Cosine Similarity Measure of Interval Valued Neutrosophic Sets
Research Article Operations on Soft Sets Revisitedtheory, fuzzy sets, and interval mathematics is...
Transcript of Research Article Operations on Soft Sets Revisitedtheory, fuzzy sets, and interval mathematics is...
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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 105752, 7 pageshttp://dx.doi.org/10.1155/2013/105752
Research ArticleOperations on Soft Sets Revisited
Ping Zhu1,2 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; [email protected]
Received 14 November 2012; Accepted 7 January 2013
Academic Editor: Han H. Choi
Copyright Β© 2013 P. Zhu and Q. Wen. This is an open access article distributed under the Creative Commons Attribution License,which 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 introduction,the concept of soft sets has gained considerable attention(see, e.g., [2β31]), including some successful applications ininformation processing [32β36], decision [37β41], 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 Morganβs 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 DeMorganβs 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 end,we 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 sets,including identity laws, domination laws, idempotent laws,commutative laws, associative laws, distributive laws, and DeMorganβs laws, hold for soft sets with respect to the newlydefined operations.
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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 sets.An 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 theory,the symbols 0, π΄π, π΄ βͺ π΅, π΄ β© π΅, π΄ \ π΅ denote, respectively,the empty set, the complement of π΄ with respect to someuniversal set π, the union of sets π΄ and π΅, the intersectionof sets π΄ and π΅, and the difference of sets π΄ and π΅ whoseelements belong to π΄ but not to π΅. In what follows, we writeP(π) for the power set of a universal set π and denoteP(π) \ {0} byPβ(π).
Throughout this paper, let π be a universal set and πΈ bethe set of all possible parameters under consideration withrespect to π. Usually, parameters are attributes, characteris-tics, or properties of objects in π. We now recall the notionof soft sets due to Molodtsov [1].
Definition 1 (seeMolodtsov [1]). Letπ be a universe andπΈ theset of parameters. A soft set over π is a pair (πΉ, π΄) consistingof a subset π΄ of πΈ and a mapping πΉ : π΄ β Pβ(π).
Note that the above definition is slightly different fromthe original one in [1] where πΉ has P(π) as its codomain.In other words, we remove the parameters having the emptyset as images under πΉ. It seems rational since this meansthat if there exists a parameter π β π΄ which is not theattribute, characteristic, or property of any object in π, thenthis parameter has no interest with respect to the knowledgestored in the soft set. As a result, a soft set of π in the sense ofDefinition 1 is a parameterized family of nonempty subsets ofπ.
Clearly, any soft set (πΉ, π΄) over π gives a partial functionπΉ
: πΈ β Pβ(π) defined by
πΉ
(π) = {πΉ (π) , if π β π΄,undefined, otherwise,
(1)
for all π β πΈ. Conversely, any partial function π from πΈ toPβ(π) gives rise to a soft set (πΉ
π, π΄π), where π΄
π= {π β πΈ |
π(π) is defined} and πΉπis the restriction of π on π΄
π.
To illustrate the above definition, Molodtsov consideredseveral examples in [1], one of which we present blow.
Example 2. Suppose that π is the set of houses underconsideration, say π = {β
1, β2, . . . , β
5}. Let πΈ be the set of
some attributes of such houses, say πΈ = {π1, π2, . . . , π
8}, where
π1, π2, . . . , π
8stand for the attributes βexpensive,β βbeautiful,β
βwooden,β βcheap,β βin the green surroundings,β βmodern,ββin good repair,β and βin bad repair,β respectively.
In this case, to define a soft set means to point outexpensive houses, beautiful houses, and so on. For example,the soft set (πΉ, π΄) that describes the βattractiveness of thehousesβ in the opinion of a buyer, say Alice, may be definedas:
π΄ = {π2, π3, π4, π5, π7} ;
πΉ (π2) = {β
2, β3, β5} , πΉ (π
3) = {β
2, β4} ,
πΉ (π4) = {β
1} , πΉ (π
5) = π, πΉ (π
7) = {β
3, β5} .
