Soft sets combined with fuzzy sets and rough sets: a tentative approach

ORIGINAL PAPER

Soft sets combined with fuzzy sets and rough sets:a tentative approach

Feng Feng Æ Changxing Li Æ B. Davvaz ÆM. Irfan Ali

Published online: 27 June 2009

� Springer-Verlag 2009

Abstract Theories of fuzzy sets and rough sets are

powerful mathematical tools for modelling various types of

uncertainty. Dubois and Prade investigated the problem of

combining fuzzy sets with rough sets. Soft set theory was

proposed by Molodtsov as a general framework for rea-

soning about vague concepts. The present paper is devoted

to a possible fusion of these distinct but closely related soft

computing approaches. Based on a Pawlak approximation

space, the approximation of a soft set is proposed to obtain

a hybrid model called rough soft sets. Alternatively, a soft

set instead of an equivalence relation can be used to

granulate the universe. This leads to a deviation of Pawlak

approximation space called a soft approximation space, in

which soft rough approximations and soft rough sets can

be introduced accordingly. Furthermore, we also consider

approximation of a fuzzy set in a soft approximation space,

and initiate a concept called soft–rough fuzzy sets, which

extends Dubois and Prade’s rough fuzzy sets. Further

research will be needed to establish whether the notions put

forth in this paper may lead to a fruitful theory.

Keywords Soft set � Fuzzy set � Rough set � Rough

fuzzy set � Approximation space � Approximation operator

1 Introduction

In some sense almost all concepts we are meeting in

everyday life are vague rather than precise. On the con-

trary, it is interesting to see that classical mathematics

requires that all mathematical notions must be exact,

otherwise precise reasoning would be impossible (Pawlak

and Skowron 2007). This gap between the real word full of

vagueness and the traditional mathematics purely con-

cerning precise concepts becomes smaller in recent years.

In fact, philosophers and recently scientists as well as

engineers are showing increasing interests in vague con-

cepts, due to the fact that many practical problems

emerging within fields such as economics, ecology, engi-

neering, environmental science, social science, and medi-

cal science require us to deal with the complexity of data

containing uncertainties. The nature of the vagueness

arising in these fields can be very different. Among many

mathematical theories designed for modelling various

types of vague concepts, fuzzy and rough sets have

received much attention and been actively studied by a

number of researchers worldwide. While some authors

argue that one theory is more general then the other, it is

accepted by majority that these two theories are closely

related, but distinct in essence because they model different

types of uncertainties. In general, a fuzzy set may be

viewed as a class with unsharp boundaries, whereas a

rough set is a coarsely described crisp set (Yao 1998).

Over the years, the theories of fuzzy sets and rough

sets have become much closer to each other for practical

needs to use both of these two theories complementarily

F. Feng (&) � C. Li

Department of Applied Mathematics and Applied Physics,

Xi’an Institute of Posts and Telecommunications,

710061 Xi’an, People’s Republic of China

e-mail: [email protected]

B. Davvaz

Department of Mathematics, Yazd University, Yazd, Iran


M. I. Ali

Department of Mathematics, Quaid-i-Azam University,

Islamabad, Pakistan


123

Soft Comput (2010) 14:899–911

DOI 10.1007/s00500-009-0465-6

for managing uncertainty that arises from inexact, noisy,

or incomplete information. Hybrid models combing fuzzy

set with rough sets have arisen in various guises in dif-

ferent settings. For instance, based on an equivalence

relation, Dubois and Prade introduced the lower and

upper approximations of fuzzy sets in a Pawlak approxi-

mation space to obtain an extended notion called rough

fuzzy sets (Dubois and Prade 1990). Alternatively, a

fuzzy similarity relation can be used to replace an

equivalence relation, and the result notion is called fuzzy

rough sets (Dubois and Prade 1990). In general, a rough

fuzzy set is the approximation of a fuzzy set in a crisp

approximation space, whereas a fuzzy rough set is the

approximation of a crisp set or a fuzzy set in a fuzzy

approximation space.

Molodtsov (1999) initiated a novel concept called soft

sets as a new mathematical tool for dealing with uncer-

tainties. The soft set theory is free from many difficulties

that have troubled the usual theoretical approaches. It

has been found that fuzzy sets, rough sets, and soft sets

are closely related concepts (Aktas and Cagman 2007).

Soft set theory has potential applications in many different

fields including the smoothness of functions, game

theory, operational research, Perron integration, probability

theory, and measurement theory (Molodtsov 1999, 2004).

Research works on soft sets are very active and progressing

rapidly in these years. Maji et al. (2002) discussed the

application of soft set theory to a decision-making problem.

Maji et al. (2001) investigated the fuzzification of a soft set

and obtained many useful results on fuzzy soft sets. Based

on fuzzy soft sets, Roy and Maji (2007) presented a method

of object recognition from an imprecise multi-observer

data. Chen et al. (2005) presented a new definition of soft

set parametrization reduction, and compared it with attri-

butes reduction in rough set theory. Kong et al. (2008)

introduced the notion called normal parameter reduction

of soft sets, by which they investigated the problem of

suboptimal choice and added parameter set in soft set

parametrization reduction. Zou and Xiao (2008) discussed

data analysis approaches of soft sets as well as fuzzy soft

sets under incomplete information. Maji et al. (2003)

defined and studied several operations on soft sets. Aktas

and Cagman (2007) related soft sets to fuzzy sets and rough

sets, providing examples to clarify their differences. They

also defined soft groups, derived some basic properties, and

showed that soft groups extended fuzzy groups. Jun (2008)

introduced and investigated soft BCK/BCI-algebras. Jun

and Park (2008) discussed the applications of soft sets to

study the ideal theory of BCK/BCI-algebras. Furthermore,

Feng et al. (2008) applied soft set theory to the study of

semirings (Feng et al. 2005, 2007; Feng and Jun 2009) and

initiated the notion called soft semirings. The present paper

aims at providing a framework to combine fuzzy sets,

rough sets, and soft sets all together, which gives rise to

several interesting new concepts such as rough soft sets,

soft rough sets, and soft–rough fuzzy sets. Although many

results reported here are only concerned with basic prop-

erties about these new notions, one could see that this study

presents a very preliminary, but potentially interesting

research direction. It will be necessary to carry out further

research to establish whether the notions put forth in this

paper may lead to a fruitful theory.

