=Paper=
{{Paper
|id=Vol-1492/Paper_31
|storemode=property
|title=Information Systems and Soft Sets
|pdfUrl=https://ceur-ws.org/Vol-1492/Paper_31.pdf
|volume=Vol-1492
|dblpUrl=https://dblp.org/rec/conf/csp/MachnickaP15
}}
==Information Systems and Soft Sets==
https://ceur-ws.org/Vol-1492/Paper_31.pdf
Information Systems and Soft Sets
Zofia Machnicka and Marek Palasinski
University of Information Technology and Management
Rzeszow, Poland
zmachnicka@wsiz.rzeszow.pl
mpalasinski@wsiz.rzeszow.pl
Abstract. It will be shown that each information system can be considered a soft
set and each finite soft 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 of objects and a finite
nonempty set of attributes. Each attribute a assigns to each object some value a(x),
which is a members of a specific finite set Va , called the range of the attribute a. Thus
an attribute can be thought of as a function a : V → Va , or as a sequence of elements
of Va . 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 is n × p, where n is the number of objects and p the
number of attribues, see Table 1 below.
Table 1. Information system
a0 · · · ak · · · ap
x0 a00 · · · ak0 · · · ap0
··· ··· ··· ··· ··· ···
xi a0i · · · aki · · · api
··· ··· ··· ··· ··· ···
xn a0n · · · akn · · · apn
Here V = {x0 , . . . , xn } , A = {a0 , , ap } , and aki is the value of the attribute ak
on xi , i.e., aki = ak (xi ). Thus for each a ∈ A, we have that a ∈ VaV and A ⊆ {VaV :
S
a ∈ A}, where the union is disjoint. Similarly, as each object x ∈ V labels a row in
the matrix, it can be identified with a tuple of values of attributes from A on this object.
Hence the set of objects V can be identified with a subset of the Cartesian product of
sets Va (see [4]) Y
V ⊆ Va .
a∈A
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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 each a ∈ A there exists a finite set Va such that
a : V → Va .
The elements of V are called objects of the system S and the elements of A attributes
of the system. An element x ∈ V is identified with a tuple
x = ha(x) : a ∈ Ai.
An information system S is called two valued iff for every a ∈ A, the set Va has two
elements, which we denote by 0 and 1. Expressions of the form
ai1 = bi1 ∧ · · · ∧ aik = bik −→ ait = bit
or
ai1 = bi1 ∧ · · · ∧ aik = bik −→ ait 6= bit
where ai1 . . . aik , ait are attributes and bi1 , . . . , bik , bit are their possible values, i.e.,
bij ∈ Vij , are called rules. A rule of the first form is called a deterministic association
rule while the one of the second form is called an inhibitory association rule. Mathe-
maticaly, any rule is an implication and a true rule is a rule that is true as an implication
for any object from V . A rule is realizable 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 a soft set has 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.
Let U be a set called a universe and let E be another set, disjoint with U , called the set
of parameters.
Definition 2. (Following [3]) Let U be a set. A pair hF, Ei is called a soft set over U
if and only if F is a mapping from E into the set of all subsets of the set U .
A soft set then can be seen as a parametrized family {F (ε) : ε ∈ E} of subsets of
the set U and the elements of E are called parameters. In the terminology of [3], for
each ε ∈ E, the set F (ε) is called an ε-approximation of the soft set. If both U and E
are finite then we will call the soft set hF, Ei finite. 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 system S (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 each i = 1, . . . , n
71
let the set of values of the attribute ai be {0, . . . , ki − 1}. We define the associated
two-valued system S (2) by setting its attributes set to be
{a10 , . . . , a1ki −1 , . . . , an0 , . . . , ankn −1 },
each one assuming one of the two values 0, 1. The set of objects remains the same.
For any rule r for S we define the corresponding rule r(2) for S (2) , replacing each
expression of the form ai = j by aij = 1 and aij = 0 in other case and in case the rule
is an inhibitory one expression ai 6= j by aij = 0. We then have the following lemma.
Lemma 1. For any rule r, if r is true and realizable in S then r(2) is true and realizable
in S (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 in S 2 , using the inverse
translation, leads to a rule of a new kind ).
Remark 1. In case of binary information systems, if two systems S1 and S2 are different
then the sets of true and realizable implications associated with S1 and S2 , respectively,
are different (see [1, 2]).
From now on let S (2) denote a binary information system. So S (2) can be presented as
in Table 1 above, where the value aki of the attribute ak on xi is 0 or 1.
It is now easy to see that each information system hV, Ai can be considered a soft
set. It is enough to take into account that any subset W ⊆V can be uniquely identified
with its characteristic function f : V → {0, 1} such that
1, when x ∈ W
f (x) =
0, otherwise.
For k = 1, . . . , p, the k th column in the table above can be considered a function
ak : V → {0, 1}, with ak (xi ) = aki , for each j = 1, . . . , n, so each column determines
a subset of V . Take the function F : A → P(V ) such that F (ak ) is the subset of V
determined by the column in the table corresponding to the attribute ak ∈ A. Then
hF, Ai is a soft set over U .
On the other hand, consider a finite soft set hF, Ei over a universe V . Assume that
V = {x0 , . . . , xn } and E = {ε0 , . . . , εp }, for some n and p. To show that this soft
set determines the unique information system, take A = {F (ε) : ε ∈ E}. For each
k = 0, . . . , p and i = 0, . . . , n let ak be the characteristic function of the set F (εi ).
Then the values of ak form the k-th column in table 1. So the soft set hF, Ei over
V is identified with the information system hV, 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 set V there is a one-one correspondence between the infor-
mation systems of the form hV, Ai and finite soft sets over the universe V .
As the finite set V in the theorem is arbitrary, the theorem can be restated as
72
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 family A of zero-one sequences of
a common length n, one can treat this family as the set of parameters E. Let U be any
set of n elements. Then each parameter ε ∈ E, being a sequence of zeros and ones is a
characteristic function of some subset of U , call this subset F (ε). Then F : E → P(U )
and the pair hF, Ei is a soft set. The original set of sequences A can be recovered from
this soft set as the family of characteristic sequences of sets F (ε), for all ε ∈ E.
References
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(2008)
2. B. Fryc, Z. Machnicka, M. PaÅĆasiÅĎski, Remarks on Two Valued Information Systems,
in: T. Pardela, B. Wilamowski (eds.) RzeszÃşw (3rd International Conference on Human
System Interaction), pp 775-777 (2010)
3. D. Molodtsov, Soft set theory - First results, Comput. Math. Appl., 37, pp 19-31 (1999)
4. K.Pancerz,Zastosowanie zbiorów przybliżonych do identyfikacji modeli systemów współ
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