(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 (πΉ, π΄) is called a null soft setif πΉ(π) = 0 for all π β π΄. Because 0 does not belong to thecodomain of πΉ in our framework, we redefine the concept ofempty soft set as follows.
Definition 3. A soft set (πΉ, π΄) over π is said to be emptywhenever π΄ = 0. Symbolically, we write (0, 0) for the emptysoft set over π.
Definition 4. A soft set (πΉ, π΄) over π is called a universal softset if π΄ = πΈ and πΉ(π) = π for all π β π΄. Symbolically, we write(π, πΈ) for the universal soft set over π.
Let us now define the subsets of a soft set.
Definition 5. Let (πΉ, π΄) and (πΊ, π΅) be two soft sets over π. Wesay that (πΉ, π΄) is a subset of (πΊ, π΅), denoted (πΉ, π΄) β (πΊ, π΅)if either (πΉ, π΄) = (0, 0) or π΄ β π΅ and πΉ(π) β πΊ(π) for everyπ β π΄. Two soft sets (πΉ, π΄) and (πΊ, π΅) are said to be equal,denoted (πΉ, π΄) = (πΊ, π΅), if (πΉ, π΄) β (πΊ, π΅) and (πΊ, π΅) β (πΉ, π΄).
By definition, two soft sets (πΉ, π΄) and (πΊ, π΅) are equal ifand only if π΄ = π΅ and πΉ(π) = πΊ(π) for all π β π΄. In [22],a similar notion, called soft subset, was defined by requiringthat π΄ β π΅ and πΉ(π) = πΊ(π) for every π β π΄. By Definition 5,the 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 (π, πΈ). Formally, we have the following proposition.
Proposition 6. For any soft set (πΉ, π΄) over π,
(0, 0) β (πΉ, π΄) β (π, πΈ) . (3)
We are now in the position to introduce some operationson soft sets.
Definition 7. Let (πΉ, π΄) and (πΊ, π΅) be two soft sets overπ.Theintersection of (πΉ, π΄) and (πΊ, π΅), denoted by (πΉ, π΄) β© (πΊ, π΅), isdefined as (πΉ β© πΊ, πΆ), where
πΆ = {π β π΄ β© π΅ | πΉ (π) β© πΊ (π) ΜΈ= 0} , βπ β πΆ,
(πΉ β© πΊ) (π) = πΉ (π) β© πΊ (π) .
(4)
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In particular, if π΄ β© π΅ = 0 or πΉ(π) β© πΊ(π) = 0 for everyπ β π΄ β© π΅, then we see that (πΉ, π΄) β© (πΊ, π΅) = (0, 0).
The following definition of union of soft sets is the sameas in [22].
Definition 8 (see [22], Definition 2.11). Let (πΉ, π΄) and (πΊ, π΅)be two soft sets over π. The union of (πΉ, π΄) and (πΊ, π΅),denoted by (πΉ, π΄) βͺ (πΊ, π΅), is defined as (πΉ βͺ πΊ, πΆ), where
πΆ = π΄ βͺ π΅, βπ β πΆ,
(πΉ βͺ πΊ) (π) =
{{
{{
{
πΉ (π) , if π β π΄ \ π΅πΊ (π) , if π β π΅ \ π΄πΉ (π) βͺ πΊ (π) , 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 (πΉ, π΄) be a soft set over π. The complementof (πΉ, π΄) with respect to the universal soft set (π, πΈ), denotedby (πΉ, π΄)π, is defined as (πΉπ, πΆ), where
πΆ = πΈ \ {π β π΄ | πΉ (π) = π}
= {π β π΄ | πΉ (π) = π}π
, βπ β πΆ,
πΉπ
(π) = {π \ πΉ (π) , if π β π΄π, otherwise.
(6)
In certain settings, the difference of two soft sets (πΉ, π΄)and (πΊ, π΅) is useful.