2 Fuzzy sets and rough sets

In this section, we recall some basic notions relevant to

fuzzy sets and rough sets. The following notations will be

used in what follows. Let U be a nonempty set, called

universe. The family of all subsets of U [resp. all fuzzy sets

in U] is denoted by PðUÞ [resp. FðUÞ].The theory of fuzzy sets initiated by Zadeh (1965)

provides an appropriate framework for representing and

processing vague concepts by allowing partial member-

ships. Since established, this theory has been actively

studied by both mathematicians and computer scientists.

Many applications of fuzzy set theory have arisen over the

years, for instance, fuzzy logic, fuzzy cellular neural

networks, fuzzy automata, fuzzy control systems, and

so on.

A fuzzy set l in a universe U is defined by a member-

ship function l : U ! ½0; 1�: For x 2 U; the membership

value l(x) essentially specifies the degree to which x 2 U

belongs to the fuzzy set l. There are many different defi-

nitions for fuzzy set operations. With the min–max system

proposed by Zadeh, fuzzy set intersection, union, and

complement are defined componentwise as follows:

• ðl \ mÞðxÞ ¼ lðxÞ ^ mðxÞ;• ðl [ mÞðxÞ ¼ lðxÞ _ mðxÞ;• lcðxÞ ¼ 1� lðxÞ;where l; m 2FðUÞ and x 2 U: By l � m; we mean that

lðxÞ� mðxÞ for all x 2 U: Clearly l ¼ m if both l � m and

m � l; i.e. lðxÞ ¼ mðxÞ for all x 2 U:

A fuzzy set can be related to a family of crisp sets by

means of level sets. Given a number t 2 ½0; 1�; a t-level set

(or t-cut) of a fuzzy set l 2FðUÞ is a crisp subset of U

defined by lt ¼ fx 2 U : lðxÞ>tg: By taking t 2 ½0; 1�; a

fuzzy set l determines a family of nested subsets of U; i.e.,

flt � U : t 2 ½0; 1�g: Conversely, a fuzzy set can be

reconstructed from its t-level sets by means of the fol-

lowing formula lðxÞ ¼Wft : x 2 ltg; where l 2FðUÞ

and x 2 U: This observation is usually summarized by a

representation theorem in fuzzy set theory, which estab-

lishes a one-to-one correspondence between a fuzzy set and

a family of crisp sets satisfying certain conditions.

900 F. Feng et al.

123

One of the most important applications of fuzzy set theory

is the concept of linguistic variables. The value of a linguistic

variable is defined as an element of its term set, a predefined

set of appropriate linguistic terms. Linguistic terms are

essentially subjective categories for a linguistic variable,

which do not hold exact meaning, however, and may be

understood differently by different people. The boundaries of

a given term are rather subjective, and may also depend on the

situation. Linguistic terms therefore cannot be expressed by

ordinary set theory; rather, each linguistic term is associated

with a fuzzy set (Aktas and Cagman 2007).

The rough set theory proposed by Pawlak (1982)

provides a systematic method for dealing with vague

concepts caused by indiscernibility in situation with

incomplete information or a lack of knowledge. The rough

set philosophy is founded on the assumption that with

every object in the universe, we associate some informa-

tion (data, knowledge). From a practical point of view, it

is better to define basic concepts of rough set theory in

terms of data. In fact, information and knowledge are

stored and represented in a data table in many data anal-

ysis applications. This data table containing rows labelled

by objects and columns labelled by attributes is called an

information system (also known as a knowledge repre-

sentation system) which can be formulated in the fol-

lowing way:

Definition 1 (Pawlak and Skowron 2007) An information

system is a pair I ¼ ðU;AÞ, where U is a nonempty finite

set of objects and A is a nonempty finite set of attributes.

Every attribute a 2 A is a function a : U!Va; where Va is

the set of values of attribute a.

Let R be an equivalence relation on the universe U.

Then the pair (U, R) is usually called a Pawlak approx-

imation space. Conventionally, we refer to R as an in-

discernibility relation (Pawlak and Skowron 2007) for the

reason that it is often obtained from an information sys-

tem (see Definition 1) and gives a partition of U due to

the indiscernibility of objects in U. For x; y 2 U; x and y

are said to be R-indiscernible if ðx; yÞ 2 R: The family of

all equivalence classes of R, i.e., the partition determined

by the equivalence relation R, will be denoted by U/R. An

equivalence class of R, i.e., the block of the partition U/R,

containing x will be denoted by ½x�R: These equivalence

classes of R are referred to as R-elementary sets

(or R-elementary granules). The elementary sets repre-

sent the basic building blocks (concepts) of our knowl-

edge about reality.

Using the indiscernibility relation R, we can define the

following two operations:

R�X ¼ fx 2 U : ½x�R � Xg;R�X ¼ fx 2 U : ½x�R \ X 6¼ ;g;

assigning to every subset X � U two sets R�X and R�Xcalled the lower and upper approximations of X with

respect to ðU;RÞ: In addition,

PosRX ¼ R�X;

NegRX ¼ U � R�X;

BndRX ¼ R�X � R�X

are called the positive, negative, and boundary regions of

X, respectively. Now, we are ready to give the definition of

rough sets:

Definition 2 (Pawlak and Skowron 2007) Let (U, R) be a

Pawlak approximation space. A subset X � U is called

definable if R�X ¼ R�X; in the opposite case, i.e., if

BndRX 6¼ ;; X is said to be rough (or inexact).

