((119865 119860) cap (119866 119861))119888

= (119865 cap 119866 1198621)119888

= ((119865 cap 119866)119888

119890 isin 1198621

| (119865 cap 119866) (119890) = 119880119888

)

= ((119865 cap 119866)119888

119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 119880119888

)

= ((119865 cap 119866)119888

119862119888

0)

(14)

On the other hand we have that

(119865 119860)119888

cup (119866 119861)119888

= (119865119888

119860119888

0) cup (119866

119888

119861119888

0)

= (119865119888

cup 119866119888

119860119888

0cup 119861119888

0)

= (119865119888

cup 119866119888

(1198600

cap 1198610)119888

)

= (119865119888

cup 119866119888

119862119888

0)

(15)

Therefore to prove (1) it suffices to show that (119865 cap 119866)119888

(119890) =

(119865119888

cup119866119888

)(119890) for all 119890 isin 119862119888

0 In fact since119862

119888

0= (11986211198620)cup119862119888

1and

(11986211198620)cap119862119888

1= 0 we need only to consider two casesThefirst

case is that 119890 isin 11986211198620 In this case 119890 isin 119860

119888

0cap119861119888

0 and thuswe get

that (119865cap119866)119888

(119890) = (119865(119890)cap119866(119890))119888

= 119865(119890)119888

cup119866(119890)119888

= (119865119888

cup119866119888

)(119890)The other case is that 119890 isin 119862

119888

1 In this case we always have by

definition that (119865 cap 119866)119888

(119890) = 119880 = (119865119888

cup 119866119888

)(119890) Consequently(119865 cap 119866)

119888

(119890) = (119865119888

cup 119866119888

)(119890) for all 119890 isin 119862119888

0 as desired

(2) By Corollary 21 and the first assertion we find that

((119865 119860) cup (119866 119861))119888

= (((119865 119860)119888

)119888

cup ((119866 119861)119888

)119888

)119888

= (((119865 119860)119888

cap (119866 119861)119888

)119888

)119888

= (119865 119860)119888

cap (119866 119861)119888

(16)

Hence the second assertion holds as well This completes theproof of the proposition

Let us end this section with an observation on thedifference of two soft sets

Proposition 23 For any soft sets (119865 119860) and (119866 119861) over 119880 wehave that

(119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888

(17)

Proof We set 1198610

= 119890 isin 119861 | 119866(119890) = 119880 and write (119865119866 119862) for(119865 119860) (119866 119861) Then we see that 119862 = 119860 119890 isin 119860 cap 119861 | 119865(119890) sube

6 Journal of Applied Mathematics

119866(119890) and (119866 119861)119888

= (119866119888

119861119888

0) As a result (119865 119860) cap (119866 119861)

119888

=

(119865 119860) cap (G119888 1198611198880) = (119865 cap 119866

119888

1198611) where

1198611

= 119890 isin 119860 cap 119861119888

0| 119865 (119890) cap 119866

119888

(119890) = 0

= (119860 119861) cup 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119862

(18)

as desired It remains to show that (119865 119866)(119890) = (119865 cap 119866119888

)(119890)

for all 119890 isin 119862 = 1198611 In fact if 119890 isin 119862 119861 then we have that

(119865 119866)(119890) = 119865(119890) = 119865(119890) cap 119880 = (119865 cap 119866119888

)(119890) if 119890 isin 119862 cap 119861then (119865 119866)(119890) = 119865(119890) 119866(119890) = 119865(119890) cap 119866

119888

(119890) = (119865 cap 119866119888

)(119890)We thus get that (119865 119866)(119890) = (119865 cap 119866

119888

)(119890) for all 119890 isin 119862 = 1198611

Consequently (119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888 finishing the

proof

5 Conclusion

In this paper we have redefined the intersection comple-ment and difference of soft sets These operations togetherwith an existing union operation form the fundamentaloperations for constructing new soft sets from given soft setsBy examining the algebraic properties of these operations wefind that all basic properties of operations on classical setssuch as identity laws domination laws distributive laws andDe Morganrsquos laws hold for the newly defined operations onsoft sets From this point of view the new operations on softsets are reasonable Motivated by the notion of Not set of aparameter set in [22] we will investigate the operations onsoft sets by introducing more conjunctions including ANDand OR into a parameter set In addition it is interesting toextend the notions of intersection complement difference ofsoft sets developed here to other soft structures such as fuzzysoft sets [29 41] vague soft sets [28] and soft rough sets [42]

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61070251 61170270 and61121061 and the Fundamental Research Funds for theCentralUniversities underGrant 2012RC0710The authorswould liketo thank the reviewers for their helpful suggestions

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] M I Ali ldquoA note on soft sets rough soft sets and fuzzy soft setsrdquoApplied Soft Computing Journal vol 11 no 4 pp 3329ndash33322011

[4] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing vol 12 no 6 pp 1814ndash1821 2012

[5] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[6] S Alkhazaleh and A R Salleh ldquoSoft expert setsrdquo Advances inDecision Sciences vol 2011 Article ID 757868 12 pages 2011

[7] S Alkhazaleh and A R Salleh ldquoGeneralised interval-valuedfuzzy soft setrdquo Journal of AppliedMathematics vol 2012 ArticleID 870504 18 pages 2012

[8] K V Babitha and J J Sunil ldquoTransitive closures and orderingon soft setsrdquo Computers amp Mathematics with Applications vol62 no 5 pp 2235ndash2239 2011

[9] T Deng and X Wang ldquoParameter significance and reductionsof soft setsrdquo International Journal of ComputerMathematics vol89 no 15 pp 1979ndash1995 2012

[10] F Feng Y B Jun and X Zhao ldquoSoft semiringsrdquo Computers ampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[11] X Ge Z Li and Y Ge ldquoTopological spaces and soft setsrdquoJournal of Computational Analysis and Applications vol 13 no5 pp 881ndash885 2011

[12] A A Hazaymeh I B Abdullah Z T Balkhi and R IIbrahim ldquoGeneralized fuzzy soft expert setrdquo Journal of AppliedMathematics vol 2012 Article ID 328195 22 pages 2012

[13] Y Jiang Y Tang Q Chen and Z Cao ldquoSemantic operationsof multiple soft sets under conflictrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1923ndash1939 2011

[14] Y Jiang Y Tang Q Chen JWang and S Tang ldquoExtending softsets with description logicsrdquo Computers amp Mathematics withApplications vol 59 no 6 pp 2087ndash2096 2010

[15] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[16] Y B Jun and S S Ahn ldquoDouble-framed soft sets with appli-cations in BCKBCI-algebrasrdquo Journal of Applied Mathematicsvol 2012 Article ID 178159 15 pages 2012

[17] Y B Jun K J Lee and A Khan ldquoSoft ordered semigroupsrdquoMathematical Logic Quarterly vol 56 no 1 pp 42ndash50 2010

[18] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun S Z Song and K S So ldquoSoft set theory appliedto p-ideals of BCI-algebras related to fuzzy pointsrdquo NeuralComputing and Applications vol 20 no 8 pp 1313ndash1320 2011

[21] Z Kong L Gao L Wang and S Li ldquoThe normal parameterreduction of soft sets and its algorithmrdquo Computers amp Mathe-matics with Applications vol 56 no 12 pp 3029ndash3037 2008

[22] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[23] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[24] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[25] J H Park O H Kim and Y C Kwun ldquoSome properties ofequivalence soft set relationsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1079ndash1088 2012

[26] B Tanay and M B Kandemir ldquoTopological structure of fuzzysoft setsrdquo Computers amp Mathematics with Applications vol 61no 10 pp 2952ndash2957 2011

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

The remainder of this paper is structured as follows InSection 2 we briefly recall the notion of soft sets Section 3is devoted to the definitions of new operations on soft setsAn example is also provided to illustrate the newly definedoperations in this section We address the basic propertiesof the operations on soft sets in Section 4 and conclude thepaper in Section 5

2 Soft Sets

For subsequent need let us review the notion of soft sets Fora detailed introduction to the soft set theory the reader mayrefer to [1 22]

We begin with some notations For classical set theorythe symbols 0 119860

119888 119860 cup 119861 119860 cap 119861 119860 119861 denote respectivelythe empty set the complement of 119860 with respect to someuniversal set 119880 the union of sets 119860 and 119861 the intersectionof sets 119860 and 119861 and the difference of sets 119860 and 119861 whoseelements belong to 119860 but not to 119861 In what follows we writeP(119880) for the power set of a universal set 119880 and denoteP(119880) 0 byPlowast(119880)

Throughout this paper let 119880 be a universal set and 119864 bethe set of all possible parameters under consideration withrespect to 119880 Usually parameters are attributes characteris-tics or properties of objects in 119880 We now recall the notionof soft sets due to Molodtsov [1]

Definition 1 (seeMolodtsov [1]) Let119880 be a universe and119864 theset of parameters A soft set over 119880 is a pair (119865 119860) consistingof a subset 119860 of 119864 and a mapping 119865 119860 rarr Plowast(119880)

Note that the above definition is slightly different fromthe original one in [1] where 119865 has P(119880) as its codomainIn other words we remove the parameters having the emptyset as images under 119865 It seems rational since this meansthat if there exists a parameter 119890 isin 119860 which is not theattribute characteristic or property of any object in 119880 thenthis parameter has no interest with respect to the knowledgestored in the soft set As a result a soft set of 119880 in the sense ofDefinition 1 is a parameterized family of nonempty subsets of119880

Clearly any soft set (119865 119860) over 119880 gives a partial function1198651015840

119864 rarr Plowast(119880) defined by

1198651015840

(119890) = 119865 (119890) if 119890 isin 119860

undefined otherwise(1)

for all 119890 isin 119864 Conversely any partial function 119891 from 119864 toPlowast(119880) gives rise to a soft set (119865

119891 119860119891) where 119860

119891= 119890 isin 119864 |

119891(119890) is defined and 119865119891is the restriction of 119891 on 119860

119891

To illustrate the above definition Molodtsov consideredseveral examples in [1] one of which we present blow

Example 2 Suppose that 119880 is the set of houses underconsideration say 119880 = ℎ

1 ℎ2 ℎ

5 Let 119864 be the set of

some attributes of such houses say 119864 = 1198901 1198902 119890

8 where

1198901 1198902 119890

8stand for the attributes ldquoexpensiverdquo ldquobeautifulrdquo

ldquowoodenrdquo ldquocheaprdquo ldquoin the green surroundingsrdquo ldquomodernrdquoldquoin good repairrdquo and ldquoin bad repairrdquo respectively

In this case to define a soft set means to point outexpensive houses beautiful houses and so on For examplethe soft set (119865 119860) that describes the ldquoattractiveness of thehousesrdquo in the opinion of a buyer say Alice may be definedas

119860 = 1198902 1198903 1198904 1198905 1198907

119865 (1198902) = ℎ

2 ℎ3 ℎ5 119865 (119890

3) = ℎ

2 ℎ4

119865 (1198904) = ℎ

1 119865 (119890

5) = 119880 119865 (119890

7) = ℎ

3 ℎ5

(2)

3 Operations on Soft Sets

In this section we generalize the basic operations on classicalsets to soft sets The examination of more properties of theseoperations is deferred to the next section

Let us start with the notions of empty and universal softsets Recall that in [22] a soft set (119865 119860) is called a null soft setif 119865(119890) = 0 for all 119890 isin 119860 Because 0 does not belong to thecodomain of 119865 in our framework we redefine the concept ofempty soft set as follows

Definition 3 A soft set (119865 119860) over 119880 is said to be emptywhenever 119860 = 0 Symbolically we write (0 0) for the emptysoft set over 119880

Definition 4 A soft set (119865 119860) over 119880 is called a universal softset if 119860 = 119864 and 119865(119890) = 119880 for all 119890 isin 119860 Symbolically we write(119880 119864) for the universal soft set over 119880

