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Applied Soft Computing 11 (2011) 3329–3332

Contents lists available at ScienceDirect

Applied Soft Computing

journa l homepage: www.e lsev ier .com/ locate /asoc

note on soft sets, rough soft sets and fuzzy soft sets

uhammad Irfan Aliepartment of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

r t i c l e i n f o

rticle history:eceived 21 May 2008

a b s t r a c t

Concept of an approximation space associated with each parameter in a soft set is discussed and anapproximation space associated with the soft set is defined. For any subset X of the universe U, there is

eceived in revised form 3 October 2010ccepted 3 January 2011vailable online 12 January 2011

eywords:oft sets

a fuzzy subset of U associated with each parameter of the soft set, also there is a fuzzy subset associatedwith the soft set. Furthermore each soft set over a set U, gives rise to a fuzzy soft set over P(U) and inducesa soft equivalence relation over P(U).

© 2011 Elsevier B.V. All rights reserved.

ough setsuzzy sets

. Introduction

Theory of Rough sets was introduced by Pawlak [15]. It is anxtension of the set theory to study ill posed data, that is it studieshe data with incomplete information. In rough set theory a subsetf the universe is described by two approximations, called lowernd upper approximations. The lower approximation of a set is thenion of all equivalence classes contained in the set, whereas thepper approximation is the union of those classes which have nonmpty intersection with the set. Equivalence classes are the basicuilding blocks in rough set theory, for upper and lower approxima-ions. A partition of a set induces an equivalence relation and viceersa. Therefore we can view the properties of a rough set eithery the partition of the set or by the equivalence relation. Applica-ions of rough sets in algebraic structures can be found in [2,5,8].ince the inception of rough set theory, theoretical enrichment andpplications are the main interest of researchers.

Fuzzy set theory was initiated by Zadeh in 1965 [16]. Dubois andrade studied both rough sets and fuzzy sets, to them these twoheories are different tools to deal uncertainty. These are not rivalheories, but can be combined in a fruitful way [6]. They defineduzzy rough sets and rough fuzzy sets. In [3] Chakrabarty et aliscussed fuzziness in rough sets. They introduced the concept ofeasure of fuzziness in rough sets.Theory of soft sets is introduced by Molodtsov [14]. This the-

ry is a relatively new approach to discuss vagueness. It is gettingopularity among the researchers and a good number of papers iseing published every year. In [11] Maji et al discussed theoreti-al aspect of soft sets and they introduced several operations for

E-mail address: mirfanali13@yahoo.com

568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2011.01.003

soft sets. Some applications of soft sets are discussed in [4,10,12].In [13] concept of fuzzy soft sets is introduced. Application of softsets in algebra was started by Aktas and Cagman [1]. Jun and Parkdiscussed soft ideal theory in BCK/BCI algebra [7].

In soft set theory membership is decided by adequate parame-ters, rough set theory employes equivalence classes, whereas fuzzyset theory depends upon grade of membership. Although three the-ories are quite distinct yet deal with vagueness. Joint application ofthese theories may result in a fruitful way. There is a natural ques-tion, “is there any relation in these three theories?” The relationshipbetween fuzzy set theory and rough set theory has been establishedin [6]. Here we endeavour to establish link between soft sets andfuzzy soft sets and soft rough sets. Finally we will see how theseconcepts can be helpful in decision making problems.

2. Preliminaries

Throughout in this paper U denotes a non empty finite set unlessstated otherwise. Molodtsov defined soft set as

Definition 1. [14] Let U be a non empty finite set and let E be a setof parameters. A pair (F, A) is called a soft set over U if F is a functionfrom a subset A of E to P(U) the power set of U. That is a soft set isa parameterized family of subsets of U.

Definition 2. [11] Let (F, A) and (G, B) be any two soft sets over U,then (G, B) is called a soft subset of (F, A) if

(1) B ⊆ A(2) G(ˇ) is a subset of F(ˇ) for all ˇ ∈ B.

Any subset of U × U is called a binary relation on U. If R is a binaryrelation on U, then R is called

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330 M. Irfan Ali / Applied Soft C

1) Reflexive if (x, x) ∈ R for all x ∈ U.2) Symmetric if (x, y) ∈ R ⇒ (y, x) ∈ R, for all x, y ∈ U.3) Transitive if (x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, for all x, y, z ∈ U.