Note that sometimes a pair ðA;BÞ 2 PðUÞ �PðUÞ is

also called a rough set if A ¼ R�X and B ¼ R�X for some

X � U (Radzikowska and Kerre 2002). If the set X � U is

defined by a predicate P and x 2 U; then we have the

following:

• x 2 R�X means that x certainly has property P,

• x 2 R�X means that x possibly has property P,

• x 2 NegRX means that X definitely does not have

property P:

Theorem 1 Suppose that (U, R) is a Pawlak approxi-

mation space and A;B � U: Then we have

1. R�ðAÞ � A � R�ðAÞ;2. R�ð;Þ ¼ ; ¼ R�ð;Þ;3. R�ðUÞ ¼ U ¼ R�ðUÞ;4. R�ðR�ðAÞÞ ¼ R�ðAÞ;5. R�ðR�ðAÞÞ ¼ R�ðAÞ;6. R�ðR�ðAÞÞ ¼ R�ðAÞ;7. R�ðR�ðAÞÞ ¼ R�ðAÞ;8. R�ðAÞ ¼ ðR�ðAcÞÞc;9. R�ðAÞ ¼ ðR�ðAcÞÞc;

10. R�ðA \ BÞ ¼ R�ðAÞ \ R�ðBÞ;11. R�ðA \ BÞ � R�ðAÞ \ R�ðBÞ;12. R�ðA [ BÞ � R�ðAÞ [ R�ðBÞ;13. R�ðA [ BÞ ¼ R�ðAÞ [ R�ðBÞ;14. A � B) R�ðAÞ � R�ðBÞ;R�ðAÞ � R�ðBÞ:

Proof See Pawlak (1991).

3 Soft sets and fuzzy soft sets

Let U be an initial universe set and EU (simply denoted

by E) be the set of all possible parameters with respect to

U. Usually, parameters are attributes, characteristics, or

properties of the objects in U. The notion of a soft set is

defined as follows:

Soft sets combined with fuzzy sets and rough sets: a tentative approach 901

123

Definition 3 (Molodtsov 1999) A pair S ¼ ðF;AÞ is

called a soft set over U; where A � E and F : A! PðUÞ is

a set-valued mapping.

In other words, a soft set over U is a parameterized

family of subsets of the universe U. For � 2 A; Fð�Þ may be

considered as the set of � -approximate elements in S ¼ðF;AÞ: For illustration, Molodtsov (1999) considered sev-

eral concrete examples of soft sets. The following is one

type of illuminating examples considered by many authors

(Aktas and Cagman 2007; Chen et al. 2005; Maji et al.

2002, 2003).

Example 1 Suppose that U ¼ fh1; h2; h3; h4; h5g is the

universe consisting of five houses and the set of para-

meters is given by E ¼ fe1; e2; e3; e4; e5g; where ei

(i ¼ 1; 2; . . .; 5) stand for ‘‘beautiful’’, ‘‘modern’’, ‘‘cheap’’,

‘‘in green surroundings’’ and ‘‘in good repair’’, respec-

tively. Now, let us consider a soft set ðF;EÞ which

describes the ‘‘attractiveness of houses’’ that Mr. X is

considering for purchase. In this case, to define the soft set

means to point out beautiful houses, modern houses and so

on. Consider the mapping F given by ‘‘houses(�)’’, where

dot (�) is to be filled in by one of the parameters ei 2 E: For

instance, Fðe1Þ means ‘‘houses(beautiful)’’, and its func-

tional value is the set consisting of all the beautiful houses

in U. Let Fðe1Þ ¼ fh5g; Fðe2Þ ¼ fh1; h4g; Fðe3Þ ¼fh1; h2; h3g and Fðe4Þ ¼ fh3; h5g: Then the soft set ðF;EÞcan be seen as a collection of approximations where each

approximation has two parts, a predicate and an approxi-

mate value set. For instance, ðbeautiful houses; fh5gÞ is

one of the approximations. Table 1 is the tabular repre-

sentation of the soft set ðF;EÞ: If hi 2 FðejÞ; then hij ¼ 1;

otherwise hij ¼ 0; where hij are the entries in the table. We

shall reconsider this example in the section concerning

soft–rough approximations of fuzzy sets.

As is shown in Aktas and Cagman (2007), both fuzzy sets

and rough sets may be considered as soft sets. Thus, one may

expect that soft set theory could provide a more general

mathematical framework for dealing with uncertain data. In

fact, the way of setting (or describing) any object in soft set

theory differs in principle from the way used in traditional

mathematics (Molodtsov 1999). In soft set theory, the initial

description of the object has an approximate nature, and we

do not need to introduce the notion of exact solution as

required in classical mathematics. The absence of any

restrictions on the approximate description in soft set theory

makes it in practice very convenient and easy to apply.

Definition 4 Let (F, A) and (G, B) be two soft sets over

U. Then (G, B) is called a soft subset of (F, A), denoted by

ðF;AÞ � ðG;BÞ; if B � A and GðbÞ � FðbÞ for all b 2 B:

Two soft sets ðF;AÞ and ðG;BÞ over U are said to be equal,

denoted by ðF;AÞ ¼ ðG;BÞ; if ðF;AÞ � ðG;BÞ and

ðG;BÞ � ðF;AÞ:

Molodtsov (1999) also proposed a general way to define

binary operations over soft sets. Assume that we have a

binary operation on PðUÞ; which is denoted by : Let

ðF;AÞ and ðG;BÞ be soft sets over U: Then, the operation

for soft sets is defined by

ðF;AÞ ðG;BÞ ¼ ðH;A� BÞ;

where Hða; bÞ ¼ FðaÞ GðbÞ; a 2 A; b 2 B and A�B is

the Cartesian product of A and B:

Maji et al. (2003) introduced the following operations

over soft sets, which can be seen as the implementations of

Molodtsov’s idea above.