Let us now define the subsets of a soft set

Definition 5 Let (119865 119860) and (119866 119861) be two soft sets over 119880 Wesay that (119865 119860) is a subset of (119866 119861) denoted (119865 119860) sube (119866 119861)

if either (119865 119860) = (0 0) or 119860 sube 119861 and 119865(119890) sube 119866(119890) for every119890 isin 119860 Two soft sets (119865 119860) and (119866 119861) are said to be equaldenoted (119865 119860) = (119866 119861) if (119865 119860) sube (119866 119861) and (119866 119861) sube (119865 119860)

By definition two soft sets (119865 119860) and (119866 119861) are equal ifand only if 119860 = 119861 and 119865(119890) = 119866(119890) for all 119890 isin 119860 In [22]a similar notion called soft subset was defined by requiringthat 119860 sube 119861 and 119865(119890) = 119866(119890) for every 119890 isin 119860 By Definition 5the empty soft set (0 0) is a subset of any soft set It also followsfrom Definition 5 that any soft set is a subset of the universalsoft set (119880 119864) Formally we have the following proposition

Proposition 6 For any soft set (119865 119860) over 119880

(0 0) sube (119865 119860) sube (119880 119864) (3)

We are now in the position to introduce some operationson soft sets

Definition 7 Let (119865 119860) and (119866 119861) be two soft sets over119880Theintersection of (119865 119860) and (119866 119861) denoted by (119865 119860) cap (119866 119861) isdefined as (119865 cap 119866 119862) where

119862 = 119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 0 forall119890 isin 119862

(119865 cap 119866) (119890) = 119865 (119890) cap 119866 (119890)

(4)

Journal of Applied Mathematics 3

In particular if 119860 cap 119861 = 0 or 119865(119890) cap 119866(119890) = 0 for every119890 isin 119860 cap 119861 then we see that (119865 119860) cap (119866 119861) = (0 0)

The following definition of union of soft sets is the sameas in [22]

Definition 8 (see [22] Definition 211) Let (119865 119860) and (119866 119861)

be two soft sets over 119880 The union of (119865 119860) and (119866 119861)denoted by (119865 119860) cup (119866 119861) is defined as (119865 cup 119866 119862) where

119862 = 119860 cup 119861 forall119890 isin 119862

(119865 cup 119866) (119890) =

119865 (119890) if 119890 isin 119860 119861

119866 (119890) if 119890 isin 119861 119860

119865 (119890) cup 119866 (119890) otherwise

(5)

We now define the notion of complement in soft set the-ory It is worth noting that this is rather different from thosein the existing literature [22 46] where the complement isusually based on the so-called NOT set of a parameter set andthe union of a soft set and its complement is not exactly thewhole universal soft set in general

Definition 9 Let (119865 119860) be a soft set over 119880 The complementof (119865 119860) with respect to the universal soft set (119880 119864) denotedby (119865 119860)

119888 is defined as (119865119888

119862) where

119862 = 119864 119890 isin 119860 | 119865 (119890) = 119880

= 119890 isin 119860 | 119865 (119890) = 119880119888

forall119890 isin 119862

119865119888

(119890) = 119880 119865 (119890) if 119890 isin 119860

119880 otherwise

(6)

In certain settings the difference of two soft sets (119865 119860)

and (119866 119861) is useful

Definition 10 Let (119865 119860) and (119866 119861) be two soft sets over 119880The difference of (119865 119860) and (119866 119861) denoted by (119865 119860) (119866 119861)is defined as (119865 119866 119862) where

119862 = 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890) forall119890 isin 119862

(119865 119866) (119890) = 119865 (119890) 119866 (119890) if 119890 isin 119860 cap 119861

119865 (119890) otherwise

(7)

By Definitions 9 and 10 we find that (119865 119860)119888

= (119880 119864)

(119865 119860) holds for any soft set (119865 119860) That is the complement of(119865 119860)with respect to the universal soft set (119880 119864) is exactly thedifference of (119880 119864) and (119865 119860) In light of this (119865 119860)(119866 119861) isalso called the relative complement of (119866 119861) in (119865 119860) while(119865 119860)

119888 is also called the absolute complement of (119865 119860)

Let us illustrate the previous operations on soft sets by asimple example

Example 11 Let us revisit Example 2 Recall that the soft set(119865 119860) describing the ldquoattractiveness of the housesrdquo in Alicersquosopinion was defined by

119860 = 1198902 1198903 1198904 1198905 1198907

119865 (1198902) = ℎ

2 ℎ3 ℎ5 119865 (119890

3) = ℎ

2 ℎ4

119865 (1198904) = ℎ

1 119865 (119890

5) = 119880 119865 (119890

7) = ℎ

3 ℎ5

(8)

In addition we assume that the ldquoattractiveness of the housesrdquoin the opinion of another buyer say Bob is described by thesoft set (119866 119861) where

119861 = 1198901 1198902 119890

7

119866 (1198901) = ℎ

3 ℎ5 119866 (119890

2) = ℎ

4

119866 (1198903) = ℎ

2 ℎ4 119866 (119890

4) = ℎ

1

119866 (1198905) = ℎ

2 ℎ3 ℎ4 ℎ5 119866 (119890

6) = 119866 (119890

7) = ℎ

3

(9)

Then by a direct computation we can readily obtain(119865 119860) cap (119866 119861) (119865 119860) cup (119866 119861) (119865 119860)

119888 and (119865 119860) (119866 119861) asfollows

(i) (119865 119860) cap (119866 119861) = (119865 cap 119866 1198903 1198904 1198905 1198907) where (119865 cap

119866)(1198903) = ℎ

2 ℎ4 (119865 cap 119866)(119890

4) = ℎ

1 (119865 cap 119866)(119890

5) =

ℎ2 ℎ3 ℎ4 ℎ5 and (119865 cap 119866)(119890

7) = ℎ

3 This means

that both Alice and Bob think that ℎ2and ℎ

4are

wooden ℎ1is cheap ℎ

2 ℎ3 ℎ4 ℎ5are in the green

surroundings and ℎ3is in the good repair

(ii) (119865 119860) cup (119866 119861) = (119865 cup 119866 1198901 1198902 119890

7) where (119865 cup

119866)(1198901) = ℎ

3 ℎ5 (119865 cup 119866)(119890

2) = ℎ

2 ℎ3 ℎ4 ℎ5 (119865 cup

119866)(1198903) = ℎ

2 ℎ4 (119865 cup 119866)(119890

4) = ℎ

1 (119865 cup 119866)(119890

5) =

119880 (119865 cup 119866)(1198906) = ℎ

3 and (119865 cup 119866)(119890

7) = ℎ

3 ℎ5 This

means that either Alice or Bob thinks that ℎ3is expen-

sive either Alice or Bob thinks that ℎ5is expensive

either Alice or Bob thinks that ℎ2is beautiful either

Alice or Bob thinks that ℎ3is beautiful and so on

(iii) (119865 119860)119888

= (119865119888

1198901 1198902 1198903 1198904 1198906 1198907 1198908) where 119865

119888

(1198901) =

119880 119865119888

(1198902) = ℎ

1 ℎ4 119865119888

(1198903) = ℎ

1 ℎ3 ℎ5 119865119888

(1198904) =

ℎ2 ℎ3 ℎ4 ℎ5 119865119888

(1198906) = 119880 119865

119888

(1198907) = ℎ

1 ℎ2 ℎ4 and

119865119888

(1198908) = 119880 This means that Alice thinks that none

of these houses is expensive neither ℎ1nor ℎ

4is

beautiful ℎ1 ℎ3 ℎ5are not wooden and so on

(iv) (119865 119860)(119866 119861) = (119865119866 1198902 1198905 1198907) where (119865119866)(119890

2) =

ℎ2 ℎ3 ℎ5 (119865 119866)(119890

5) = ℎ

1 and (119865 119866)(119890

7) =

ℎ5 This means that Alice thinks of ℎ

2 ℎ3 and ℎ

5

as beautiful but Bob does not think that these arebeautiful and so on

4 Algebraic Properties of Soft Set Operations

This section is devoted to some algebraic properties of soft setoperations defined in the last section

Let us begin with some properties involving intersectionsand unions The first four laws are obvious We omit theirproofs here since the proofs follow directly from the defini-tions of intersection and union of soft sets

4 Journal of Applied Mathematics

Proposition 12 (Identity laws) For any soft set (119865 119860) over 119880we have that

(1) (119865 119860) cap (119880 119864) = (119865 119860)

(2) (119865 119860) cup (0 0) = (119865 119860)

Proposition 13 (Domination laws) For any soft set (119865 119860)

over 119880 we have that

(1) (119865 119860) cap (0 0) = (0 0)

(2) (119865 119860) cup (119880 119864) = (119880 119864)

Proposition 14 (Idempotent laws) For any soft set (119865 119860) over119880 we have that

(1) (119865 119860) cap (119865 119860) = (119865 119860)

(2) (119865 119860) cup (119865 119860) = (119865 119860)

Proposition 15 (Commutative laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)

(2) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)

Now we turn our attention to the associative laws

Proposition 16 (Associative laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) ((119865 119860) cap (119866B)) cap (119867 119862) = (119865 119860) cap ((119866 119861) cap (119867 119862))

(2) ((119865 119860) cup (119866 119861)) cup (119867 119862) = (119865 119860) cup ((119866 119861) cup (119867 119862))

Proof We only prove the first assertion since the secondone is the same as Proposition 25(i) in [22] For simplicitywe write (119871 119860

1015840

) (119877 1198611015840

) and (119865 cap 119866 1198601) for ((119865 119860) cap

(119866 119861))cap(119867 119862) (119865 119860)cap((119866 119861)cap(119867 119862)) and (119865 119860)cap(119866 119861)respectively We thus get by definition that

1198601015840

= 119890 isin 1198601

cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 1198601

| (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) = 0 (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) cap 119867 (119890) = 0

(10)

By the same token we have that 1198611015840

= 119890 isin 119860 cap 119861 cap 119862 | 119865(119890) cap

119866(119890) cap 119867(119890) = 0 and thus 1198601015840

= 1198611015840 Moreover for any 119890 isin 119860

1015840we have that

119871 (119890) = (119865 cap 119866) (119890) cap 119867 (119890)

= 119865 (119890) cap 119866 (119890) cap 119867 (119890)

= 119865 (119890) cap (119866 (119890) cap 119867 (119890))

= 119865 (119890) cap (119866 cap 119867) (119890)

= 119877 (119890)

(11)

namely 119871(119890) = 119877(119890) Therefore the assertion (1) holds

Proposition 17 (Distributive laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) (119865 119860)cap((119866 119861)cup(119867 119862)) = ((119865 119860)cap(119866 119861))cup((119865 119860)cap

(119867 119862))(2) (119865 119860)cup((119866 119861)cap(119867 119862)) = ((119865 119860)cup(119866 119861))cap((119865 119860)cup

(119867 119862))

Proof We only verify the first assertion the second one canbe verified similarly For simplicity we write (119871 119860

1015840

) and(119877 1198611015840

) for (119865 119860) cap ((119866 119861) cup (119867 119862)) and ((119865 119860) cap (119866 119861)) cup

((119865 119860) cap (119867 119862)) respectively We thus see that

1198601015840

= 119890 isin 119860 cap (119861 cup 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin (119860 cap 119861) cup (119860 cap 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap (119866 (119890) cup 119867 (119890)) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 1198611015840

(12)

namely 1198601015840 = 1198611015840 Furthermore for any 119890 isin 119860

1015840 one can checkthat 119871(119890) = 119865(119890) cap (119866 cup 119867)(119890) = ((119865 cap 119866) cup (119865 cap 119867))(119890) = 119877(119890)

by a routine computation We do not go into the details hereHence the assertion (1) holds