Definition 10. Let (πΉ, π΄) and (πΊ, π΅) be two soft sets over π.The difference of (πΉ, π΄) and (πΊ, π΅), denoted by (πΉ, π΄) \ (πΊ, π΅),is defined as (πΉ \ πΊ, πΆ), where
πΆ = π΄ \ {π β π΄ β© π΅ | πΉ (π) β πΊ (π)} , βπ β πΆ,
(πΉ \ πΊ) (π) = {πΉ (π) \ πΊ (π) , if π β π΄ β© π΅πΉ (π) , otherwise.
(7)
By Definitions 9 and 10, we find that (πΉ, π΄)π = (π, πΈ) \(πΉ, π΄) holds for any soft set (πΉ, π΄). That is, the complement of(πΉ, π΄)with respect to the universal soft set (π, πΈ) is exactly thedifference of (π, πΈ) and (πΉ, π΄). In light of this, (πΉ, π΄)\(πΊ, π΅) isalso called the relative complement of (πΊ, π΅) in (πΉ, π΄), while(πΉ, π΄)
π is also called the absolute complement of (πΉ, π΄).
Let us illustrate the previous operations on soft sets by asimple example.
Example 11. Let us revisit Example 2. Recall that the soft set(πΉ, π΄) describing the βattractiveness of the housesβ in Aliceβsopinion was defined by
π΄ = {π2, π3, π4, π5, π7} ;
πΉ (π2) = {β
2, β3, β5} , πΉ (π
3) = {β
2, β4} ,
πΉ (π4) = {β
1} , πΉ (π
5) = π, πΉ (π
7) = {β
3, β5} .
(8)
In addition, we assume that the βattractiveness of the housesβin the opinion of another buyer, say Bob, is described by thesoft set (πΊ, π΅), where
π΅ = {π1, π2, . . . , π
7} ;
πΊ (π1) = {β
3, β5} , πΊ (π
2) = {β
4} ,
πΊ (π3) = {β
2, β4} , πΊ (π
4) = {β
1} ,
πΊ (π5) = {β
2, β3, β4, β5} , πΊ (π
6) = πΊ (π
7) = {β
3} .
(9)
Then by a direct computation, we can readily obtain(πΉ, π΄) β© (πΊ, π΅), (πΉ, π΄) βͺ (πΊ, π΅), (πΉ, π΄)π, and (πΉ, π΄) \ (πΊ, π΅) asfollows.
(i) (πΉ, π΄) β© (πΊ, π΅) = (πΉ β© πΊ, {π3, π4, π5, π7}), where (πΉ β©
πΊ)(π3) = {β
2, β4}, (πΉ β© πΊ)(π
4) = {β
1}, (πΉ β© πΊ)(π
5) =
{β2, β3, β4, β5}, and (πΉ β© πΊ)(π
7) = {β
3}. This means
that both Alice and Bob think that β2and β
4are
wooden, β1is cheap, β
2, β3, β4, β5are in the green
surroundings, and β3is in the good repair.
(ii) (πΉ, π΄) βͺ (πΊ, π΅) = (πΉ βͺ πΊ, {π1, π2, . . . , π
7}), where (πΉ βͺ
πΊ)(π1) = {β
3, β5}, (πΉ βͺ πΊ)(π
2) = {β
2, β3, β4, β5}, (πΉ βͺ
πΊ)(π3) = {β
2, β4}, (πΉ βͺ πΊ)(π
4) = {β
1}, (πΉ βͺ πΊ)(π
5) =
π, (πΉ βͺ πΊ)(π6) = {β
3}, and (πΉ βͺ πΊ)(π
7) = {β
3, β5}. This
means that either Alice or Bob thinks that β3is expen-
sive, either Alice or Bob thinks that β5is expensive,
either Alice or Bob thinks that β2is beautiful, either
Alice or Bob thinks that β3is beautiful, and so on.
(iii) (πΉ, π΄)π = (πΉπ, {π1, π2, π3, π4, π6, π7, π8}), where πΉπ(π
1) =
π, πΉπ(π2) = {β
1, β4}, πΉπ(π
3) = {β
1, β3, β5}, πΉπ(π
4) =
{β2, β3, β4, β5}, πΉπ(π
6) = π, πΉπ(π
7) = {β
1, β2, β4}, and
πΉπ
(π8) = π. This means that Alice thinks that none
of these houses is expensive, neither β1nor β
4is
beautiful, β1, β3, β5are not wooden, and so on.