Definition 5 (Maji et al. 2003) If (F, A) and (G, B) are

two soft sets over a common universe U, then (F,A) AND

(G,B) denoted by ðF;AÞ ^ ðG;BÞ is defined by ðF;AÞ ^ðG;BÞ ¼ ðH;A� BÞ; where Hðx; yÞ ¼ FðxÞ \ GðyÞ for all

ðx; yÞ 2 A� B:

Definition 6 (Maji et al. 2003) If ðF;AÞ and ðG;BÞ are

two soft sets over a common universe U; then ‘‘ðF;AÞ OR

ðG;BÞ’’ denoted by ðF;AÞ _ ðG;BÞ is defined by ðF;AÞ _ðG;BÞ ¼ ðH;A� BÞ; where Hðx; yÞ ¼ FðxÞ [ GðyÞ for all

ðx; yÞ 2 A� B:

It is worth noting that differing from Molodtsov’s above

idea, we can define binary operations for soft sets in the

following way:

Definition 7 Suppose that E : PðEÞ �PðEÞ ! PðEÞ is

a binary operation on PðEÞ; and U : PðUÞ �PðUÞ !PðUÞ is a binary operation on PðUÞ: Then for any two soft

sets S ¼ ðF;AÞ and T ¼ ðG;BÞ over U; S T is defined

as the soft set R ¼ ðH;CÞ where C ¼ AE B and HðxÞ ¼FðxÞ U GðxÞ for all x 2 C:

Definition 8 (Ali et al. 2009) Let (F, A) and (G, B) be

two soft sets over a common universe U.

(1) The extended intersection of ðF;AÞ and ðG;BÞ;denoted by ðF;AÞ uE ðG;BÞ; is defined as the soft

set ðH;CÞ; where C ¼ A [ B; and 8e 2 C;

HðeÞ ¼FðeÞ; if e 2 A� B;GðeÞ; if e 2 B� A;FðeÞ \ GðeÞ; if e 2 A \ B:

8<

:

Table 1 Tabular representation of the soft set ðF;EÞ

h1 h2 h3 h4 h5

e1 0 0 0 0 1

e2 1 0 0 1 0

e3 1 1 1 0 0

e4 0 0 1 0 1

902 F. Feng et al.

123

(2) The restricted intersection of ðF;AÞ and ðG;BÞ;denoted by ðF;AÞeðG;BÞ; is defined as the soft set

ðH;CÞ; where C ¼ A \ B and HðcÞ ¼ FðcÞ \ GðcÞfor all c 2 C:

(3) The extended union of ðF;AÞ and ðG;BÞ; denoted by

ðF;AÞ e[ðG;BÞ; is defined as the soft set ðH;CÞ; where

C ¼ A [ B; and 8e 2 C;

HðeÞ ¼FðeÞ; if e 2 A� B;GðeÞ; if e 2 B� A;FðeÞ [ GðeÞ; if e 2 A \ B:

8<

:

(4) The restricted union of ðF;AÞ and ðG;BÞ; denoted by

ðF;AÞ [R ðG;BÞ; is defined as the soft set (H,C),

where C ¼ A \ B and HðcÞ ¼ FðcÞ [ GðcÞ for all

c 2 C:

Note that restricted intersection was also known as

bi-intersection in Feng et al. (2008), and extended union was

at first introduced and called union by Maji et al. (2003).

Definition 9 (Ali et al. 2009) Let U be an initial universe

set, E be the universe set of parameters, and A;B � E:

(1) fN ðU;AÞ ¼ ðN;AÞ is called a relative null soft set (with

respect to the parameter set A) if NðxÞ ¼ ; for all

x 2 A:

(2) fWðU;BÞ ¼ ðW ;BÞ is called a relative whole soft set

(with respect to the parameter set B) if WðxÞ ¼ U for

all x 2 B:

The relative whole soft set with respect to the universe

set of parameters E is called the absolute soft set (or whole

soft set) over U and simply denoted by eAU instead offWðU;EÞ: Dually fN ðU;EÞ is called the null soft set over U and

denoted by fN U :

Definition 10 (Ali et al. 2009) The relative complement

of a soft set (F, A), denoted by ðF;AÞr; is defined as the soft

set ðFr;AÞ where FrðxÞ ¼ U � FðxÞ for all x 2 A:

The following result indicates that soft sets and binary

relations are closely related.

Theorem 2 Let S ¼ ðf ;AÞ be a soft set over U. Then S

induces a binary relation RS � A� U; which is defined by

ðx; yÞ 2 RS , y 2 f ðxÞ;

where x 2 A; y 2 U:

Conversely, assume that R is a binary relation from A to

U. Define a set-valued mapping fR : A! PðUÞ by

fRðxÞ ¼ fy 2 U : ðx; yÞ 2 Rg;

where x 2 A: Then SR ¼ ðfR;AÞ is a soft set over U.

Moreover, we have that SRS¼ S and RSR

¼ R:

Proof One easily sees that the first part holds. Thus it

suffices to show that SRS¼ S and RSR

¼ R: Suppose that

S ¼ ðf ;AÞ is a soft set over U and x 2 A: Then for any

y 2 U; by definition we have that

y 2 fRSðxÞ , ðx; yÞ 2 RS , y 2 f ðxÞ:

That is, fRSðxÞ ¼ f ðxÞ for all x 2 A: Hence, fRS

¼ f ; and so

ðfRS;AÞ ¼ ðf ;AÞ: That is, SRS

¼ S:

Next, assume that R � A� U; x 2 A and y 2 U: Then

by definition, it is clear that

ðx; yÞ 2 RSR, y 2 fRðxÞ , ðx; yÞ 2 R:

Therefore we conclude that RSR¼ R as required.

In what follows, RS will be called the canonical relation

of the soft set S; and SR will be called the canonical soft

set of the binary relation R:

Definition 11 (Ganter and Wille 1999) A formal context

is a triple (U, V, I) where U, V are two finite sets called

object set and property set, and I is a binary relation

between U and V.