Like usual sets soft sets are monotonic with respect tointersection and union

Journal of Applied Mathematics 5

Proposition 18 Let (119865119894 119860119894) and (119866

119894 119861119894) 119894 = 1 2 be soft sets

over 119880 If (119865119894 119860119894) sube (119866

119894 119861119894) 119894 = 1 2 then we have that

(1) (1198651 1198601) cap (119865

2 1198602) sube (119866

1 1198611) cap (119866

2 1198612)

(2) (1198651 1198601) cup (119865

2 1198602) sube (119866

1 1198611) cup (119866

2 1198612)

Proof It is clear by the definitions of intersection union andsubset of soft sets

Recall that in classical set theory we have that 119883 sube 119884 ifand only if 119883 cap 119884 = 119883 which is also equivalent to 119883 cup 119884 = 119884For soft sets we have the following observation

Proposition 19 Let (119865 119860) and (119866 119861) be soft sets over119880Thenthe following are equivalent

(1) (119865 119860) sube (119866 119861)(2) (119865 119860) cap (119866 119861) = (119865 119860)(3) (119865 119860) cup (119866 119861) = (119866 119861)

Proof Again it is obvious by the definitions of intersectionunion and subset of soft sets

The following several properties are concerned with thecomplement of soft sets

Proposition 20 Let (119865 119860) and (119866 119861) be two soft sets over 119880Then (119866 119861) = (119865 119860)

119888 if and only if (119865 119860) cap (119866 119861) = (0 0) and(119865 119860) cup (119866 119861) = (119880 119864)

Proof If (119866 119861) = (119865 119860)119888 then we see by definition that

(119865 119860)cap(119865 119860)119888

= (0 0) and (119865 119860)cup(119865 119860)119888

= (119865 119860)cup(119865119888

119890 isin

119860 | 119865(119890) = 119880119888

) = (119880 119864) Whence the necessity is trueConversely assume that (119865 119860) cap (119866 119861) = (0 0) and

(119865 119860) cup (119866 119861) = (119880 119864) The latter means that 119860 cup 119861 = 119864Moreover we obtain that 119865(119890) = 119880 for all 119890 isin 119860 119861 and119866(119890) = 119880 for all 119890 isin 119861 119860 For any 119890 isin 119860 cap 119861 it followsfrom (119865 119860) cap (119866 119861) = (0 0) and (F 119860) cup (119866 119861) = (119880 119864) that119865(119890) cup 119866(119890) = 119880 and 119865(119890) cap 119866(119890) = 0 As neither 119865(119890) nor119866(119890) is empty this forces that 119861 = 119890 isin 119860 | 119865(119890) = 119880

119888For any 119890 isin 119861 if 119890 isin 119860 then 119866(119890) = 119865(119890)

119888

= 119865119888

(119890)if 119890 isin 119861 119860 then 119866(119890) = 119880 = 119865

119888

(119890) This implies that(119865 119860)

119888

= (119865119888

119890 isin 119860 | 119865(119890) = 119880119888

) = (119866 119861) finishing theproof

The following fact follows immediately fromProposition 20

Corollary 21 For any soft set (119865 119860) over 119880 we have that

((119865 119860)119888

)119888

= (119865 119860) (13)

Proof Note that (119865 119860)119888

cap(119865 119860) = (0 0) and (119865 119860)119888

cup(119865 119860) =

(119880 119864) It therefore follows from Proposition 20 that (119865 119860) =

((119865 119860)119888

)119888 as desired

With the above corollary we can prove the De Morganrsquoslaws of soft sets

Proposition 22 (De Morganrsquos laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) ((119865 119860) cap (119866 119861))119888

= (119865 119860)119888

cup (119866 119861)119888

(2) ((119865 119860) cup (119866 119861))119888

= (119865 119860)119888

cap (119866 119861)119888

Proof (1) For convenience let 1198600

= 119890 isin 119860 | 119865(119890) = 1198801198610

= 119890 isin 119861 | 119866(119890) = 119880 1198620

= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 119880and 119862

1= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 0 Then we have that

((119865 119860) cap (119866 119861))119888

= (119865 cap 119866 1198621)119888

= ((119865 cap 119866)119888

119890 isin 1198621

| (119865 cap 119866) (119890) = 119880119888

)

= ((119865 cap 119866)119888

119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 119880119888

)

= ((119865 cap 119866)119888

119862119888

0)

(14)

On the other hand we have that

(119865 119860)119888

cup (119866 119861)119888

= (119865119888

119860119888

0) cup (119866

119888

119861119888

0)

= (119865119888

cup 119866119888

119860119888

0cup 119861119888

0)

= (119865119888

cup 119866119888

(1198600

cap 1198610)119888

)

= (119865119888

cup 119866119888

119862119888

0)

(15)

Therefore to prove (1) it suffices to show that (119865 cap 119866)119888

(119890) =

(119865119888

cup119866119888

)(119890) for all 119890 isin 119862119888

0 In fact since119862

119888

0= (11986211198620)cup119862119888

1and

(11986211198620)cap119862119888

1= 0 we need only to consider two casesThefirst

case is that 119890 isin 11986211198620 In this case 119890 isin 119860

119888

0cap119861119888

0 and thuswe get

that (119865cap119866)119888

(119890) = (119865(119890)cap119866(119890))119888

= 119865(119890)119888

cup119866(119890)119888

= (119865119888

cup119866119888

)(119890)The other case is that 119890 isin 119862

119888

1 In this case we always have by

definition that (119865 cap 119866)119888

(119890) = 119880 = (119865119888

cup 119866119888

)(119890) Consequently(119865 cap 119866)

119888

(119890) = (119865119888

cup 119866119888

)(119890) for all 119890 isin 119862119888

0 as desired

(2) By Corollary 21 and the first assertion we find that

((119865 119860) cup (119866 119861))119888

= (((119865 119860)119888

)119888

cup ((119866 119861)119888

)119888

)119888

= (((119865 119860)119888

cap (119866 119861)119888

)119888

)119888

= (119865 119860)119888

cap (119866 119861)119888

(16)

Hence the second assertion holds as well This completes theproof of the proposition

Let us end this section with an observation on thedifference of two soft sets

Proposition 23 For any soft sets (119865 119860) and (119866 119861) over 119880 wehave that

(119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888

(17)

Proof We set 1198610

= 119890 isin 119861 | 119866(119890) = 119880 and write (119865119866 119862) for(119865 119860) (119866 119861) Then we see that 119862 = 119860 119890 isin 119860 cap 119861 | 119865(119890) sube

6 Journal of Applied Mathematics

119866(119890) and (119866 119861)119888

= (119866119888

119861119888

0) As a result (119865 119860) cap (119866 119861)

119888

=

(119865 119860) cap (G119888 1198611198880) = (119865 cap 119866

119888

1198611) where

1198611

= 119890 isin 119860 cap 119861119888

0| 119865 (119890) cap 119866

119888

(119890) = 0

= (119860 119861) cup 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119862

(18)

as desired It remains to show that (119865 119866)(119890) = (119865 cap 119866119888

)(119890)

for all 119890 isin 119862 = 1198611 In fact if 119890 isin 119862 119861 then we have that

(119865 119866)(119890) = 119865(119890) = 119865(119890) cap 119880 = (119865 cap 119866119888

)(119890) if 119890 isin 119862 cap 119861then (119865 119866)(119890) = 119865(119890) 119866(119890) = 119865(119890) cap 119866

119888

(119890) = (119865 cap 119866119888

)(119890)We thus get that (119865 119866)(119890) = (119865 cap 119866

119888

)(119890) for all 119890 isin 119862 = 1198611

Consequently (119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888 finishing the

proof

5 Conclusion

In this paper we have redefined the intersection comple-ment and difference of soft sets These operations togetherwith an existing union operation form the fundamentaloperations for constructing new soft sets from given soft setsBy examining the algebraic properties of these operations wefind that all basic properties of operations on classical setssuch as identity laws domination laws distributive laws andDe Morganrsquos laws hold for the newly defined operations onsoft sets From this point of view the new operations on softsets are reasonable Motivated by the notion of Not set of aparameter set in [22] we will investigate the operations onsoft sets by introducing more conjunctions including ANDand OR into a parameter set In addition it is interesting toextend the notions of intersection complement difference ofsoft sets developed here to other soft structures such as fuzzysoft sets [29 41] vague soft sets [28] and soft rough sets [42]

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61070251 61170270 and61121061 and the Fundamental Research Funds for theCentralUniversities underGrant 2012RC0710The authorswould liketo thank the reviewers for their helpful suggestions

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] M I Ali ldquoA note on soft sets rough soft sets and fuzzy soft setsrdquoApplied Soft Computing Journal vol 11 no 4 pp 3329ndash33322011

[4] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing vol 12 no 6 pp 1814ndash1821 2012

[5] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[6] S Alkhazaleh and A R Salleh ldquoSoft expert setsrdquo Advances inDecision Sciences vol 2011 Article ID 757868 12 pages 2011

[7] S Alkhazaleh and A R Salleh ldquoGeneralised interval-valuedfuzzy soft setrdquo Journal of AppliedMathematics vol 2012 ArticleID 870504 18 pages 2012

[8] K V Babitha and J J Sunil ldquoTransitive closures and orderingon soft setsrdquo Computers amp Mathematics with Applications vol62 no 5 pp 2235ndash2239 2011

[9] T Deng and X Wang ldquoParameter significance and reductionsof soft setsrdquo International Journal of ComputerMathematics vol89 no 15 pp 1979ndash1995 2012

[10] F Feng Y B Jun and X Zhao ldquoSoft semiringsrdquo Computers ampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[11] X Ge Z Li and Y Ge ldquoTopological spaces and soft setsrdquoJournal of Computational Analysis and Applications vol 13 no5 pp 881ndash885 2011

[12] A A Hazaymeh I B Abdullah Z T Balkhi and R IIbrahim ldquoGeneralized fuzzy soft expert setrdquo Journal of AppliedMathematics vol 2012 Article ID 328195 22 pages 2012

[13] Y Jiang Y Tang Q Chen and Z Cao ldquoSemantic operationsof multiple soft sets under conflictrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1923ndash1939 2011

[14] Y Jiang Y Tang Q Chen JWang and S Tang ldquoExtending softsets with description logicsrdquo Computers amp Mathematics withApplications vol 59 no 6 pp 2087ndash2096 2010

[15] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[16] Y B Jun and S S Ahn ldquoDouble-framed soft sets with appli-cations in BCKBCI-algebrasrdquo Journal of Applied Mathematicsvol 2012 Article ID 178159 15 pages 2012

[17] Y B Jun K J Lee and A Khan ldquoSoft ordered semigroupsrdquoMathematical Logic Quarterly vol 56 no 1 pp 42ndash50 2010

[18] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun S Z Song and K S So ldquoSoft set theory appliedto p-ideals of BCI-algebras related to fuzzy pointsrdquo NeuralComputing and Applications vol 20 no 8 pp 1313ndash1320 2011

[21] Z Kong L Gao L Wang and S Li ldquoThe normal parameterreduction of soft sets and its algorithmrdquo Computers amp Mathe-matics with Applications vol 56 no 12 pp 3029ndash3037 2008

[22] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[23] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[24] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[25] J H Park O H Kim and Y C Kwun ldquoSome properties ofequivalence soft set relationsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1079ndash1088 2012

[26] B Tanay and M B Kandemir ldquoTopological structure of fuzzysoft setsrdquo Computers amp Mathematics with Applications vol 61no 10 pp 2952ndash2957 2011

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

In particular if 119860 cap 119861 = 0 or 119865(119890) cap 119866(119890) = 0 for every119890 isin 119860 cap 119861 then we see that (119865 119860) cap (119866 119861) = (0 0)