(iv) (πΉ, π΄)\(πΊ, π΅) = (πΉ\πΊ, {π2, π5, π7}), where (πΉ\πΊ)(π
2) =
{β2, β3, β5}, (πΉ \ πΊ)(π
5) = {β
1}, and (πΉ \ πΊ)(π
7) =
{β5}. This means that Alice thinks of β
2, β3, and β
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.
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Proposition 12 (Identity laws). For any soft set (πΉ, π΄) over π,we have that
(1) (πΉ, π΄) β© (π, πΈ) = (πΉ, π΄),
(2) (πΉ, π΄) βͺ (0, 0) = (πΉ, π΄).
Proposition 13 (Domination laws). For any soft set (πΉ, π΄)over π, we have that
(1) (πΉ, π΄) β© (0, 0) = (0, 0),
(2) (πΉ, π΄) βͺ (π, πΈ) = (π, πΈ).
Proposition 14 (Idempotent laws). For any soft set (πΉ, π΄) overπ, we have that
(1) (πΉ, π΄) β© (πΉ, π΄) = (πΉ, π΄),
(2) (πΉ, π΄) βͺ (πΉ, π΄) = (πΉ, π΄).
Proposition 15 (Commutative laws). For any soft sets (πΉ, π΄)and (πΊ, π΅) over π, we have that
(1) (πΉ, π΄) β© (πΊ, π΅) = (πΊ, π΅) β© (πΉ, π΄),
(2) (πΉ, π΄) βͺ (πΊ, π΅) = (πΊ, π΅) βͺ (πΉ, π΄).
Now we turn our attention to the associative laws.
Proposition 16 (Associative laws). For any soft sets (πΉ, π΄),(πΊ, π΅), and (π», πΆ) over π, we have that
(1) ((πΉ, π΄) β© (πΊ,B)) β© (π», πΆ) = (πΉ, π΄) β© ((πΊ, π΅) β© (π», πΆ)),
(2) ((πΉ, π΄) βͺ (πΊ, π΅)) βͺ (π», πΆ) = (πΉ, π΄) βͺ ((πΊ, π΅) βͺ (π», πΆ)).
Proof. We only prove the first assertion since the secondone is the same as Proposition 2.5(i) in [22]. For simplicity,we write (πΏ, π΄), (π , π΅), and (πΉ β© πΊ, π΄
1) for ((πΉ, π΄) β©
(πΊ, π΅))β©(π», πΆ), (πΉ, π΄)β©((πΊ, π΅)β©(π», πΆ)), and (πΉ, π΄)β©(πΊ, π΅),respectively. We thus get by definition that
π΄
= {π β π΄1
β© πΆ | (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
= {π β π΄1
| (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
β© {π β πΆ | (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
= {π β π΄ β© π΅ | (πΉ β© πΊ) (π) ΜΈ= 0, (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
β© {π β πΆ | (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
= {π β π΄ β© π΅ | (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
β© {π β πΆ | (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
= {π β π΄ β© π΅ β© πΆ | (πΉ β© πΊ) (π) β© π» (π) ΜΈ= 0}
= {π β π΄ β© π΅ β© πΆ | πΉ (π) β© πΊ (π) β© π» (π) ΜΈ= 0} .
(10)
By the same token, we have that π΅ = {π β π΄ β© π΅ β© πΆ | πΉ(π) β©πΊ(π) β© π»(π) ΜΈ= 0}, and thus π΄ = π΅. Moreover, for any π β π΄,we have that
πΏ (π) = (πΉ β© πΊ) (π) β© π» (π)
= πΉ (π) β© πΊ (π) β© π» (π)
= πΉ (π) β© (πΊ (π) β© π» (π))
= πΉ (π) β© (πΊ β© π») (π)
= π (π) ,
(11)
namely, πΏ(π) = π (π). Therefore, the assertion (1) holds.