Corollary 1 Every formal context can be seen as a soft

set and vice versa.

Proof This follows immediately from Theorem 2.

As pointed out by several researchers, information sys-

tems and soft sets are closely related (Chen et al. 2005;

Zou and Xiao 2008). Given a soft set S ¼ ðF;AÞ over a

universe U. If U and A are both nonempty finite sets, then

S could induce an information system in a natural way. In

fact, for any attribute a 2 A; one can define a function

a : U ! Va ¼ f0; 1g by

aðxÞ ¼ 1; if x 2 FðaÞ;0; otherwise:

�

Therefore, every soft set may be considered as an

information system. This justifies the tabular represen-

tation of soft sets used widely in the literature. Conversely,

it is worth noting that a soft set can also be applied

to express an information system. Let I ¼ ðU;AÞ be an

information system. Taking

B ¼[

a2A

fag � Va;

as the parameter set, then a soft set (F, B) can be defined by

setting

Fða; vÞ ¼ fx 2 U : aðxÞ ¼ vg;

where a 2 A and v 2 Va:Maji et al. (2001) initiated the study on hybrid structures

involving both fuzzy sets and soft sets. In Maji et al. (2001)

the notion of fuzzy soft sets was introduced as a fuzzy

generalization of (classical) soft sets and some basic

properties are discussed in detail.


123

Definition 12 (Roy and Maji 2007) Let FðUÞ be the set

of all fuzzy sets in U. Let E be the parameter set and

A � E: A pair (F,A) is called a fuzzy soft set over U, where

F is a mapping given by F : A!FðUÞ:

In the definition of a fuzzy soft set, fuzzy sets in the

universe U are used as substitutes for the crisp subsets of U.

Hence, every soft set may be considered as a fuzzy soft set.

In addition, by analogy with soft sets, one easily sees that

every fuzzy soft set can be viewed as an (fuzzy) informa-

tion system and be represented by a corresponding data

table with entries belonging to the unit interval [0, 1]

instead of 0 and 1 only.

4 Rough fuzzy sets

Based on an equivalence relation on the universe of dis-

course, Dubois and Prade (1990) introduced the lower and

upper approximations of fuzzy sets in a Pawlak approxi-

mation space, and obtained a new notion called rough

fuzzy sets.

Definition 13 (Dubois and Prade 1990) Let ðU;RÞ be a

Pawlak approximation space. For a fuzzy set l 2FðUÞ;the lower and upper rough approximations of l in ðU;RÞare denoted by RðlÞ and RðlÞ; respectively, which are

fuzzy sets in U defined by

RðlÞðxÞ ¼^flðyÞ : y 2 ½x�Rg;

and

RðlÞðxÞ ¼_flðyÞ : y 2 ½x�RÞg;

for all x 2 U: The operators R and R are called the lower

and upper rough approximation operators on fuzzy sets. If

RðlÞ ¼ RðlÞ; the fuzzy set l is said to be definable;

otherwise, l is called a rough fuzzy set.

By definition, it is easy to see that if l ¼ X is a crisp

subset of U; then we have that

RðXÞ ¼ R�X ¼ fx 2 U : ½x�R � Xg;

and

RðXÞ ¼ R�X ¼ fx 2 U : ½x�R \ X 6¼ ;g:

In this case, it follows that X 2 PðUÞ is a rough fuzzy set if

and only if X is a rough set. Thus, one can say that rough

fuzzy sets are natural extensions of rough sets.

In general, a rough fuzzy set can be seen as an

approximation of a fuzzy set in a crisp approximation

space. This hybrid model combining the concepts of both

fuzzy sets and rough sets may be used to deal with

knowledge acquisition in information systems with fuzzy

decisions (Slowinski and Stefanowski 1999).

5 Rough soft sets

Motivated by Dubois and Prade’s original idea about rough

fuzzy sets, we consider the lower and upper approxima-

tions of a soft set in a Pawlak approximation space, which

gives rise to the following notions in a natural way.

Definition 14 Let ðU;RÞ be a Pawlak approximation

space and S ¼ ðF;AÞ be a soft set over U: The lower and

upper rough approximations of S ¼ ðF;AÞ with respect to

ðU;RÞ are denoted by R�ðSÞ ¼ ðF�;AÞ and R�ðSÞ ¼ðF�;AÞ; which are soft sets over U with the set-valued

mappings given by

F�ðxÞ ¼ R�ðFðxÞÞ;

and

F�ðxÞ ¼ R�ðFðxÞÞ;

where x 2 A: The operators R� and R� are called the lower

and upper rough approximation operators on soft sets. If

R�ðSÞ ¼ R�ðSÞ; the soft set S is said to be definable;

otherwise S is called a rough soft set.

One can verify the following basic properties of rough

soft sets:

Theorem 3 Suppose that ðU;RÞ is a Pawlak approxi-

mation space and S ¼ ðF;AÞ is a soft set over U: Then we

have

1. R�ðSÞ � S � R�ðSÞ;2. R�ðfN ðU;AÞÞ ¼ fN ðU;AÞ ¼ R�ðfN ðU;AÞÞ;3. R�ðfWðU;AÞÞ ¼ fWðU;AÞ ¼ R�ðfWðU;AÞÞ;4. R�ðR�ðSÞÞ ¼ R�ðSÞ;5. R�ðR�ðSÞÞ ¼ R�ðSÞ;6. R�ðR�ðSÞÞ ¼ R�ðSÞ;7. R�ðR�ðSÞÞ ¼ R�ðSÞ;8. R�ðSÞ ¼ ðR�ðSrÞÞr;9. R�ðSÞ ¼ ðR�ðSrÞÞr:

Proof This is easily obtained from Theorem 1, Definition

9, Definition 10, and Definition 14.