The following definition of union of soft sets is the sameas in [22]

Definition 8 (see [22] Definition 211) Let (119865 119860) and (119866 119861)

be two soft sets over 119880 The union of (119865 119860) and (119866 119861)denoted by (119865 119860) cup (119866 119861) is defined as (119865 cup 119866 119862) where

119862 = 119860 cup 119861 forall119890 isin 119862

(119865 cup 119866) (119890) =

119865 (119890) if 119890 isin 119860 119861

119866 (119890) if 119890 isin 119861 119860

119865 (119890) cup 119866 (119890) otherwise

(5)

We now define the notion of complement in soft set the-ory It is worth noting that this is rather different from thosein the existing literature [22 46] where the complement isusually based on the so-called NOT set of a parameter set andthe union of a soft set and its complement is not exactly thewhole universal soft set in general

Definition 9 Let (119865 119860) be a soft set over 119880 The complementof (119865 119860) with respect to the universal soft set (119880 119864) denotedby (119865 119860)

119888 is defined as (119865119888

119862) where

119862 = 119864 119890 isin 119860 | 119865 (119890) = 119880

= 119890 isin 119860 | 119865 (119890) = 119880119888

forall119890 isin 119862

119865119888

(119890) = 119880 119865 (119890) if 119890 isin 119860

119880 otherwise

(6)

In certain settings the difference of two soft sets (119865 119860)

and (119866 119861) is useful

Definition 10 Let (119865 119860) and (119866 119861) be two soft sets over 119880The difference of (119865 119860) and (119866 119861) denoted by (119865 119860) (119866 119861)is defined as (119865 119866 119862) where

119862 = 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890) forall119890 isin 119862

(119865 119866) (119890) = 119865 (119890) 119866 (119890) if 119890 isin 119860 cap 119861

119865 (119890) otherwise

(7)

By Definitions 9 and 10 we find that (119865 119860)119888

= (119880 119864)

(119865 119860) holds for any soft set (119865 119860) That is the complement of(119865 119860)with respect to the universal soft set (119880 119864) is exactly thedifference of (119880 119864) and (119865 119860) In light of this (119865 119860)(119866 119861) isalso called the relative complement of (119866 119861) in (119865 119860) while(119865 119860)

119888 is also called the absolute complement of (119865 119860)

Let us illustrate the previous operations on soft sets by asimple example

Example 11 Let us revisit Example 2 Recall that the soft set(119865 119860) describing the ldquoattractiveness of the housesrdquo in Alicersquosopinion was defined by

119860 = 1198902 1198903 1198904 1198905 1198907

119865 (1198902) = ℎ

2 ℎ3 ℎ5 119865 (119890

3) = ℎ

2 ℎ4

119865 (1198904) = ℎ

1 119865 (119890

5) = 119880 119865 (119890

7) = ℎ

3 ℎ5

(8)

In addition we assume that the ldquoattractiveness of the housesrdquoin the opinion of another buyer say Bob is described by thesoft set (119866 119861) where

119861 = 1198901 1198902 119890

7

119866 (1198901) = ℎ

3 ℎ5 119866 (119890

2) = ℎ

4

119866 (1198903) = ℎ

2 ℎ4 119866 (119890

4) = ℎ

1

119866 (1198905) = ℎ

2 ℎ3 ℎ4 ℎ5 119866 (119890

6) = 119866 (119890

7) = ℎ

3

(9)

Then by a direct computation we can readily obtain(119865 119860) cap (119866 119861) (119865 119860) cup (119866 119861) (119865 119860)

119888 and (119865 119860) (119866 119861) asfollows

(i) (119865 119860) cap (119866 119861) = (119865 cap 119866 1198903 1198904 1198905 1198907) where (119865 cap

119866)(1198903) = ℎ

2 ℎ4 (119865 cap 119866)(119890

4) = ℎ

1 (119865 cap 119866)(119890

5) =

ℎ2 ℎ3 ℎ4 ℎ5 and (119865 cap 119866)(119890

7) = ℎ

3 This means

that both Alice and Bob think that ℎ2and ℎ

4are

wooden ℎ1is cheap ℎ

2 ℎ3 ℎ4 ℎ5are in the green

surroundings and ℎ3is in the good repair

(ii) (119865 119860) cup (119866 119861) = (119865 cup 119866 1198901 1198902 119890

7) where (119865 cup

119866)(1198901) = ℎ

3 ℎ5 (119865 cup 119866)(119890

2) = ℎ

2 ℎ3 ℎ4 ℎ5 (119865 cup

119866)(1198903) = ℎ

2 ℎ4 (119865 cup 119866)(119890

4) = ℎ

1 (119865 cup 119866)(119890

5) =

119880 (119865 cup 119866)(1198906) = ℎ

3 and (119865 cup 119866)(119890

7) = ℎ

3 ℎ5 This

means that either Alice or Bob thinks that ℎ3is expen-

sive either Alice or Bob thinks that ℎ5is expensive

either Alice or Bob thinks that ℎ2is beautiful either

Alice or Bob thinks that ℎ3is beautiful and so on

(iii) (119865 119860)119888

= (119865119888

1198901 1198902 1198903 1198904 1198906 1198907 1198908) where 119865

119888

(1198901) =

119880 119865119888

(1198902) = ℎ

1 ℎ4 119865119888

(1198903) = ℎ

1 ℎ3 ℎ5 119865119888

(1198904) =

ℎ2 ℎ3 ℎ4 ℎ5 119865119888

(1198906) = 119880 119865

119888

(1198907) = ℎ

1 ℎ2 ℎ4 and

119865119888

(1198908) = 119880 This means that Alice thinks that none

of these houses is expensive neither ℎ1nor ℎ

4is

beautiful ℎ1 ℎ3 ℎ5are not wooden and so on

(iv) (119865 119860)(119866 119861) = (119865119866 1198902 1198905 1198907) where (119865119866)(119890

2) =

ℎ2 ℎ3 ℎ5 (119865 119866)(119890

5) = ℎ

1 and (119865 119866)(119890

7) =

ℎ5 This means that Alice thinks of ℎ

2 ℎ3 and ℎ

5

as beautiful but Bob does not think that these arebeautiful and so on

4 Algebraic Properties of Soft Set Operations

This section is devoted to some algebraic properties of soft setoperations defined in the last section

Let us begin with some properties involving intersectionsand unions The first four laws are obvious We omit theirproofs here since the proofs follow directly from the defini-tions of intersection and union of soft sets

4 Journal of Applied Mathematics

Proposition 12 (Identity laws) For any soft set (119865 119860) over 119880we have that

(1) (119865 119860) cap (119880 119864) = (119865 119860)

(2) (119865 119860) cup (0 0) = (119865 119860)

Proposition 13 (Domination laws) For any soft set (119865 119860)

over 119880 we have that

(1) (119865 119860) cap (0 0) = (0 0)

(2) (119865 119860) cup (119880 119864) = (119880 119864)

Proposition 14 (Idempotent laws) For any soft set (119865 119860) over119880 we have that

(1) (119865 119860) cap (119865 119860) = (119865 119860)

(2) (119865 119860) cup (119865 119860) = (119865 119860)

Proposition 15 (Commutative laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)

(2) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)

Now we turn our attention to the associative laws

Proposition 16 (Associative laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) ((119865 119860) cap (119866B)) cap (119867 119862) = (119865 119860) cap ((119866 119861) cap (119867 119862))

(2) ((119865 119860) cup (119866 119861)) cup (119867 119862) = (119865 119860) cup ((119866 119861) cup (119867 119862))

Proof We only prove the first assertion since the secondone is the same as Proposition 25(i) in [22] For simplicitywe write (119871 119860

1015840

) (119877 1198611015840

) and (119865 cap 119866 1198601) for ((119865 119860) cap

(119866 119861))cap(119867 119862) (119865 119860)cap((119866 119861)cap(119867 119862)) and (119865 119860)cap(119866 119861)respectively We thus get by definition that

1198601015840

= 119890 isin 1198601

cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 1198601

| (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) = 0 (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) cap 119867 (119890) = 0

(10)

By the same token we have that 1198611015840

= 119890 isin 119860 cap 119861 cap 119862 | 119865(119890) cap

119866(119890) cap 119867(119890) = 0 and thus 1198601015840

= 1198611015840 Moreover for any 119890 isin 119860

1015840we have that

119871 (119890) = (119865 cap 119866) (119890) cap 119867 (119890)

= 119865 (119890) cap 119866 (119890) cap 119867 (119890)

= 119865 (119890) cap (119866 (119890) cap 119867 (119890))

= 119865 (119890) cap (119866 cap 119867) (119890)

= 119877 (119890)

(11)

namely 119871(119890) = 119877(119890) Therefore the assertion (1) holds

Proposition 17 (Distributive laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) (119865 119860)cap((119866 119861)cup(119867 119862)) = ((119865 119860)cap(119866 119861))cup((119865 119860)cap

(119867 119862))(2) (119865 119860)cup((119866 119861)cap(119867 119862)) = ((119865 119860)cup(119866 119861))cap((119865 119860)cup

(119867 119862))

Proof We only verify the first assertion the second one canbe verified similarly For simplicity we write (119871 119860

1015840

) and(119877 1198611015840

) for (119865 119860) cap ((119866 119861) cup (119867 119862)) and ((119865 119860) cap (119866 119861)) cup

((119865 119860) cap (119867 119862)) respectively We thus see that

1198601015840

= 119890 isin 119860 cap (119861 cup 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin (119860 cap 119861) cup (119860 cap 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap (119866 (119890) cup 119867 (119890)) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 1198611015840

(12)

namely 1198601015840 = 1198611015840 Furthermore for any 119890 isin 119860

1015840 one can checkthat 119871(119890) = 119865(119890) cap (119866 cup 119867)(119890) = ((119865 cap 119866) cup (119865 cap 119867))(119890) = 119877(119890)

by a routine computation We do not go into the details hereHence the assertion (1) holds

Like usual sets soft sets are monotonic with respect tointersection and union

Journal of Applied Mathematics 5

Proposition 18 Let (119865119894 119860119894) and (119866

119894 119861119894) 119894 = 1 2 be soft sets

over 119880 If (119865119894 119860119894) sube (119866

119894 119861119894) 119894 = 1 2 then we have that

(1) (1198651 1198601) cap (119865

2 1198602) sube (119866

1 1198611) cap (119866

2 1198612)

(2) (1198651 1198601) cup (119865

2 1198602) sube (119866

1 1198611) cup (119866

2 1198612)

Proof It is clear by the definitions of intersection union andsubset of soft sets

Recall that in classical set theory we have that 119883 sube 119884 ifand only if 119883 cap 119884 = 119883 which is also equivalent to 119883 cup 119884 = 119884For soft sets we have the following observation

Proposition 19 Let (119865 119860) and (119866 119861) be soft sets over119880Thenthe following are equivalent

(1) (119865 119860) sube (119866 119861)(2) (119865 119860) cap (119866 119861) = (119865 119860)(3) (119865 119860) cup (119866 119861) = (119866 119861)

Proof Again it is obvious by the definitions of intersectionunion and subset of soft sets

The following several properties are concerned with thecomplement of soft sets

Proposition 20 Let (119865 119860) and (119866 119861) be two soft sets over 119880Then (119866 119861) = (119865 119860)

119888 if and only if (119865 119860) cap (119866 119861) = (0 0) and(119865 119860) cup (119866 119861) = (119880 119864)

Proof If (119866 119861) = (119865 119860)119888 then we see by definition that

(119865 119860)cap(119865 119860)119888

= (0 0) and (119865 119860)cup(119865 119860)119888

= (119865 119860)cup(119865119888

119890 isin

119860 | 119865(119890) = 119880119888

) = (119880 119864) Whence the necessity is trueConversely assume that (119865 119860) cap (119866 119861) = (0 0) and