A binary relation R is called an equivalence relation if it is reflex-ve, symmetric and transitive. Now we define an approximationpace in Pawlak’s sense.

efinition 3. [3] Let U be a set called universe and let R be anquivalence relation on U, called indiscernibility relation. The pairU, R) is called an approximation space.

efinition 4. A fuzzy subset of U is a function � from U to [0, 1].

. Roughness associated with soft sets

In the following we see that a soft set gives rise to an approxi-ation space in Pawlak’s sense.

efinition 5. Let (�, A) be a soft set over U × U. Then (�, A) is calledsoft binary relation over U.

In fact (�, A) is a parametrized collection of binary relations on, that is for each parameter ˛i ∈ A, we have a binary relation �(˛i)n U.

efinition 6. A soft binary relation (�, A) over a set U, is calledsoft equivalence relation over U, if �(˛i) /= ∅ is an equivalence

elation on U for all ˛i ∈ A.

It is well known that each equivalence relation on a set partitionshe set into disjoint classes and each partition of the set providess an equivalence relation on the set. Therefore a soft equivalenceelation over U, provides us a parametrized collection of partitionsf U. To elaborate this concept consider the following example.

xample 1. Let U ={

h1, h2, h3, h4, h5, h6}

be a set ofouses under consideration where E = {e1, e2, e3, e4, e5} and= {e1, e2, e3, e4} be a subset of parameters for selection of theouse. Let

1 stands for expensive houses,2 stands for wooden houses,3 stands for houses located in green surroundings,4 stands for houses located in the urban area,5 stands for the low cost houses.

Let (F, A) be the soft set to categorize the houses with respecto parameters given by set A, such that F(e1) =

{h1, h3

}, F(e2) =

h1, h3, h6}

, F(e3) ={

h1, h3, h4, h5}

, F(e4) ={

h1, h2, h3}

. Foromputer applications it is more appropriate to represent a softet in tabular form

h1 h2 h3 h4 h5 h6

e1 1 0 1 0 0 0e2 1 0 1 0 0 1e3 1 0 1 1 1 0e4 1 1 1 0 0 0

Now it is easy to see from the above table that each of the param-ter ei;i = 1, 2, 3, 4 induces an equivalence relation on U. So we havesoft equivalence relation say (�, A) over U. Hence we get the fol-

owing equivalence classes for each of the equivalence relation asollowing,

for �(e1) the equivalence classes are{

h1, h3}

,{

h2, h4, h5, h6}

,

for �(e ) the equivalence classes are{

h , h , h}

,{

h , h , h}

,

2 1 3 6 2 4 5

for �(e3) the equivalence classes are{

h1, h3, h4, h5}

,{

h2, h6}

,

for �(e4) the equivalence classes are{

h1, h2, h3}

,{

h4, h5, h6}

.We also observe that there is an indiscernibility relation defined

y the soft set (F, A) itself. This indiscernibility relation is obtained

ting 11 (2011) 3329–3332

by intersection of all the equivalence relations induced by param-eters. Let us say

IND(F, A) =⋂

ei ∈ A

�(ei) = �.

So the partition of U obtained by indiscernibility relation IND(F,A) is

{h1, h3

},{

h2}

,{

h4, h5}

,{

h6}

. It is evident that for each�(ei) where i = 1, 2, 3, 4, (U, �(ei)) give us an approximation spacesin Pawlak’s sense. Also (U, �) is an approximation space.

In view of the above example any subset X of U, can be approx-imated by the equivalence relation �(˛i). The equivalence classcontaining an element x ∈ U determined by the equivalence relation�(˛i) is denoted by [x]�(˛i)

. The parametrized collection of subsets

denoted by (�X, A) defined as �X (˛i) =⋃x ∈ X

{[x]�(˛i)

: [x]�(˛i)⊆ X

}for all ˛i ∈ A, is called soft lower approximation of X with respect tosoft equivalence relations (�, A).

The parametrized collection of subsets of U, denoted by (�̄X, A)

defined as �̄X (˛i) =⋃x ∈ X

{[x]�(˛i)

: [x]�(˛i)∩ X /= ∅

}for all ˛i ∈ A, is

called soft upper approximation of X with respect to soft equiva-lence relations (�, A).