Theorem 4 Suppose that ðU;RÞ is a Pawlak approxi-

mation space and S ¼ ðF;AÞ;T ¼ ðG;BÞ are soft sets

over U: Then we have

1. R�ðSeTÞ ¼ R�ðSÞeR�ðTÞ;2. R�ðS uE TÞ ¼ R�ðSÞ uE R�ðTÞ;3. R�ðSeTÞ � R�ðSÞeR�ðTÞ;4. R�ðS uE TÞ � R�ðSÞ uE R�ðTÞ;5. R�ðS [R TÞ � R�ðSÞ [R R�ðTÞ;6. R�ðS e[TÞ � R�ðSÞ e[ R�ðTÞ;7. R�ðS [R TÞ ¼ R�ðSÞ [R R�ðTÞ;8. R�ðS e[TÞ ¼ R�ðSÞ e[ R�ðTÞ;9. S � T) R�ðSÞ � R�ðTÞ;R�ðSÞ � R�ðTÞ:

904 F. Feng et al.

123

Proof Let ðU;RÞ be a Pawlak approximation space, and

let S ¼ ðF;AÞ; T ¼ ðG;BÞ be soft sets over U.

(1) Let SeT ¼ R ¼ ðH;CÞ: Then C ¼ A \ B and

HðxÞ ¼ FðxÞ \ GðxÞ; 8x 2 C: Using Definition (Maji

et al. 2002), R�ðRÞ ¼ ðH�;CÞ; where

H�ðxÞ ¼ R�ðHðxÞÞ ¼ R�ðFðxÞ \ GðxÞÞ;

for all x 2 C: Now by Theorem 1,

R�ðFðxÞ \ GðxÞÞ ¼ R�ðFðxÞÞ \ R�ðGðxÞÞ;

and so we deduce that H�ðxÞ ¼ F�ðxÞ \ G�ðxÞ for all

x 2 C: Hence, R�ðRÞ ¼ R�ðSÞeR�ðTÞ:(2) Let S uE T ¼ R ¼ ðH;CÞ: Then C ¼ A [ B and for

all e 2 C;

HðeÞ ¼FðeÞ; if e 2 A� B;GðeÞ; if e 2 B� A;FðeÞ \ GðeÞ; if e 2 A \ B:

8<

:

Using Definition (Maji et al. 2002), R�ðRÞ ¼ ðH�;CÞ;where

H�ðeÞ ¼R�ðFðeÞÞ; if e 2 A� B;R�ðGðeÞÞ; if e 2 B� A;R�ðFðeÞ \ GðeÞÞ; if e 2 A \ B;

8<

:

for all e 2 C: Now by Theorem 1,

R�ðFðeÞ \ GðeÞÞ ¼ R�ðFðeÞÞ \ R�ðGðeÞÞ;

and so we deduce that

H�ðeÞ ¼F�ðeÞ; if e 2 A� B;G�ðeÞ; if e 2 B� A;F�ðeÞ \ G�ðeÞ; if e 2 A \ B;

8<

:

for all e 2 C: Hence, R�ðRÞ ¼ R�ðSÞ uE R�ðTÞ:(3) This is similar to the proof of (1).

(4) This is similar to the proof of (2).

(5) Let S [R T ¼ R ¼ ðH;CÞ: Then C ¼ A \ B and

HðxÞ ¼ FðxÞ [ GðxÞ; 8x 2 C: Using Definition (Maji

et al. 2002), R�ðRÞ ¼ ðH�;CÞ; where

H�ðxÞ ¼ R�ðHðxÞÞ ¼ R�ðFðxÞ [ GðxÞÞ;

for all x 2 C: Now by Theorem 1,

R�ðFðxÞ \ GðxÞÞ � R�ðFðxÞÞ \ R�ðGðxÞÞ;

and so we deduce that H�ðxÞ � F�ðxÞ \ G�ðxÞ for all

x 2 C: Hence, R�ðRÞ � R�ðSÞ [R R�ðTÞ:(6) Let S e[T ¼ R ¼ ðH;CÞ: Then C ¼ A [ B and for all

e 2 C;

HðeÞ ¼FðeÞ; if e 2 A� B;GðeÞ; if e 2 B� A;FðeÞ [ GðeÞ; if e 2 A \ B:

8<

:

Using Definition 14, R�ðRÞ ¼ ðH�;CÞ; where

H�ðeÞ ¼R�ðFðeÞÞ; if e 2 A� B;R�ðGðeÞÞ; if e 2 B� A;R�ðFðeÞ [ GðeÞÞ; if e 2 A \ B;

8<

:

for all e 2 C: Now by Theorem 1,

R�ðFðeÞ [ GðeÞÞ � R�ðFðeÞÞ \ R�ðGðeÞÞ;

and so we deduce that R�ðRÞ � R�ðSÞ e[R�ðTÞ:(7) This is similar to the proof of (5).


(9) Assume that S � T: Then by Definition 4, we have

A � B and FðxÞ � GðxÞ; 8x 2 A: Now it follows from

Theorem 1 that R�ðFðxÞÞ � R�ðGðxÞÞ and R�ðFðxÞÞ �R�ðGðxÞÞ for all x 2 C: Hence, we conclude that

R�ðSÞ � R�ðTÞ and R�ðSÞ � R�ðTÞ: This completes

our proof.

6 Soft rough sets

In the last section, based on an equivalence relation on the

universe of discourse, we introduced rough approximations

of soft sets and obtained a hybrid concept called rough soft

sets. We shall discuss another way to combine soft sets

with rough sets in this section. In fact, a soft set instead of

an equivalence relation can be used to granulate the uni-

verse of discourse. The result is a deviation of Pawlak

approximation space called a soft approximation space, in

which soft rough approximations and soft rough sets can be

introduced accordingly.

Definition 15 Let S ¼ ðf ;AÞ be a soft set over U: Then

the pair P ¼ ðU;SÞ is called a soft approximation space.