(119865 119860) cup (119866 119861) = (119880 119864) The latter means that 119860 cup 119861 = 119864Moreover we obtain that 119865(119890) = 119880 for all 119890 isin 119860 119861 and119866(119890) = 119880 for all 119890 isin 119861 119860 For any 119890 isin 119860 cap 119861 it followsfrom (119865 119860) cap (119866 119861) = (0 0) and (F 119860) cup (119866 119861) = (119880 119864) that119865(119890) cup 119866(119890) = 119880 and 119865(119890) cap 119866(119890) = 0 As neither 119865(119890) nor119866(119890) is empty this forces that 119861 = 119890 isin 119860 | 119865(119890) = 119880

119888For any 119890 isin 119861 if 119890 isin 119860 then 119866(119890) = 119865(119890)

119888

= 119865119888

(119890)if 119890 isin 119861 119860 then 119866(119890) = 119880 = 119865

119888

(119890) This implies that(119865 119860)

119888

= (119865119888

119890 isin 119860 | 119865(119890) = 119880119888

) = (119866 119861) finishing theproof

The following fact follows immediately fromProposition 20

Corollary 21 For any soft set (119865 119860) over 119880 we have that

((119865 119860)119888

)119888

= (119865 119860) (13)

Proof Note that (119865 119860)119888

cap(119865 119860) = (0 0) and (119865 119860)119888

cup(119865 119860) =

(119880 119864) It therefore follows from Proposition 20 that (119865 119860) =

((119865 119860)119888

)119888 as desired

With the above corollary we can prove the De Morganrsquoslaws of soft sets

Proposition 22 (De Morganrsquos laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) ((119865 119860) cap (119866 119861))119888

= (119865 119860)119888

cup (119866 119861)119888

(2) ((119865 119860) cup (119866 119861))119888

= (119865 119860)119888

cap (119866 119861)119888

Proof (1) For convenience let 1198600

= 119890 isin 119860 | 119865(119890) = 1198801198610

= 119890 isin 119861 | 119866(119890) = 119880 1198620

= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 119880and 119862

1= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 0 Then we have that

((119865 119860) cap (119866 119861))119888

= (119865 cap 119866 1198621)119888

= ((119865 cap 119866)119888

119890 isin 1198621

| (119865 cap 119866) (119890) = 119880119888

)

= ((119865 cap 119866)119888

119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 119880119888

)

= ((119865 cap 119866)119888

119862119888

0)

(14)

On the other hand we have that

(119865 119860)119888

cup (119866 119861)119888

= (119865119888

119860119888

0) cup (119866

119888

119861119888

0)

= (119865119888

cup 119866119888

119860119888

0cup 119861119888

0)

= (119865119888

cup 119866119888

(1198600

cap 1198610)119888

)

= (119865119888

cup 119866119888

119862119888

0)

(15)

Therefore to prove (1) it suffices to show that (119865 cap 119866)119888

(119890) =

(119865119888

cup119866119888

)(119890) for all 119890 isin 119862119888

0 In fact since119862

119888

0= (11986211198620)cup119862119888

1and

(11986211198620)cap119862119888

1= 0 we need only to consider two casesThefirst

case is that 119890 isin 11986211198620 In this case 119890 isin 119860

119888

0cap119861119888

0 and thuswe get

that (119865cap119866)119888

(119890) = (119865(119890)cap119866(119890))119888

= 119865(119890)119888

cup119866(119890)119888

= (119865119888

cup119866119888

)(119890)The other case is that 119890 isin 119862

119888

1 In this case we always have by

definition that (119865 cap 119866)119888

(119890) = 119880 = (119865119888

cup 119866119888

)(119890) Consequently(119865 cap 119866)

119888

(119890) = (119865119888

cup 119866119888

)(119890) for all 119890 isin 119862119888

0 as desired

(2) By Corollary 21 and the first assertion we find that

((119865 119860) cup (119866 119861))119888

= (((119865 119860)119888

)119888

cup ((119866 119861)119888

)119888

)119888

= (((119865 119860)119888

cap (119866 119861)119888

)119888

)119888

= (119865 119860)119888

cap (119866 119861)119888

(16)

Hence the second assertion holds as well This completes theproof of the proposition

Let us end this section with an observation on thedifference of two soft sets

Proposition 23 For any soft sets (119865 119860) and (119866 119861) over 119880 wehave that

(119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888

(17)

Proof We set 1198610

= 119890 isin 119861 | 119866(119890) = 119880 and write (119865119866 119862) for(119865 119860) (119866 119861) Then we see that 119862 = 119860 119890 isin 119860 cap 119861 | 119865(119890) sube

6 Journal of Applied Mathematics

119866(119890) and (119866 119861)119888

= (119866119888

119861119888

0) As a result (119865 119860) cap (119866 119861)

119888

=

(119865 119860) cap (G119888 1198611198880) = (119865 cap 119866

119888

1198611) where

1198611

= 119890 isin 119860 cap 119861119888

0| 119865 (119890) cap 119866

119888

(119890) = 0

= (119860 119861) cup 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119862

(18)

as desired It remains to show that (119865 119866)(119890) = (119865 cap 119866119888

)(119890)

for all 119890 isin 119862 = 1198611 In fact if 119890 isin 119862 119861 then we have that

(119865 119866)(119890) = 119865(119890) = 119865(119890) cap 119880 = (119865 cap 119866119888

)(119890) if 119890 isin 119862 cap 119861then (119865 119866)(119890) = 119865(119890) 119866(119890) = 119865(119890) cap 119866

119888

(119890) = (119865 cap 119866119888

)(119890)We thus get that (119865 119866)(119890) = (119865 cap 119866

119888

)(119890) for all 119890 isin 119862 = 1198611

Consequently (119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888 finishing the

proof

5 Conclusion

In this paper we have redefined the intersection comple-ment and difference of soft sets These operations togetherwith an existing union operation form the fundamentaloperations for constructing new soft sets from given soft setsBy examining the algebraic properties of these operations wefind that all basic properties of operations on classical setssuch as identity laws domination laws distributive laws andDe Morganrsquos laws hold for the newly defined operations onsoft sets From this point of view the new operations on softsets are reasonable Motivated by the notion of Not set of aparameter set in [22] we will investigate the operations onsoft sets by introducing more conjunctions including ANDand OR into a parameter set In addition it is interesting toextend the notions of intersection complement difference ofsoft sets developed here to other soft structures such as fuzzysoft sets [29 41] vague soft sets [28] and soft rough sets [42]

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61070251 61170270 and61121061 and the Fundamental Research Funds for theCentralUniversities underGrant 2012RC0710The authorswould liketo thank the reviewers for their helpful suggestions

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] M I Ali ldquoA note on soft sets rough soft sets and fuzzy soft setsrdquoApplied Soft Computing Journal vol 11 no 4 pp 3329ndash33322011

[4] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing vol 12 no 6 pp 1814ndash1821 2012

[5] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[6] S Alkhazaleh and A R Salleh ldquoSoft expert setsrdquo Advances inDecision Sciences vol 2011 Article ID 757868 12 pages 2011

[7] S Alkhazaleh and A R Salleh ldquoGeneralised interval-valuedfuzzy soft setrdquo Journal of AppliedMathematics vol 2012 ArticleID 870504 18 pages 2012

[8] K V Babitha and J J Sunil ldquoTransitive closures and orderingon soft setsrdquo Computers amp Mathematics with Applications vol62 no 5 pp 2235ndash2239 2011

[9] T Deng and X Wang ldquoParameter significance and reductionsof soft setsrdquo International Journal of ComputerMathematics vol89 no 15 pp 1979ndash1995 2012

[10] F Feng Y B Jun and X Zhao ldquoSoft semiringsrdquo Computers ampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[11] X Ge Z Li and Y Ge ldquoTopological spaces and soft setsrdquoJournal of Computational Analysis and Applications vol 13 no5 pp 881ndash885 2011

[12] A A Hazaymeh I B Abdullah Z T Balkhi and R IIbrahim ldquoGeneralized fuzzy soft expert setrdquo Journal of AppliedMathematics vol 2012 Article ID 328195 22 pages 2012

[13] Y Jiang Y Tang Q Chen and Z Cao ldquoSemantic operationsof multiple soft sets under conflictrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1923ndash1939 2011

[14] Y Jiang Y Tang Q Chen JWang and S Tang ldquoExtending softsets with description logicsrdquo Computers amp Mathematics withApplications vol 59 no 6 pp 2087ndash2096 2010

[15] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[16] Y B Jun and S S Ahn ldquoDouble-framed soft sets with appli-cations in BCKBCI-algebrasrdquo Journal of Applied Mathematicsvol 2012 Article ID 178159 15 pages 2012

[17] Y B Jun K J Lee and A Khan ldquoSoft ordered semigroupsrdquoMathematical Logic Quarterly vol 56 no 1 pp 42ndash50 2010

[18] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun S Z Song and K S So ldquoSoft set theory appliedto p-ideals of BCI-algebras related to fuzzy pointsrdquo NeuralComputing and Applications vol 20 no 8 pp 1313ndash1320 2011

[21] Z Kong L Gao L Wang and S Li ldquoThe normal parameterreduction of soft sets and its algorithmrdquo Computers amp Mathe-matics with Applications vol 56 no 12 pp 3029ndash3037 2008

[22] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[23] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[24] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[25] J H Park O H Kim and Y C Kwun ldquoSome properties ofequivalence soft set relationsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1079ndash1088 2012

[26] B Tanay and M B Kandemir ldquoTopological structure of fuzzysoft setsrdquo Computers amp Mathematics with Applications vol 61no 10 pp 2952ndash2957 2011

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Proposition 12 (Identity laws) For any soft set (119865 119860) over 119880we have that

(1) (119865 119860) cap (119880 119864) = (119865 119860)

(2) (119865 119860) cup (0 0) = (119865 119860)

Proposition 13 (Domination laws) For any soft set (119865 119860)

over 119880 we have that

(1) (119865 119860) cap (0 0) = (0 0)

(2) (119865 119860) cup (119880 119864) = (119880 119864)

Proposition 14 (Idempotent laws) For any soft set (119865 119860) over119880 we have that

(1) (119865 119860) cap (119865 119860) = (119865 119860)

(2) (119865 119860) cup (119865 119860) = (119865 119860)

Proposition 15 (Commutative laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)

(2) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)

Now we turn our attention to the associative laws

Proposition 16 (Associative laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) ((119865 119860) cap (119866B)) cap (119867 119862) = (119865 119860) cap ((119866 119861) cap (119867 119862))

(2) ((119865 119860) cup (119866 119861)) cup (119867 119862) = (119865 119860) cup ((119866 119861) cup (119867 119862))

Proof We only prove the first assertion since the secondone is the same as Proposition 25(i) in [22] For simplicitywe write (119871 119860

1015840

) (119877 1198611015840

) and (119865 cap 119866 1198601) for ((119865 119860) cap

(119866 119861))cap(119867 119862) (119865 119860)cap((119866 119861)cap(119867 119862)) and (119865 119860)cap(119866 119861)respectively We thus get by definition that

1198601015840

= 119890 isin 1198601

cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 1198601

| (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) = 0 (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

cap 119890 isin 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | (119865 cap 119866) (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) cap 119867 (119890) = 0

(10)

By the same token we have that 1198611015840

= 119890 isin 119860 cap 119861 cap 119862 | 119865(119890) cap

119866(119890) cap 119867(119890) = 0 and thus 1198601015840

= 1198611015840 Moreover for any 119890 isin 119860

1015840we have that

119871 (119890) = (119865 cap 119866) (119890) cap 119867 (119890)

= 119865 (119890) cap 119866 (119890) cap 119867 (119890)

= 119865 (119890) cap (119866 (119890) cap 119867 (119890))