The soft set (B�X, A) defined by B�X (˛i) = �̄X (˛i) − �X (˛i) forall ˛i ∈ A is called soft boundary region of X, with respect to softequivalence relation (�, A).

A subset X of U is called totally rough with respect to soft equiv-alence relation (�, A) if B�X(˛i) /= ∅ for all ˛i ∈ A.

A subset X of U is called partly rough or partly definable withrespect to soft equivalence relation (�, A) if B�X(˛i) =∅ for some˛ ∈ A.

A subset X of U is called totally definable with respect to softequivalence relation (�, A) if B�X(˛i) =∅ for all ˛i ∈ A.

4. Fuzziness associated with soft sets

In the last section we have seen that a soft set can be transformedinto a soft approximation space in Pawlak’s sense. In this section wediscuss for any subset X of U there is an associated fuzzy subset �i ofU for each ˛i ∈ A; further we can also find a fuzzy subset associatedwith a subset X of U for IND(F, A) = �.

As we know that each soft set (F, A) over U, induces a soft equiv-alence relation (�, A) over U. Now each �(˛i) where ˛i ∈ A, gives usa partition of U, for any X ⊆ U the soft rough set of X is denotedby

(�X (˛i), �̄X (˛i)

)where ˛i ∈ A. The set X is approximated by

two approximations, one the lower approximation, which is anapproximation from the inside, the other is the upper approxi-mation of X that is approximation from outside. The degree ofrough belongingness of each x ∈ U gives rise to a fuzzy subset foreach ˛i ∈ A, where the degree of rough belongingness is definedas (

∣∣{[x]�(˛i)∩ X

}∣∣/∣∣[x]�(˛i)

∣∣). It is obvious that degree of roughbelongingness is a number from the interval [0, 1]. Hence wecan define a fuzzy subset �i of U for each �(ei). Let �i : U → [0, 1]defined as �i(x) = (

∣∣{[x]�(˛i)∩ X

}∣∣/∣∣[x]�(˛i)

∣∣). Also we can find afuzzy subset � of U for IND(F, A) = �. It can be defined in the sim-ilar fashion as explained above that is � : U → [0, 1] defined as�(x) = (

∣∣{[x]� ∩ X}∣∣/

∣∣[x]�

∣∣).Definition 7. [13] Let U be an initial universe and E be a set ofparameters. Let FP(U) denotes the set of all fuzzy subsets of U. LetA ⊆ E. A pair (F, A) is called a fuzzy soft set over U, where F is a

mapping given by F : A → FP(U).

In the following example we see that for any subset of the uni-verse U there is a fuzzy subset of U for each parameter. Hence wehave a fuzzy soft set in the sense [13].

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M. Irfan Ali / Applied Soft C

xample 2. Let U ={

1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

and= {e1, e2, e3, e4, e5} be the set of parameters where e1 denotesivisibility by 2, e2 denotes divisibility by 3, e3 denotes divis-

bility by 4, e4 denotes divisibility by 5 and e5 denotes therimeness. If A = {e1, e2, e3, e4}, then the soft set (F, A) isiven by F(e1) =

{2, 4, 6, 8, 10

}, F(e2) =

{3, 6, 9

}, F(e3) =

4, 8}

, F(e4) ={

5, 10}

. The tabular representation of the soft setF, A) is

1 2 3 4 5 6 7 8 9 10

e1 0 1 0 1 0 1 0 1 0 1e2 0 0 1 0 0 1 0 0 1 0e3 0 0 0 1 0 0 0 1 0 0e4 0 0 0 0 1 0 0 0 0 1

It is clear from the Table 2 that the given soft set has induced aoft equivalence relation say (�, A) over U and equivalence classesetermined by each equivalence relation are as following

For �(e1) equivalence classes are{

1, 3, 5, 7, 9}

,{

2, 4, 6, 8, 10}

.

For �(e2) equivalence classes are{

1, 2, 4, 5, 7, 8, 10}

,{

3, 6, 9}

.

For �(e3) equivalence classes are{

1, 2, 3, 5, 6, 7, 9, 10}

,{

4, 8}

.

For �(e4) equivalence classes are{

1, 2, 3, 4, 6, 7, 8, 9}

,{

5, 10}

.