Based on P; we define the following two operations:

aprPðXÞ ¼ fu 2 U : 9a 2 A½u 2 f ðaÞ � X�g;

aprPðXÞ ¼ fu 2 U : 9a 2 A½u 2 f ðaÞ; f ðaÞ \ X 6¼ ;�g;

assigning to every subset X � U two sets aprPðXÞ and

aprPðXÞ called the lower and upper soft rough

approximations of X in P; respectively. Moreover,

PosPðXÞ ¼ aprPðXÞ;

NegPðXÞ ¼ U � aprPðXÞ;BndPðXÞ ¼ aprPðXÞ � apr

PðXÞ

are called the soft positive, soft negative and soft boundary

regions of X; respectively. If aprPðXÞ ¼ aprPðXÞ; X is said

to be soft definable; otherwise X is called a soft rough set.

By definition, we immediately have that X � U is a soft

definable set if BndPðXÞ ¼ ;: Also, it is clear that

aprPðXÞ � X and apr

PðXÞ � aprPðXÞ for all X � U:


123

Nevertheless, it is worth noting that X � aprPðXÞ does not

hold in general as illuminated by the following example.

Example 2 Suppose that U ¼ fu1; u2; u3; u4; u5; u6g; E ¼fe1; e2; e3; e4; e5; e6g and A ¼ fe1; e2; e3; e4g � E: Let S ¼ðf ;AÞ be a soft set over U given by Table 2 and the soft

approximation space P ¼ ðU;SÞ:

For X ¼ fu3; u4; u5g � U; we have aprPðXÞ ¼ fu3g;

and aprPðXÞ ¼ fu1; u2; u3; u5g: Thus, aprPðXÞ 6¼ aprPðXÞ

and X is a soft rough set. It is clear that X ¼ fu3; u4; u5g* aprPðXÞ ¼ fu1; u2; u3; u5g: Moreover, we have

PosPðXÞ ¼ fu3g; NegPðXÞ ¼ fu4; u6g and BndPðXÞ ¼fu1; u2; u5g: On the other hand, let us consider X1 ¼fu3; u4g � U: Since apr

PðX1Þ ¼ fu3g ¼ aprPðX1Þ; by

definition X1 is a soft definable set.

Proposition 1 Let S ¼ ðf ;AÞ be a soft set over U and

P ¼ ðU;SÞ be a soft approximation space. Then,

aprPðXÞ ¼

[

a2A

ff ðaÞ : f ðaÞ � Xg

and

aprPðXÞ ¼[

a2A

ff ðaÞ : f ðaÞ \ X 6¼ ;g

for all X � U:

Proof This is easily obtained from the definition of soft

rough approximations.

Let S ¼ ðf ;AÞ be a soft set over U; P ¼ ðU;SÞ be a

soft approximation space and X � U: We can define four

basic classes of soft rough sets:

• X is said to be roughly soft definable if aprPðXÞ 6¼ ;

and aprPðXÞ 6¼ U;

• X is internally soft indefinable if aprPðXÞ ¼ ; and

aprPðXÞ 6¼ U;

• X is externally soft indefinable if aprPðXÞ 6¼ ; and

aprPðXÞ ¼ U;

• X is totally soft indefinable if aprPðXÞ ¼ ; and

aprPðXÞ ¼ U:

The intuitive meaning of this classification is as follows:

If X is roughly soft definable, this means that we are able to

decide for some elements of U that they belong to X; and

meanwhile for some elements of U we are able to decide that

they belong to U � X; using available knowledge from the

soft approximation space P: If X is internally soft indefin-

able, this means that we are able to decide about some ele-

ments of U that they belong to U � X; but we are unable to

decide for any element of U that it belongs to X; by

employing P: If X is externally soft indefinable, this means

that we are able to decide for some elements of U that they

belong to X; but we are unable to decide, for any element of

U that it belongs to U � X; by employing P: If X is totally

soft indefinable, we are unable to decide for any element of

U whether it belongs to X or U � X; by employing P:

Theorem 5 Let S ¼ ðf ;AÞ be a soft set over U;P ¼ðU;SÞ be a soft approximation space and X; Y � U: Then

we have

1. aprPð;Þ ¼ aprPð;Þ ¼ ;;

2. aprPðUÞ ¼ aprPðUÞ ¼

Sa2A f ðaÞ;

3. X � Y ) aprPðXÞ � apr

PðYÞ;

4. X � Y ) aprPðXÞ � aprPðYÞ;5. apr

PðX \ YÞ � apr

PðXÞ \ apr

PðYÞ;

6. aprPðX [ YÞ � apr

PðXÞ [ apr

PðYÞ;

7. aprPðX [ YÞ ¼ aprPðXÞ [ aprPðYÞ;8. aprPðX \ YÞ � aprPðXÞ \ aprPðYÞ:

Proof

(1) Straightforward.

(2) This follows directly from Proposition 1 by substi-

tuting U for X:

(3) Assume that X � Y: If u 2 aprPðXÞ; then by defini-

tion, there exists some a 2 A such that u 2 f ðaÞ � X:

Hence, it follows that u 2 f ðaÞ � Y; and so u 2apr

PðYÞ: This shows that apr

PðXÞ � apr

PðYÞ:


(5) Since X \ Y � X; we deduce from (3) that aprPðX \

YÞ � aprPðXÞ: Similarly we have apr

PðX \ YÞ �

aprPðYÞ; and so apr

PðX \ YÞ � apr

PðXÞ \ apr

PðYÞ:

(6) Since X � X [ Y; by (3) we have aprPðXÞ �

aprPðX [ YÞ: Similarly apr

PðYÞ � apr

PðX [ YÞ; and

so aprPðX [ YÞ � apr

PðXÞ [ apr

PðYÞ:

(7) Let u 2 aprPðX [ YÞ: By definition, there exists some

a 2 A such that u 2 f ðaÞ and f ðaÞ \ ðX [ YÞ 6¼ ;:Thus we have that either f ðaÞ \ X 6¼ ; or f ðaÞ \ Y 6¼;; indicating that u 2 aprPðXÞ or u 2 aprPðYÞ: This

shows that aprPðX [ YÞ � aprPðXÞ [ aprPðYÞ:To prove the reverse inclusion, note that X � X [ Y

and so it follows from (4) that aprPðXÞ � aprPðX [YÞ . Dually we have that aprPðYÞ � aprPðX [ YÞ .