Proposition 17 (Distributive laws). For any soft sets (πΉ, π΄),(πΊ, π΅), and (π», πΆ) over π, we have that
(1) (πΉ, π΄)β©((πΊ, π΅)βͺ(π», πΆ)) = ((πΉ, π΄)β©(πΊ, π΅))βͺ((πΉ, π΄)β©(π», πΆ)),
(2) (πΉ, π΄)βͺ((πΊ, π΅)β©(π», πΆ)) = ((πΉ, π΄)βͺ(πΊ, π΅))β©((πΉ, π΄)βͺ(π», πΆ)).
Proof. We only verify the first assertion; the second one canbe verified similarly. For simplicity, we write (πΏ, π΄) and(π , π΅
) for (πΉ, π΄) β© ((πΊ, π΅) βͺ (π», πΆ)) and ((πΉ, π΄) β© (πΊ, π΅)) βͺ((πΉ, π΄) β© (π», πΆ)), respectively. We thus see that
π΄
= {π β π΄ β© (π΅ βͺ πΆ) | πΉ (π) β© (πΊ βͺ π») (π) ΜΈ= 0}
= {π β (π΄ β© π΅) βͺ (π΄ β© πΆ) | πΉ (π) β© (πΊ βͺ π») (π) ΜΈ= 0}
= {π β π΄ β© π΅ | πΉ (π) β© (πΊ βͺ π») (π) ΜΈ= 0}
βͺ {π β π΄ β© πΆ | πΉ (π) β© (πΊ βͺ π») (π) ΜΈ= 0}
= {π β π΄ β© π΅ β© πΆπ
| πΉ (π) β© πΊ (π) ΜΈ= 0}
βͺ {π β π΄ β© π΅ β© πΆ | πΉ (π) β© (πΊ (π) βͺ π» (π)) ΜΈ= 0}
βͺ {π β π΄ β© π΅π
β© πΆ | πΉ (π) β© π» (π) ΜΈ= 0}
= {π β π΄ β© π΅ β© πΆπ
| πΉ (π) β© πΊ (π) ΜΈ= 0}
βͺ {π β π΄ β© π΅ β© πΆ | πΉ (π) β© πΊ (π) ΜΈ= 0}
βͺ {π β π΄ β© π΅π
β© πΆ | πΉ (π) β© π» (π) ΜΈ= 0}
βͺ {π β π΄ β© π΅ β© πΆ | πΉ (π) β© π» (π) ΜΈ= 0}
= {π β π΄ β© π΅ | πΉ (π) β© πΊ (π) ΜΈ= 0}
βͺ {π β π΄ β© πΆ | πΉ (π) β© π» (π) ΜΈ= 0}
= π΅
,
(12)
namely, π΄ = π΅. Furthermore, for any π β π΄, one can checkthat πΏ(π) = πΉ(π) β© (πΊ βͺ π»)(π) = ((πΉ β© πΊ) βͺ (πΉ β© π»))(π) = π (π)by a routine computation. We do not go into the details here.Hence, the assertion (1) holds.
Like usual sets, soft sets are monotonic with respect tointersection and union.
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Journal of Applied Mathematics 5
Proposition 18. Let (πΉπ, π΄π) and (πΊ
π, π΅π), π = 1, 2, be soft sets
over π. If (πΉπ, π΄π) β (πΊ
π, π΅π), π = 1, 2, then we have that
(1) (πΉ1, π΄1) β© (πΉ
2, π΄2) β (πΊ
1, π΅1) β© (πΊ
2, π΅2),
(2) (πΉ1, π΄1) βͺ (πΉ
2, π΄2) β (πΊ
1, π΅1) βͺ (πΊ
2, π΅2).
Proof. It is clear by the definitions of intersection, union, andsubset of soft sets.
Recall that in classical set theory, we have that π β π ifand only if π β© π = π, which is also equivalent to π βͺ π = π.For soft sets, we have the following observation.