Further equivalence classes determined by IND(F, A) =4

i=1

�(ei) = � are{

1, 7}

,{

2}

,{

3, 9}

,{

4, 8}

,{

5}

,{

6}

,{

10}

.

et X ={

3, 4, 5, 6, 9}

then �X (e1) = ∅ and �̄X (e1) = U, �X (e2) =3, 6, 9

}and �̄X (e2) = U, �X (e3) = ∅ and �̄X (e3) = U,�X (e4) = ∅

nd �̄X (e4) = U. Now �X ={

3, 5, 6, 9}

and �̄X ={

3, 4, 5, 6, 8, 9}

.Therefore for each parameter ei where i = 1, 2, 3, 4 fuzzy subsets

f U for X ={

3, 4, 5, 6, 9}

are given below

For �(e1) the fuzzy subset �1 is0.61 , 0.4

2 , 0.63 , 0.4

4 , 0.65 , 0.4

6 , 0.67 , 0.4

8 , 0.69 , 0.4

10 }.For �(e2) the fuzzy subset �2 is

0.2851 , 0.285

2 , 13 , 0.285

4 , 0.2855 , 1

6 , 0.2857 , 0.285

8 , 19 , 0.285

10 }.For �(e3) the fuzzy subset �3 is

0.51 , 0.5

2 , 0.53 , 0.5

4 , 0.55 , 0.5

6 , 0.57 , 0.5

8 , 0.59 , 0.5

10 }.For �(e4) the fuzzy subset �4 is

0.51 , 0.5

2 , 0.53 , 0.5

4 , 0.55 , 0.5

6 , 0.57 , 0.5

8 , 0.59 , 0.5

10 }.Further for � we have the fuzzy subset �

01 , 0

2 , 13 , 0.5

4 , 15 , 1

6 , 07 , 0.5

8 , 91 , 0

10 }.It is clear that if X = U then �i(x) = 1 for all x ∈ U. If X =∅ then

i(x) = 0 for all x ∈ U.

. Application of soft sets in decision making problems

In this section we study the fuzzy subsets of P(U), associated withach parameter of a soft set (F, A) over U. These fuzzy subsets of P(U)ive rise to certain equivalence relations. These fuzzy subsets andquivalence relations play an important role in decision making.

Let (F, A) be a soft set over a set U. Then F : A → P(U) is a mapping.or all a ∈ A define a map Da : P(U) → [0, 1] such that

a (X) =

⎧⎨⎩

∣∣F (a) ∩ X∣∣∣∣F (a)

∣∣ if F (a) /= ∅

0 if F (a) = ∅

Then clearly Da is a fuzzy set over P(U) for each a ∈ A. Therefore

F, A) induces a fuzzy soft set over P (U).

roposition 1. Let (F, A) be a soft set over a set U, then for any∈ A, Da (X) = Da (Y) if and only if

∣∣F (a) ∩ X∣∣ =

∣∣F (a) ∩ Y∣∣, where X,

∈ P (Y).

ting 11 (2011) 3329–3332 3331

Above proposition leads us to a soft binary relation over P(U),induced by the soft set set (F, A). We denote this binary relation by(

ı, A)

and it is defined as (X, Y) ∈ ı (a) if and only if Da (X) = Da (Y),where X, Y ∈ P(U), a ∈ A.

Theorem 1. The soft binary relation(

ı, A)

over P(U) is a soft equiv-alence relation and each partition P(U)/ı (a) maintains a strict orderamong its equivalence classes for all a ∈ A.

Proof. By definition of(

ı, A)

it is easy to see that ı (a) is an equiv-

alence relation for each a ∈ A. Hence(

ı, A)

is a soft equivalencerelation over P (U). Therefore for each a ∈ A, P(U)/ı (a) is a parti-tion of P(U). If for some a ∈ A, a class in P(U)/ı (a) containing someelement X ∈ P(U) is denoted by [X]ı(a), then by definition, for eachY ∈ [X]ı(a) we have Da (X) = Da (Y). This means that each class inP(U)/ı (a) can be characterized by a unique real number from [0, 1].For the class [X]ı(a), let this number be x and we call it characteristicof [X]ı(a). Since each class in P(U)/ı (a) has a unique characteris-tic belonging to [0, 1], therefore there is a strict order among theclasses. Therefore we define this order as [Z]ı(a) ≺ [X]ı(a) if and onlyif z < x, where z is the characteristic of the class [Z]ı(a). �

From above discussion we conclude that for a soft set over theuniverse U, there is an associated fuzzy soft set over P(U) whichgives rise to a soft equivalence relation on P(U). In the following wepresent an example to depict how above mentioned notions can behelpful in decision making problems.

