Information Systems and Soft Sets

Information Systems and Soft Sets

Zofia Machnicka and Marek Palasinski University of Information Technology and Management

Rzeszow, Poland

[email protected] [email protected]

Abstract. It will be shown that each information system can be considered a soft set and eachfinitesoft set can be considered an information system.

1 Basic Information

The notion of an information system is well established in the literature, e.g. [1, 4]. In- formally, an information system consists of a finite nonempty set ofobjectsand a finite nonempty set of attributes. Each attributeaassigns to each object some valuea(x), which is a members of a specific finite setV_a, called therangeof the attributea. Thus an attribute can be thought of as a functiona:V →Va, or as a sequence of elements ofVa. Such a tuple is represented as a column in a matrix (or table) representing the information system. The rows of the matrix are labelled by objects and the columns by attributes. The matrix dimention isn×p, wherenis the number of objects andpthe number of attribues, see Table 1 below.

Table 1.Information system a0 · · · a_k · · · a_p x0 a00 · · · ak0 · · · ap0

· · · · xi a0i · · · aki · · · api

· · · · xn a0n · · · akn · · · apn

HereV ={x0, . . . , xn},A ={a0, , ap}, andakiis the value of the attributeak

onxi, i.e.,aki=ak(xi).Thus for eacha∈A, we have thata∈V_a^V andA⊆S {V_a^V : a ∈ A}, where the union is disjoint. Similarly, as each objectx∈ V labels a row in the matrix, it can be identified with a tuple of values of attributes fromAon this object.

Hence the set of objectsV can be identified with a subset of the Cartesian product of setsV_a(see [4])

V ⊆ Y

a∈A

Va.

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Definition 1. An information system is a pair S = hV, Ai, where V and A are nonempty finite sets, such that for eacha ∈ Athere exists a finite set Va such that a:V →Va.

The elements ofV are called objects of the systemSand the elements ofAattributes of the system. An elementx∈V is identified with a tuple

x=ha(x) :a∈Ai.

An information systemSis calledtwo valuediff for everya∈ A, the setV_ahas two elements, which we denote by0and1. Expressions of the form

ai₁=bi₁∧ · · · ∧aik=bik−→ait =bit

or

a_i1=b_i1∧ · · · ∧a_ik=b_ik−→a_it 6=b_it

whereai₁. . . aik, ait are attributes andbi₁, . . . , bik, bit are their possible values, i.e., bij ∈Vij, are calledrules. A rule of the first form is called adeterministic association rulewhile the one of the second form is called aninhibitory association rule. Mathe- maticaly, any rule is an implication and atruerule is a rule that is true as an implication for any object from V. A rule isrealizable if its predecessor is true for at least one object from V.It was shown in [2] that every information system can be equivalently replaced by a two valued information system. We recall the procedure in Section3.

2 Soft Sets

The notion of asoft sethas been proposed in [3]as a mathematical approach to uncer- tainty, alternative to that of a fuzzy set. It has subsequently provoked a lot of research.

LetUbe a set called auniverseand letEbe another set, disjoint withU, called the set ofparameters.

Definition 2. (Following [3]) LetU be a set. A pairhF, Eiis called a soft set overU if and only ifFis a mapping fromEinto the set of all subsets of the setU.

A soft set then can be seen as a parametrized family{F(ε) :ε∈E}of subsets of the setU and the elements ofE are called parameters. In the terminology of [3], for eachε∈E, the setF(ε)is called anε-approximationof the soft set. If bothU andE are finite then we will call the soft sethF, Eifinite. In the next section we show that there is a natural connection between soft sets and information systems.

3 Connection Between Soft Sets and Information Systems

The following procedure of getting a two valued systemS⁽²⁾from a given information system S was presented at the HSI conference in 2010 ([2]) and we recall it here for completeness. Let S = hV, Ai, where A = (a1, . . . , an)and for eachi = 1, . . . , n

71 let the set of values of the attribute ai be{0, . . . , ki −1}. We define the associated two-valued systemS⁽²⁾by setting its attributes set to be

{a10, . . . , a_1ki−1, . . . , a_n0, . . . , a_nkn−1},

each one assuming one of the two values0,1. The set of objects remains the same.

For any rule r for S we define the corresponding ruler⁽²⁾ for S⁽²⁾, replacing each expression of the formai=jbyaij = 1andaij = 0in other case and in case the rule is an inhibitory one expressionai6=jbyaij = 0. We then have the following lemma.

Lemma 1. For any ruler, ifris true and realizable inSthenr⁽²⁾is true and realizable inS⁽²⁾.

The proof of the lemma is straightforward, by contradiction. To show the converse, one needs to define new rules (each rule true and realizable inS², using the inverse translation, leads to a rule of a new kind ).

Remark 1. In case of binary information systems, if two systemsS₁andS₂are different then the sets of true and realizable implications associated withS₁andS₂, respectively, are different (see [1, 2]).

From now on letS⁽²⁾denote a binary information system. SoS⁽²⁾can be presented as in Table 1 above, where the valueakiof the attributeakonxiis0or1.

It is now easy to see that each information systemhV, Aican be considered a soft set. It is enough to take into account that any subsetW ⊆V can be uniquely identified with its characteristic functionf :V → {0,1}such that

f(x) =

1, whenx∈W 0, otherwise.

Fork = 1, . . . , p, thekth column in the table above can be considered a function ak :V → {0,1}, withak(xi) =aki, for eachj= 1, . . . , n, so each column determines a subset ofV. Take the functionF : A → P(V)such thatF(ak)is the subset ofV determined by the column in the table corresponding to the attributea_k ∈ A. Then hF, Aiis a soft set overU.

On the other hand, consider a finite soft sethF, Eiover a universeV. Assume that V = {x0, . . . , x_n}andE = {ε0, . . . , ε_p}, for somenandp. To show that this soft set determines the unique information system, takeA = {F(ε) : ε ∈ E}. For each k = 0, . . . , pandi = 0, . . . , nleta_k be the characteristic function of the setF(εi).

Then the values of ak form thek-th column in table 1. So the soft sethF, Ei over V is identified with the information systemhV, Ai. Clearly, the two identifications are mutual inverses, so each soft set determines a unique information system and vice versa.

This allows us to state the main result of this note.

Theorem 1. Given a finite setV there is a one-one correspondence between the infor- mation systems of the formhV, Aiand finite soft sets over the universeV.

As the finite setV in the theorem is arbitrary, the theorem can be restated as

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Corollary 1. Every information system can be regarded as a finite soft set and every finite soft set can be regarded as an information system.

Remark 2. As the anonymous referee points out, the main result can be alternatively explained in more general terms as follows. Given a familyAof zero-one sequences of a common lengthn, one can treat this family as the set of parametersE. LetUbe any set ofnelements. Then each parameterε∈E, being a sequence of zeros and ones is a characteristic function of some subset ofU, call this subsetF(ε). ThenF:E→ P(U) and the pairhF, Eiis a soft set. The original set of sequencesAcan be recovered from this soft set as the family of characteristic sequences of setsF(ε), for allε∈E.

References

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3. D. Molodtsov,Soft set theory - First results, Comput. Math. Appl., 37, pp 19-31 (1999) 4. K.Pancerz,Zastosowanie zbiorów przybli˙zonych do identyfikacji modeli systemów współ

bie˙znych,Rozprawa doktorska, IPI PAN, Warszawa (2006)