= 119865 (119890) cap (119866 cap 119867) (119890)

= 119877 (119890)

(11)

namely 119871(119890) = 119877(119890) Therefore the assertion (1) holds

Proposition 17 (Distributive laws) For any soft sets (119865 119860)(119866 119861) and (119867 119862) over 119880 we have that

(1) (119865 119860)cap((119866 119861)cup(119867 119862)) = ((119865 119860)cap(119866 119861))cup((119865 119860)cap

(119867 119862))(2) (119865 119860)cup((119866 119861)cap(119867 119862)) = ((119865 119860)cup(119866 119861))cap((119865 119860)cup

(119867 119862))

Proof We only verify the first assertion the second one canbe verified similarly For simplicity we write (119871 119860

1015840

) and(119877 1198611015840

) for (119865 119860) cap ((119866 119861) cup (119867 119862)) and ((119865 119860) cap (119866 119861)) cup

((119865 119860) cap (119867 119862)) respectively We thus see that

1198601015840

= 119890 isin 119860 cap (119861 cup 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin (119860 cap 119861) cup (119860 cap 119862) | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap (119866 cup 119867) (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap (119866 (119890) cup 119867 (119890)) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 cap 119862119888

| 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119861119888

cap 119862 | 119865 (119890) cap 119867 (119890) = 0

cup 119890 isin 119860 cap 119861 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 0

cup 119890 isin 119860 cap 119862 | 119865 (119890) cap 119867 (119890) = 0

= 1198611015840

(12)

namely 1198601015840 = 1198611015840 Furthermore for any 119890 isin 119860

1015840 one can checkthat 119871(119890) = 119865(119890) cap (119866 cup 119867)(119890) = ((119865 cap 119866) cup (119865 cap 119867))(119890) = 119877(119890)

by a routine computation We do not go into the details hereHence the assertion (1) holds

Like usual sets soft sets are monotonic with respect tointersection and union

Journal of Applied Mathematics 5

Proposition 18 Let (119865119894 119860119894) and (119866

119894 119861119894) 119894 = 1 2 be soft sets

over 119880 If (119865119894 119860119894) sube (119866

119894 119861119894) 119894 = 1 2 then we have that

(1) (1198651 1198601) cap (119865

2 1198602) sube (119866

1 1198611) cap (119866

2 1198612)

(2) (1198651 1198601) cup (119865

2 1198602) sube (119866

1 1198611) cup (119866

2 1198612)

Proof It is clear by the definitions of intersection union andsubset of soft sets

Recall that in classical set theory we have that 119883 sube 119884 ifand only if 119883 cap 119884 = 119883 which is also equivalent to 119883 cup 119884 = 119884For soft sets we have the following observation

Proposition 19 Let (119865 119860) and (119866 119861) be soft sets over119880Thenthe following are equivalent

(1) (119865 119860) sube (119866 119861)(2) (119865 119860) cap (119866 119861) = (119865 119860)(3) (119865 119860) cup (119866 119861) = (119866 119861)

Proof Again it is obvious by the definitions of intersectionunion and subset of soft sets

The following several properties are concerned with thecomplement of soft sets

Proposition 20 Let (119865 119860) and (119866 119861) be two soft sets over 119880Then (119866 119861) = (119865 119860)

119888 if and only if (119865 119860) cap (119866 119861) = (0 0) and(119865 119860) cup (119866 119861) = (119880 119864)

Proof If (119866 119861) = (119865 119860)119888 then we see by definition that

(119865 119860)cap(119865 119860)119888

= (0 0) and (119865 119860)cup(119865 119860)119888

= (119865 119860)cup(119865119888

119890 isin

119860 | 119865(119890) = 119880119888

) = (119880 119864) Whence the necessity is trueConversely assume that (119865 119860) cap (119866 119861) = (0 0) and

(119865 119860) cup (119866 119861) = (119880 119864) The latter means that 119860 cup 119861 = 119864Moreover we obtain that 119865(119890) = 119880 for all 119890 isin 119860 119861 and119866(119890) = 119880 for all 119890 isin 119861 119860 For any 119890 isin 119860 cap 119861 it followsfrom (119865 119860) cap (119866 119861) = (0 0) and (F 119860) cup (119866 119861) = (119880 119864) that119865(119890) cup 119866(119890) = 119880 and 119865(119890) cap 119866(119890) = 0 As neither 119865(119890) nor119866(119890) is empty this forces that 119861 = 119890 isin 119860 | 119865(119890) = 119880

119888For any 119890 isin 119861 if 119890 isin 119860 then 119866(119890) = 119865(119890)

119888

= 119865119888

(119890)if 119890 isin 119861 119860 then 119866(119890) = 119880 = 119865

119888

(119890) This implies that(119865 119860)

119888

= (119865119888

119890 isin 119860 | 119865(119890) = 119880119888

) = (119866 119861) finishing theproof

The following fact follows immediately fromProposition 20

Corollary 21 For any soft set (119865 119860) over 119880 we have that

((119865 119860)119888

)119888

= (119865 119860) (13)

Proof Note that (119865 119860)119888

cap(119865 119860) = (0 0) and (119865 119860)119888

cup(119865 119860) =

(119880 119864) It therefore follows from Proposition 20 that (119865 119860) =

((119865 119860)119888

)119888 as desired

With the above corollary we can prove the De Morganrsquoslaws of soft sets

Proposition 22 (De Morganrsquos laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) ((119865 119860) cap (119866 119861))119888

= (119865 119860)119888

cup (119866 119861)119888

(2) ((119865 119860) cup (119866 119861))119888

= (119865 119860)119888

cap (119866 119861)119888

Proof (1) For convenience let 1198600

= 119890 isin 119860 | 119865(119890) = 1198801198610

= 119890 isin 119861 | 119866(119890) = 119880 1198620

= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 119880and 119862

1= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 0 Then we have that

((119865 119860) cap (119866 119861))119888

= (119865 cap 119866 1198621)119888

= ((119865 cap 119866)119888

119890 isin 1198621

| (119865 cap 119866) (119890) = 119880119888

)

= ((119865 cap 119866)119888

119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 119880119888

)

= ((119865 cap 119866)119888

119862119888

0)

(14)

On the other hand we have that

(119865 119860)119888

cup (119866 119861)119888

= (119865119888

119860119888

0) cup (119866

119888

119861119888

0)

= (119865119888

cup 119866119888

119860119888

0cup 119861119888

0)

= (119865119888

cup 119866119888

(1198600

cap 1198610)119888

)

= (119865119888

cup 119866119888

119862119888

0)

(15)

Therefore to prove (1) it suffices to show that (119865 cap 119866)119888

(119890) =

(119865119888

cup119866119888

)(119890) for all 119890 isin 119862119888

0 In fact since119862

119888

0= (11986211198620)cup119862119888

1and

(11986211198620)cap119862119888

1= 0 we need only to consider two casesThefirst

case is that 119890 isin 11986211198620 In this case 119890 isin 119860

119888

0cap119861119888

0 and thuswe get

that (119865cap119866)119888

(119890) = (119865(119890)cap119866(119890))119888

= 119865(119890)119888

cup119866(119890)119888

= (119865119888

cup119866119888

)(119890)The other case is that 119890 isin 119862

119888

1 In this case we always have by

definition that (119865 cap 119866)119888

(119890) = 119880 = (119865119888

cup 119866119888

)(119890) Consequently(119865 cap 119866)

119888

(119890) = (119865119888

cup 119866119888

)(119890) for all 119890 isin 119862119888

0 as desired

(2) By Corollary 21 and the first assertion we find that

((119865 119860) cup (119866 119861))119888

= (((119865 119860)119888

)119888

cup ((119866 119861)119888

)119888

)119888

= (((119865 119860)119888

cap (119866 119861)119888

)119888

)119888

= (119865 119860)119888

cap (119866 119861)119888

(16)

Hence the second assertion holds as well This completes theproof of the proposition

Let us end this section with an observation on thedifference of two soft sets

Proposition 23 For any soft sets (119865 119860) and (119866 119861) over 119880 wehave that

(119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888

(17)

Proof We set 1198610

= 119890 isin 119861 | 119866(119890) = 119880 and write (119865119866 119862) for(119865 119860) (119866 119861) Then we see that 119862 = 119860 119890 isin 119860 cap 119861 | 119865(119890) sube

6 Journal of Applied Mathematics

119866(119890) and (119866 119861)119888

= (119866119888

119861119888

0) As a result (119865 119860) cap (119866 119861)

119888

=

(119865 119860) cap (G119888 1198611198880) = (119865 cap 119866

119888

1198611) where

1198611

= 119890 isin 119860 cap 119861119888

0| 119865 (119890) cap 119866

119888

(119890) = 0

= (119860 119861) cup 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119862

(18)

as desired It remains to show that (119865 119866)(119890) = (119865 cap 119866119888

)(119890)

for all 119890 isin 119862 = 1198611 In fact if 119890 isin 119862 119861 then we have that

(119865 119866)(119890) = 119865(119890) = 119865(119890) cap 119880 = (119865 cap 119866119888

)(119890) if 119890 isin 119862 cap 119861then (119865 119866)(119890) = 119865(119890) 119866(119890) = 119865(119890) cap 119866

119888

(119890) = (119865 cap 119866119888

)(119890)We thus get that (119865 119866)(119890) = (119865 cap 119866

119888

)(119890) for all 119890 isin 119862 = 1198611

Consequently (119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888 finishing the

proof

5 Conclusion

In this paper we have redefined the intersection comple-ment and difference of soft sets These operations togetherwith an existing union operation form the fundamentaloperations for constructing new soft sets from given soft setsBy examining the algebraic properties of these operations wefind that all basic properties of operations on classical setssuch as identity laws domination laws distributive laws andDe Morganrsquos laws hold for the newly defined operations onsoft sets From this point of view the new operations on softsets are reasonable Motivated by the notion of Not set of aparameter set in [22] we will investigate the operations onsoft sets by introducing more conjunctions including ANDand OR into a parameter set In addition it is interesting toextend the notions of intersection complement difference ofsoft sets developed here to other soft structures such as fuzzysoft sets [29 41] vague soft sets [28] and soft rough sets [42]

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61070251 61170270 and61121061 and the Fundamental Research Funds for theCentralUniversities underGrant 2012RC0710The authorswould liketo thank the reviewers for their helpful suggestions

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] M I Ali ldquoA note on soft sets rough soft sets and fuzzy soft setsrdquoApplied Soft Computing Journal vol 11 no 4 pp 3329ndash33322011

[4] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing vol 12 no 6 pp 1814ndash1821 2012

[5] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[6] S Alkhazaleh and A R Salleh ldquoSoft expert setsrdquo Advances inDecision Sciences vol 2011 Article ID 757868 12 pages 2011

[7] S Alkhazaleh and A R Salleh ldquoGeneralised interval-valuedfuzzy soft setrdquo Journal of AppliedMathematics vol 2012 ArticleID 870504 18 pages 2012

[8] K V Babitha and J J Sunil ldquoTransitive closures and orderingon soft setsrdquo Computers amp Mathematics with Applications vol62 no 5 pp 2235ndash2239 2011

[9] T Deng and X Wang ldquoParameter significance and reductionsof soft setsrdquo International Journal of ComputerMathematics vol89 no 15 pp 1979ndash1995 2012

[10] F Feng Y B Jun and X Zhao ldquoSoft semiringsrdquo Computers ampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[11] X Ge Z Li and Y Ge ldquoTopological spaces and soft setsrdquoJournal of Computational Analysis and Applications vol 13 no5 pp 881ndash885 2011

[12] A A Hazaymeh I B Abdullah Z T Balkhi and R IIbrahim ldquoGeneralized fuzzy soft expert setrdquo Journal of AppliedMathematics vol 2012 Article ID 328195 22 pages 2012