Proposition 19. Let (πΉ, π΄) and (πΊ, π΅) be soft sets overπ.Thenthe following are equivalent:
(1) (πΉ, π΄) β (πΊ, π΅),(2) (πΉ, π΄) β© (πΊ, π΅) = (πΉ, π΄),(3) (πΉ, π΄) βͺ (πΊ, π΅) = (πΊ, π΅).
Proof. Again, it is obvious by the definitions of intersection,union, and subset of soft sets.
The following several properties are concerned with thecomplement of soft sets.
Proposition 20. Let (πΉ, π΄) and (πΊ, π΅) be two soft sets over π.Then (πΊ, π΅) = (πΉ, π΄)π if and only if (πΉ, π΄) β© (πΊ, π΅) = (0, 0) and(πΉ, π΄) βͺ (πΊ, π΅) = (π, πΈ).
Proof. If (πΊ, π΅) = (πΉ, π΄)π, then we see by definition that(πΉ, π΄)β©(πΉ, π΄)
π
= (0, 0) and (πΉ, π΄)βͺ(πΉ, π΄)π = (πΉ, π΄)βͺ(πΉπ, {π βπ΄ | πΉ(π) = π}
π
) = (π, πΈ). Whence, the necessity is true.Conversely, assume that (πΉ, π΄) β© (πΊ, π΅) = (0, 0) and
(πΉ, π΄) βͺ (πΊ, π΅) = (π, πΈ). The latter means that π΄ βͺ π΅ = πΈ.Moreover, we obtain that πΉ(π) = π for all π β π΄ \ π΅ andπΊ(π) = π for all π β π΅ \ π΄. For any π β π΄ β© π΅, it followsfrom (πΉ, π΄) β© (πΊ, π΅) = (0, 0) and (F, π΄) βͺ (πΊ, π΅) = (π, πΈ) thatπΉ(π) βͺ πΊ(π) = π and πΉ(π) β© πΊ(π) = 0. As neither πΉ(π) norπΊ(π) is empty, this forces that π΅ = {π β π΄ | πΉ(π) = π}π.For any π β π΅, if π β π΄, then πΊ(π) = πΉ(π)π = πΉπ(π);if π β π΅ \ π΄, then πΊ(π) = π = πΉπ(π). This implies that(πΉ, π΄)
π
= (πΉπ
, {π β π΄ | πΉ(π) = π}π
) = (πΊ, π΅), finishing theproof.
The following fact follows immediately fromProposition 20.
Corollary 21. For any soft set (πΉ, π΄) over π, we have that
((πΉ, π΄)π
)π
= (πΉ, π΄) . (13)
Proof. Note that (πΉ, π΄)πβ©(πΉ, π΄) = (0, 0) and (πΉ, π΄)πβͺ(πΉ, π΄) =(π, πΈ). It therefore follows from Proposition 20 that (πΉ, π΄) =((πΉ, π΄)
π
)π, as desired.
With the above corollary, we can prove the De Morganβslaws of soft sets.
Proposition 22 (De Morganβs laws). For any soft sets (πΉ, π΄)and (πΊ, π΅) over π, we have that
(1) ((πΉ, π΄) β© (πΊ, π΅))π = (πΉ, π΄)π βͺ (πΊ, π΅)π,
(2) ((πΉ, π΄) βͺ (πΊ, π΅))π = (πΉ, π΄)π β© (πΊ, π΅)π.
Proof. (1) For convenience, let π΄0
= {π β π΄ | πΉ(π) = π},π΅0
= {π β π΅ | πΊ(π) = π}, πΆ0
= {π β π΄ β© π΅ | πΉ(π) β© πΊ(π) = π},and πΆ
1= {π β π΄ β© π΅ | πΉ(π) β© πΊ(π) ΜΈ= 0}. Then we have that
((πΉ, π΄) β© (πΊ, π΅))π
= (πΉ β© πΊ, πΆ1)π
= ((πΉ β© πΊ)π
, {π β πΆ1
| (πΉ β© πΊ) (π) = π}π
)
= ((πΉ β© πΊ)π
, {π β π΄ β© π΅ | πΉ (π) β© πΊ (π) = π}π
)
= ((πΉ β© πΊ)π
, πΆπ
0) .