Example 3. Let us suppose that Mr. X wants to buy some mobilephones for his family and friends. Let three types of mobile phonessay a, b and d are available. To select mobile phones parametersare say: price (should be low), coverage (larger), functions (moreare required) and outlook (should be beautiful). To represent theseparameters we use p, c, f and o for Price, Coverage, Functions andOutlook respectively. Let A =

{p, c, f, o

}and U =

{a, b, d

}. The soft

set (F, A) is given by the mapping F : A → P(U), which is defined as

F (p) = F (f ) ={

b}

, F (c) ={

a, d}

, F (o) ={

a, b, d}

.

If he considers p as the sole decision parameter then we have afuzzy subset of P(U) as the following:

Dp ={

0∅

,0

{a} ,1{b} ,

0{d} ,

12{

a, b} ,

12{

b, d} ,

0{a, d

} ,13{

a, b, d}

}.

This fuzzy subset give rise to equivalencerelation ı (p) whose equivalence classes are{∅, {a} ,

{d}

,{

a, d}}

,{{

b}}

,{{

a, b}

,{

b, d}}

,{{

a, b, d}}

.

Order among these classes is given by{{

b}}

>{{a, b

},{

b, d}}

>{{

a, b, d}}

>{∅, {a} ,

{d}

,{

a, d}}

. Ifhe considers c as the sole decision parameter then we have a fuzzysubset of P(U) as the following:

Dc ={

0∅

,12

{a} ,0{b} ,

12{d} ,

12{

a, b} ,

12{

b, d} ,

1{a, d

} ,13{

a, b, d}

}.

This fuzzy subset give rise to equivalencerelation ı (c) whose equivalence classes are{∅,

{b}}

,{

{a} ,{

d}

,{

a, b}

,{

b, d}}

,{{

a, d}}

,{{

a, b, d}}

.

Order among these classes is given by{{

a, d}}

>{{a} ,

{d}

,{

a, b}

,{

b, d}}

>{{

a, b, d}}

>{∅,

{b}}

. If heconsiders o as the sole decision parameter then we have a fuzzy

subset of P(U) as the following:

Do ={

0∅

,13

{a} ,13{b} ,

13{d} ,

23{

a, b} ,

23{

b, d} ,

23{

a, d} ,

1{a, b, d

}}

.

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332 M. Irfan Ali / Applied Soft C

This fuzzy subset give rise to equivalenceelation ı (o) whose equivalence classes are∅, } ,

{{a} ,

{b}

,{

d}}

,{{

a, b}

,{

b, d}

,{

a, d}}

,{{

a, b, d}}

.

rder among these classes is given by{{

a, b, d}}

>{a, b

},{

b, d}

,{

a, d}}

>{

{a} ,{

b}

,{

d}}

> {∅}.Conclusion: Theory of soft sets is a new tool to discuss uncer-

ainty. In a soft set there is an associated approximation space forach parameter because every parameter in a soft set induces anquivalence relation. Intersection of all induced equivalence rela-ions give rise to indiscernibility associated with a soft set. In anpproximation space (in Pawlak’s sense) for any subset of the uni-erse we can define a fuzzy subset. Hence for each parameter in aoft set we can define a fuzzy set, so a soft set give rise to a fuzzyoft set in Maji’s sense also. Furthermore each soft set over a set U,ives rise to a fuzzy soft set over P(U) and induces a soft equivalenceelation over P(U). These concepts can be very helpful in decisionaking problems.

cknowledgments

Author is highly grateful to anonymous referees and Professorajkumar Roy Editor in Chief for their valuable comments and sug-estions for improving this paper. Author is also thankful to hishD supervisor Dr. Muhammad Shabir for his cooperation in thisesearch.

[

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ting 11 (2011) 3329–3332

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