[13] Y Jiang Y Tang Q Chen and Z Cao ldquoSemantic operationsof multiple soft sets under conflictrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1923ndash1939 2011

[14] Y Jiang Y Tang Q Chen JWang and S Tang ldquoExtending softsets with description logicsrdquo Computers amp Mathematics withApplications vol 59 no 6 pp 2087ndash2096 2010

[15] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[16] Y B Jun and S S Ahn ldquoDouble-framed soft sets with appli-cations in BCKBCI-algebrasrdquo Journal of Applied Mathematicsvol 2012 Article ID 178159 15 pages 2012

[17] Y B Jun K J Lee and A Khan ldquoSoft ordered semigroupsrdquoMathematical Logic Quarterly vol 56 no 1 pp 42ndash50 2010

[18] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun S Z Song and K S So ldquoSoft set theory appliedto p-ideals of BCI-algebras related to fuzzy pointsrdquo NeuralComputing and Applications vol 20 no 8 pp 1313ndash1320 2011

[21] Z Kong L Gao L Wang and S Li ldquoThe normal parameterreduction of soft sets and its algorithmrdquo Computers amp Mathe-matics with Applications vol 56 no 12 pp 3029ndash3037 2008

[22] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[23] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[24] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[25] J H Park O H Kim and Y C Kwun ldquoSome properties ofequivalence soft set relationsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1079ndash1088 2012

[26] B Tanay and M B Kandemir ldquoTopological structure of fuzzysoft setsrdquo Computers amp Mathematics with Applications vol 61no 10 pp 2952ndash2957 2011

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Proposition 18 Let (119865119894 119860119894) and (119866

119894 119861119894) 119894 = 1 2 be soft sets

over 119880 If (119865119894 119860119894) sube (119866

119894 119861119894) 119894 = 1 2 then we have that

(1) (1198651 1198601) cap (119865

2 1198602) sube (119866

1 1198611) cap (119866

2 1198612)

(2) (1198651 1198601) cup (119865

2 1198602) sube (119866

1 1198611) cup (119866

2 1198612)

Proof It is clear by the definitions of intersection union andsubset of soft sets

Recall that in classical set theory we have that 119883 sube 119884 ifand only if 119883 cap 119884 = 119883 which is also equivalent to 119883 cup 119884 = 119884For soft sets we have the following observation

Proposition 19 Let (119865 119860) and (119866 119861) be soft sets over119880Thenthe following are equivalent

(1) (119865 119860) sube (119866 119861)(2) (119865 119860) cap (119866 119861) = (119865 119860)(3) (119865 119860) cup (119866 119861) = (119866 119861)

Proof Again it is obvious by the definitions of intersectionunion and subset of soft sets

The following several properties are concerned with thecomplement of soft sets

Proposition 20 Let (119865 119860) and (119866 119861) be two soft sets over 119880Then (119866 119861) = (119865 119860)

119888 if and only if (119865 119860) cap (119866 119861) = (0 0) and(119865 119860) cup (119866 119861) = (119880 119864)

Proof If (119866 119861) = (119865 119860)119888 then we see by definition that

(119865 119860)cap(119865 119860)119888

= (0 0) and (119865 119860)cup(119865 119860)119888

= (119865 119860)cup(119865119888

119890 isin

119860 | 119865(119890) = 119880119888

) = (119880 119864) Whence the necessity is trueConversely assume that (119865 119860) cap (119866 119861) = (0 0) and

(119865 119860) cup (119866 119861) = (119880 119864) The latter means that 119860 cup 119861 = 119864Moreover we obtain that 119865(119890) = 119880 for all 119890 isin 119860 119861 and119866(119890) = 119880 for all 119890 isin 119861 119860 For any 119890 isin 119860 cap 119861 it followsfrom (119865 119860) cap (119866 119861) = (0 0) and (F 119860) cup (119866 119861) = (119880 119864) that119865(119890) cup 119866(119890) = 119880 and 119865(119890) cap 119866(119890) = 0 As neither 119865(119890) nor119866(119890) is empty this forces that 119861 = 119890 isin 119860 | 119865(119890) = 119880

119888For any 119890 isin 119861 if 119890 isin 119860 then 119866(119890) = 119865(119890)

119888

= 119865119888

(119890)if 119890 isin 119861 119860 then 119866(119890) = 119880 = 119865

119888

(119890) This implies that(119865 119860)

119888

= (119865119888

119890 isin 119860 | 119865(119890) = 119880119888

) = (119866 119861) finishing theproof

The following fact follows immediately fromProposition 20

Corollary 21 For any soft set (119865 119860) over 119880 we have that

((119865 119860)119888

)119888

= (119865 119860) (13)

Proof Note that (119865 119860)119888

cap(119865 119860) = (0 0) and (119865 119860)119888

cup(119865 119860) =

(119880 119864) It therefore follows from Proposition 20 that (119865 119860) =

((119865 119860)119888

)119888 as desired

With the above corollary we can prove the De Morganrsquoslaws of soft sets

Proposition 22 (De Morganrsquos laws) For any soft sets (119865 119860)

and (119866 119861) over 119880 we have that

(1) ((119865 119860) cap (119866 119861))119888

= (119865 119860)119888

cup (119866 119861)119888

(2) ((119865 119860) cup (119866 119861))119888

= (119865 119860)119888

cap (119866 119861)119888

Proof (1) For convenience let 1198600

= 119890 isin 119860 | 119865(119890) = 1198801198610

= 119890 isin 119861 | 119866(119890) = 119880 1198620

= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 119880and 119862

1= 119890 isin 119860 cap 119861 | 119865(119890) cap 119866(119890) = 0 Then we have that

((119865 119860) cap (119866 119861))119888

= (119865 cap 119866 1198621)119888

= ((119865 cap 119866)119888

119890 isin 1198621

| (119865 cap 119866) (119890) = 119880119888

)

= ((119865 cap 119866)119888

119890 isin 119860 cap 119861 | 119865 (119890) cap 119866 (119890) = 119880119888

)

= ((119865 cap 119866)119888

119862119888

0)

(14)

On the other hand we have that

(119865 119860)119888

cup (119866 119861)119888

= (119865119888

119860119888

0) cup (119866

119888

119861119888

0)

= (119865119888

cup 119866119888

119860119888

0cup 119861119888

0)

= (119865119888

cup 119866119888

(1198600

cap 1198610)119888

)

= (119865119888

cup 119866119888

119862119888

0)

(15)

Therefore to prove (1) it suffices to show that (119865 cap 119866)119888

(119890) =

(119865119888

cup119866119888

)(119890) for all 119890 isin 119862119888

0 In fact since119862

119888

0= (11986211198620)cup119862119888

1and

(11986211198620)cap119862119888

1= 0 we need only to consider two casesThefirst

case is that 119890 isin 11986211198620 In this case 119890 isin 119860

119888

0cap119861119888

0 and thuswe get

that (119865cap119866)119888

(119890) = (119865(119890)cap119866(119890))119888

= 119865(119890)119888

cup119866(119890)119888

= (119865119888

cup119866119888

)(119890)The other case is that 119890 isin 119862

119888

1 In this case we always have by

definition that (119865 cap 119866)119888

(119890) = 119880 = (119865119888

cup 119866119888

)(119890) Consequently(119865 cap 119866)

119888

(119890) = (119865119888

cup 119866119888

)(119890) for all 119890 isin 119862119888

0 as desired

(2) By Corollary 21 and the first assertion we find that

((119865 119860) cup (119866 119861))119888

= (((119865 119860)119888

)119888

cup ((119866 119861)119888

)119888

)119888

= (((119865 119860)119888

cap (119866 119861)119888

)119888

)119888

= (119865 119860)119888

cap (119866 119861)119888

(16)

Hence the second assertion holds as well This completes theproof of the proposition

Let us end this section with an observation on thedifference of two soft sets

Proposition 23 For any soft sets (119865 119860) and (119866 119861) over 119880 wehave that

(119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888

(17)

Proof We set 1198610

= 119890 isin 119861 | 119866(119890) = 119880 and write (119865119866 119862) for(119865 119860) (119866 119861) Then we see that 119862 = 119860 119890 isin 119860 cap 119861 | 119865(119890) sube

6 Journal of Applied Mathematics

119866(119890) and (119866 119861)119888

= (119866119888

119861119888

0) As a result (119865 119860) cap (119866 119861)

119888

=

(119865 119860) cap (G119888 1198611198880) = (119865 cap 119866

119888

1198611) where

1198611

= 119890 isin 119860 cap 119861119888

0| 119865 (119890) cap 119866

119888

(119890) = 0

= (119860 119861) cup 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119862

(18)

as desired It remains to show that (119865 119866)(119890) = (119865 cap 119866119888

)(119890)

for all 119890 isin 119862 = 1198611 In fact if 119890 isin 119862 119861 then we have that

(119865 119866)(119890) = 119865(119890) = 119865(119890) cap 119880 = (119865 cap 119866119888

)(119890) if 119890 isin 119862 cap 119861then (119865 119866)(119890) = 119865(119890) 119866(119890) = 119865(119890) cap 119866

119888

(119890) = (119865 cap 119866119888

)(119890)We thus get that (119865 119866)(119890) = (119865 cap 119866

119888

)(119890) for all 119890 isin 119862 = 1198611

Consequently (119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888 finishing the

proof

5 Conclusion

In this paper we have redefined the intersection comple-ment and difference of soft sets These operations togetherwith an existing union operation form the fundamentaloperations for constructing new soft sets from given soft setsBy examining the algebraic properties of these operations wefind that all basic properties of operations on classical setssuch as identity laws domination laws distributive laws andDe Morganrsquos laws hold for the newly defined operations onsoft sets From this point of view the new operations on softsets are reasonable Motivated by the notion of Not set of aparameter set in [22] we will investigate the operations onsoft sets by introducing more conjunctions including ANDand OR into a parameter set In addition it is interesting toextend the notions of intersection complement difference ofsoft sets developed here to other soft structures such as fuzzysoft sets [29 41] vague soft sets [28] and soft rough sets [42]

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61070251 61170270 and61121061 and the Fundamental Research Funds for theCentralUniversities underGrant 2012RC0710The authorswould liketo thank the reviewers for their helpful suggestions

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] M I Ali ldquoA note on soft sets rough soft sets and fuzzy soft setsrdquoApplied Soft Computing Journal vol 11 no 4 pp 3329ndash33322011

[4] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing vol 12 no 6 pp 1814ndash1821 2012

[5] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[6] S Alkhazaleh and A R Salleh ldquoSoft expert setsrdquo Advances inDecision Sciences vol 2011 Article ID 757868 12 pages 2011

[7] S Alkhazaleh and A R Salleh ldquoGeneralised interval-valuedfuzzy soft setrdquo Journal of AppliedMathematics vol 2012 ArticleID 870504 18 pages 2012

[8] K V Babitha and J J Sunil ldquoTransitive closures and orderingon soft setsrdquo Computers amp Mathematics with Applications vol62 no 5 pp 2235ndash2239 2011

[9] T Deng and X Wang ldquoParameter significance and reductionsof soft setsrdquo International Journal of ComputerMathematics vol89 no 15 pp 1979ndash1995 2012

[10] F Feng Y B Jun and X Zhao ldquoSoft semiringsrdquo Computers ampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[11] X Ge Z Li and Y Ge ldquoTopological spaces and soft setsrdquoJournal of Computational Analysis and Applications vol 13 no5 pp 881ndash885 2011

[12] A A Hazaymeh I B Abdullah Z T Balkhi and R IIbrahim ldquoGeneralized fuzzy soft expert setrdquo Journal of AppliedMathematics vol 2012 Article ID 328195 22 pages 2012