(14)
On the other hand, we have that
(πΉ, π΄)π
βͺ (πΊ, π΅)π
= (πΉπ
, π΄π
0) βͺ (πΊ
π
, π΅π
0)
= (πΉπ
βͺ πΊπ
, π΄π
0βͺ π΅π
0)
= (πΉπ
βͺ πΊπ
, (π΄0
β© π΅0)π
)
= (πΉπ
βͺ πΊπ
, πΆπ
0) .
(15)
Therefore, to prove (1), it suffices to show that (πΉ β© πΊ)π(π) =(πΉπ
βͺπΊπ
)(π) for all π β πΆπ0. In fact, sinceπΆπ
0= (πΆ1\πΆ0)βͺπΆπ
1and
(πΆ1\πΆ0)β©πΆπ
1= 0, we need only to consider two cases.Thefirst
case is that π β πΆ1\πΆ0. In this case, π β π΄π
0β©π΅π
0, and thuswe get
that (πΉβ©πΊ)π(π) = (πΉ(π)β©πΊ(π))π = πΉ(π)πβͺπΊ(π)π = (πΉπβͺπΊπ)(π).The other case is that π β πΆπ
1. In this case, we always have by
definition that (πΉ β© πΊ)π(π) = π = (πΉπ βͺ πΊπ)(π). Consequently,(πΉ β© πΊ)
π
(π) = (πΉπ
βͺ πΊπ
)(π) for all π β πΆπ0, as desired.
(2) By Corollary 21 and the first assertion, we find that
((πΉ, π΄) βͺ (πΊ, π΅))π
= (((πΉ, π΄)π
)π
βͺ ((πΊ, π΅)π
)π
)π
= (((πΉ, π΄)π
β© (πΊ, π΅)π
)π
)π
= (πΉ, π΄)π
β© (πΊ, π΅)π
.
(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 (πΉ, π΄) and (πΊ, π΅) over π, wehave that
(πΉ, π΄) \ (πΊ, π΅) = (πΉ, π΄) β© (πΊ, π΅)π
. (17)
Proof. We set π΅0
= {π β π΅ | πΊ(π) = π} and write (πΉ\πΊ, πΆ) for(πΉ, π΄) \ (πΊ, π΅). Then we see that πΆ = π΄ \ {π β π΄ β© π΅ | πΉ(π) β
-
6 Journal of Applied Mathematics
πΊ(π)} and (πΊ, π΅)π = (πΊπ, π΅π0). As a result, (πΉ, π΄) β© (πΊ, π΅)π =
(πΉ, π΄) β© (Gπ, π΅π0) = (πΉ β© πΊ
π
, π΅1), where
π΅1
= {π β π΄ β© π΅π
0| πΉ (π) β© πΊ
π
(π) ΜΈ= 0}
= (π΄ \ π΅) βͺ {π β π΄ β© π΅ | πΉ (π) ΜΈβ πΊ (π)}
= π΄ \ {π β π΄ β© π΅ | πΉ (π) β πΊ (π)}
= πΆ,
(18)
as desired. It remains to show that (πΉ \ πΊ)(π) = (πΉ β© πΊπ)(π)for all π β πΆ = π΅
1. In fact, if π β πΆ \ π΅, then we have that
(πΉ \ πΊ)(π) = πΉ(π) = πΉ(π) β© π = (πΉ β© πΊπ
)(π); if π β πΆ β© π΅,then (πΉ \ πΊ)(π) = πΉ(π) \ πΊ(π) = πΉ(π) β© πΊπ(π) = (πΉ β© πΊπ)(π).We thus get that (πΉ \ πΊ)(π) = (πΉ β© πΊπ)(π) for all π β πΆ = π΅
1.
Consequently, (πΉ, π΄) \ (πΊ, π΅) = (πΉ, π΄) β© (πΊ, π΅)π, finishing theproof.
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 sets.By 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 Morganβs 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 2012RC0710.The authorswould liketo thank the reviewers for their helpful suggestions.
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