[13] Y Jiang Y Tang Q Chen and Z Cao ldquoSemantic operationsof multiple soft sets under conflictrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1923ndash1939 2011

[14] Y Jiang Y Tang Q Chen JWang and S Tang ldquoExtending softsets with description logicsrdquo Computers amp Mathematics withApplications vol 59 no 6 pp 2087ndash2096 2010

[15] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[16] Y B Jun and S S Ahn ldquoDouble-framed soft sets with appli-cations in BCKBCI-algebrasrdquo Journal of Applied Mathematicsvol 2012 Article ID 178159 15 pages 2012

[17] Y B Jun K J Lee and A Khan ldquoSoft ordered semigroupsrdquoMathematical Logic Quarterly vol 56 no 1 pp 42ndash50 2010

[18] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun S Z Song and K S So ldquoSoft set theory appliedto p-ideals of BCI-algebras related to fuzzy pointsrdquo NeuralComputing and Applications vol 20 no 8 pp 1313ndash1320 2011

[21] Z Kong L Gao L Wang and S Li ldquoThe normal parameterreduction of soft sets and its algorithmrdquo Computers amp Mathe-matics with Applications vol 56 no 12 pp 3029ndash3037 2008

[22] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[23] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[24] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[25] J H Park O H Kim and Y C Kwun ldquoSome properties ofequivalence soft set relationsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1079ndash1088 2012

[26] B Tanay and M B Kandemir ldquoTopological structure of fuzzysoft setsrdquo Computers amp Mathematics with Applications vol 61no 10 pp 2952ndash2957 2011

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

119866(119890) and (119866 119861)119888

= (119866119888

119861119888

0) As a result (119865 119860) cap (119866 119861)

119888

=

(119865 119860) cap (G119888 1198611198880) = (119865 cap 119866

119888

1198611) where

1198611

= 119890 isin 119860 cap 119861119888

0| 119865 (119890) cap 119866

119888

(119890) = 0

= (119860 119861) cup 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119860 119890 isin 119860 cap 119861 | 119865 (119890) sube 119866 (119890)

= 119862

(18)

as desired It remains to show that (119865 119866)(119890) = (119865 cap 119866119888

)(119890)

for all 119890 isin 119862 = 1198611 In fact if 119890 isin 119862 119861 then we have that

(119865 119866)(119890) = 119865(119890) = 119865(119890) cap 119880 = (119865 cap 119866119888

)(119890) if 119890 isin 119862 cap 119861then (119865 119866)(119890) = 119865(119890) 119866(119890) = 119865(119890) cap 119866

119888

(119890) = (119865 cap 119866119888

)(119890)We thus get that (119865 119866)(119890) = (119865 cap 119866

119888

)(119890) for all 119890 isin 119862 = 1198611

Consequently (119865 119860) (119866 119861) = (119865 119860) cap (119866 119861)119888 finishing the

proof

5 Conclusion

In this paper we have redefined the intersection comple-ment and difference of soft sets These operations togetherwith an existing union operation form the fundamentaloperations for constructing new soft sets from given soft setsBy examining the algebraic properties of these operations wefind that all basic properties of operations on classical setssuch as identity laws domination laws distributive laws andDe Morganrsquos laws hold for the newly defined operations onsoft sets From this point of view the new operations on softsets are reasonable Motivated by the notion of Not set of aparameter set in [22] we will investigate the operations onsoft sets by introducing more conjunctions including ANDand OR into a parameter set In addition it is interesting toextend the notions of intersection complement difference ofsoft sets developed here to other soft structures such as fuzzysoft sets [29 41] vague soft sets [28] and soft rough sets [42]

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grants 61070251 61170270 and61121061 and the Fundamental Research Funds for theCentralUniversities underGrant 2012RC0710The authorswould liketo thank the reviewers for their helpful suggestions

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] H Aktas and N Cagman ldquoSoft sets and soft groupsrdquo Informa-tion Sciences vol 177 no 13 pp 2726ndash2735 2007

[3] M I Ali ldquoA note on soft sets rough soft sets and fuzzy soft setsrdquoApplied Soft Computing Journal vol 11 no 4 pp 3329ndash33322011

[4] M I Ali ldquoAnother view on reduction of parameters in soft setsrdquoApplied Soft Computing vol 12 no 6 pp 1814ndash1821 2012

[5] M I Ali M Shabir and M Naz ldquoAlgebraic structures of softsets associated with new operationsrdquoComputers ampMathematicswith Applications vol 61 no 9 pp 2647ndash2654 2011

[6] S Alkhazaleh and A R Salleh ldquoSoft expert setsrdquo Advances inDecision Sciences vol 2011 Article ID 757868 12 pages 2011

[7] S Alkhazaleh and A R Salleh ldquoGeneralised interval-valuedfuzzy soft setrdquo Journal of AppliedMathematics vol 2012 ArticleID 870504 18 pages 2012

[8] K V Babitha and J J Sunil ldquoTransitive closures and orderingon soft setsrdquo Computers amp Mathematics with Applications vol62 no 5 pp 2235ndash2239 2011

[9] T Deng and X Wang ldquoParameter significance and reductionsof soft setsrdquo International Journal of ComputerMathematics vol89 no 15 pp 1979ndash1995 2012

[10] F Feng Y B Jun and X Zhao ldquoSoft semiringsrdquo Computers ampMathematics with Applications vol 56 no 10 pp 2621ndash26282008

[11] X Ge Z Li and Y Ge ldquoTopological spaces and soft setsrdquoJournal of Computational Analysis and Applications vol 13 no5 pp 881ndash885 2011

[12] A A Hazaymeh I B Abdullah Z T Balkhi and R IIbrahim ldquoGeneralized fuzzy soft expert setrdquo Journal of AppliedMathematics vol 2012 Article ID 328195 22 pages 2012

[13] Y Jiang Y Tang Q Chen and Z Cao ldquoSemantic operationsof multiple soft sets under conflictrdquo Computers amp Mathematicswith Applications vol 62 no 4 pp 1923ndash1939 2011

[14] Y Jiang Y Tang Q Chen JWang and S Tang ldquoExtending softsets with description logicsrdquo Computers amp Mathematics withApplications vol 59 no 6 pp 2087ndash2096 2010

[15] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[16] Y B Jun and S S Ahn ldquoDouble-framed soft sets with appli-cations in BCKBCI-algebrasrdquo Journal of Applied Mathematicsvol 2012 Article ID 178159 15 pages 2012

[17] Y B Jun K J Lee and A Khan ldquoSoft ordered semigroupsrdquoMathematical Logic Quarterly vol 56 no 1 pp 42ndash50 2010

[18] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun S Z Song and K S So ldquoSoft set theory appliedto p-ideals of BCI-algebras related to fuzzy pointsrdquo NeuralComputing and Applications vol 20 no 8 pp 1313ndash1320 2011

[21] Z Kong L Gao L Wang and S Li ldquoThe normal parameterreduction of soft sets and its algorithmrdquo Computers amp Mathe-matics with Applications vol 56 no 12 pp 3029ndash3037 2008

[22] P KMaji R Biswas and A R Roy ldquoSoft set theoryrdquoComputersamp Mathematics with Applications vol 45 no 4-5 pp 555ndash5622003

[23] P Majumdar and S K Samanta ldquoGeneralised fuzzy soft setsrdquoComputers amp Mathematics with Applications vol 59 no 4 pp1425ndash1432 2010

[24] W K Min ldquoSimilarity in soft set theoryrdquo Applied MathematicsLetters vol 25 no 3 pp 310ndash314 2012

[25] J H Park O H Kim and Y C Kwun ldquoSome properties ofequivalence soft set relationsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1079ndash1088 2012

[26] B Tanay and M B Kandemir ldquoTopological structure of fuzzysoft setsrdquo Computers amp Mathematics with Applications vol 61no 10 pp 2952ndash2957 2011

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

[27] Z Xiao K Gong S Xia and Y Zou ldquoExclusive disjunctive softsetsrdquo Computers amp Mathematics with Applications vol 59 no6 pp 2128ndash2137 2010

[28] W Xu J Ma S Wang and G Hao ldquoVague soft sets and theirpropertiesrdquo Computers amp Mathematics with Applications vol59 no 2 pp 787ndash794 2010

[29] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[30] Y Yin H Li and Y B Jun ldquoOn algebraic structure ofintuitionistic fuzzy soft setsrdquo Computers amp Mathematics withApplications vol 64 no 9 pp 2896ndash2911 2012

[31] P Zhu andQWen ldquoProbabilistic soft setsrdquo in Proceedings of theIEEE Conference on Granular Computing (GrCrsquo10) pp 635ndash638IEEE Press San Jose Calif USA August 2010

[32] N Cagman S Enginoglu and F Citak ldquoFuzzy soft set theoryand its applicationsrdquo Iranian Journal of Fuzzy Systems vol 8no 3 pp 137ndash147 2011

[33] T Herawan and M M Deris ldquoOn multi-soft sets constructionin information systemsrdquo in Emerging Intelligent ComputingTechnology and Applications with Aspects of Artificial Intelli-gence 5th International Conference on Intelligent Computing(ICICrsquo09) vol 5755 of Lecture Notes in Computer Science pp101ndash110 Springer Ulsan Republic of Korea September 2009

[34] T Herawan A N M Rose and M M Deris ldquoSoft set theoreticapproach for dimensionality reductionrdquo in Proceedings of theDatabase Theory and Application International Conference(DTArsquo09) pp 171ndash178 Springer Jeju Island Republic of KoreaDecember 2009

[35] M M Mushrif S Sengupta and A K Ray ldquoTexture clas-sification using a novel soft-set theory based classificationalgorithmrdquo in Computer VisionmdashACCV 2006 vol 3851 ofLecture Notes in Computer Science pp 246ndash254 2006

[36] Y Zou and Z Xiao ldquoData analysis approaches of soft sets underincomplete informationrdquo Knowledge-Based Systems vol 21 no8 pp 941ndash945 2008

[37] T M Basu N K Mahapatra and S K Mondal ldquoA balancedsolution of a fuzzy soft set based decision making problem inmedical sciencerdquo Applied Soft Computing vol 12 no 10 pp3260ndash3275 2012

[38] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[39] F Feng Y Li and N Cagman ldquoGeneralized uni-int decisionmaking schemes based on choice value soft setsrdquo EuropeanJournal of Operational Research vol 220 no 1 pp 162ndash170 2012

[40] P K Maji A R Roy and R Biswas ldquoAn application of soft setsin a decision making problemrdquo Computers ampMathematics withApplications vol 44 no 8-9 pp 1077ndash1083 2002

[41] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[42] F Feng and X Liu ldquoSoft rough sets with applications todemand analysisrdquo in Proceedings of the International Workshopon Intelligent Systems and Applications (ISArsquo09) pp 1ndash4 IEEEWuhan China May 2009

[43] H Qin X Ma J M Zain and T Herawan ldquoA novel softset approach in selecting clustering attributerdquoKnowledge-BasedSystems vol 36 pp 139ndash145 2012

[44] Z Xiao KGong andY Zou ldquoA combined forecasting approachbased on fuzzy soft setsrdquo Journal of Computational and AppliedMathematics vol 228 no 1 pp 326ndash333 2009

[45] C-F Yang ldquoA note on ldquoSoft set theoryrdquo [Computers ampMathematics with Applications 45 (2003) no 4-5 555ndash562]rdquoComputers amp Mathematics with Applications vol 56 no 7 pp1899ndash1900 2008

[46] M I Ali F Feng X Liu W K Min and M Shabir ldquoOn somenew operations in soft set theoryrdquo Computers amp Mathematicswith Applications vol 57 no 9 pp 1547ndash1553 2009

[47] A Sezgin and A O Atagun ldquoOn operations of soft setsrdquoComputers amp Mathematics with Applications vol 61 no 5 pp1457ndash1467 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of