Hence we conclude that aprPðX [ YÞ ¼ aprPðXÞ[ aprPðYÞ:


It should be noted that the inclusions in Theorem 5 may

be strict, as shown in the following example.

Table 2 Tabular representation of the soft set S

u1 u2 u3 u4 u5 u6

e1 1 0 0 0 0 1

e2 0 0 1 0 0 0

e3 0 0 0 0 0 0

e4 1 1 0 0 1 0

906 F. Feng et al.

123

Example 3 Suppose that U ¼ fu1; u2; � � � ; u8g; E ¼fe1; e2; � � � ; e6g and B ¼ fe1; e2; e3; e4g � E: Let T ¼ðg;BÞ be a soft set over U given by Table 3 and the soft

approximation space P ¼ ðU;TÞ:

For X ¼ fu2; u7; u8g � U; we have aprPðXÞ ¼ f ðe1Þ ¼

fu7; u8g; and aprPðXÞ ¼ f ðe1Þ [ f ðe3Þ [ f ðe4Þ ¼ fu1; u2;

u4; u7; u8g:For Y ¼ fu1; u4; u7g � U; we have apr

PðYÞ ¼ f ðe4Þ ¼

fu4; u7g; and aprPðYÞ ¼ f ðe1Þ [ f ðe2Þ [ f ðe3Þ [ f ðe4Þ ¼fu1; u2; u4; u5; u6; u7; u8g:

Since X \ Y ¼ fu7g and X [ Y ¼ fu1; u2; u4; u7; u8g; we

have aprPðX \ YÞ ¼ ;; aprPðX \ YÞ ¼ f ðe1Þ [ f ðe4Þ ¼

fu4; u7; u8g; and aprPðX [ YÞ ¼ f ðe1Þ [ f ðe3Þ [ f ðe4Þ ¼

fu1; u2; u4; u7; u8g:Now, we have apr

PðXÞ \ apr

PðYÞ ¼ fu7g and so

aprPðX \ YÞ ¼ ; apr

PðXÞ \ apr

PðYÞ; which shows that

the inclusion (5) in Theorem 5 may hold strictly.

From the observation that aprPðXÞ [ apr

PðYÞ ¼ f ðe1Þ [

f ðe4Þ ¼ fu4; u7; u8g aprPðX [ YÞ; we deduce that the

inclusion (6) in Theorem 5 could be strict.

Moreover, aprPðXÞ \ aprPðYÞ ¼ fu1; u2; u4; u7; u8g and

aprPðX \ YÞ ¼ fu4; u7; u8g: This shows that the inclusion

(8) in Theorem 5 could be strict.

Definition 16 Let S ¼ ðf ;UÞ be a soft set over U and

P ¼ ðU;SÞ be a soft approximation space. For all X; Y �U; we define

X�� P

Y , aprPðXÞ ¼ apr

PðYÞ;

X�� PY , aprPðXÞ ¼ aprPðYÞ;

and

X �P Y , X�� P

Y and X�� PY:

These binary relations are called the lower soft rough equal

relation, the upper soft rough equal relation, and the soft

rough equal relation, respectively.

It is easy to verify that the relations defined above are all

equivalence relations over PðUÞ:

Theorem 6 Let S ¼ ðf ;AÞ be a soft set over U and P ¼ðU;SÞ be the soft approximation space. Then,

1. X�� PY , X�� PðX [ YÞ�� PY;

2. X�� PX1; Y �� PY1 ) ðX [ YÞ�� PðX1 [ Y1Þ;

3. X�� PY ) X [ ðU � YÞ�� PU;

4. X � Y; Y �� P; ) X�� P;;5. X � Y;X�� PU ) Y �� PU;

where X; Y ;X1; Y1 � U:

Proof

(1) Suppose that X�� PY: By definition we have

aprPðXÞ ¼ aprPðYÞ: But from Theorem 5, we know

that aprPðX [ YÞ ¼ aprPðXÞ [ aprPðYÞ: Hence,

aprPðX [ YÞ ¼ aprPðXÞ ¼ aprPðYÞ and so X�� PðX [YÞ�� PY: Conversely, if X�� PðX [ YÞ�� PY; then it

follows that X�� PY using the transitivity of �� P:

(2) Assume that X�� PX1 and Y �� PY1: By definition

aprPðXÞ ¼ aprPðX1Þ and aprPðYÞ ¼ aprPðY1Þ: Also

by Theorem 5, we have aprPðX [ YÞ ¼ aprPðXÞ [aprPðYÞ and aprPðX1 [ Y1Þ ¼ aprPðX1Þ [ aprPðY1Þ:Thus we deduce that aprPðX [ YÞ ¼ aprPðX1 [ Y1Þ;and so ðX [ YÞ�� PðX1 [ Y1Þ:

(3) Suppose that X�� PY: By definition we have

aprPðXÞ ¼ aprPðYÞ: Also by Theorem 5, we have

aprPðX [ ðU � YÞÞ ¼ aprPðXÞ [ aprPðU � YÞ and

aprPðUÞ ¼ aprPðYÞ [ aprPðU � YÞ: It follows that

aprPðX [ ðU � YÞÞ ¼ aprPðYÞ[ aprPðU � YÞ ¼aprPðUÞ: Hence, X [ ðU � YÞ�� PU:

(4) Suppose that X � Y and Y �� P;: From Theorem 5, we

have aprPðXÞ � aprPðYÞ ¼ aprPð;Þ ¼ ;: Hence,

aprPðXÞ ¼ ; ¼ aprPð;Þ and so X�� P;:(5) Assume that X � Y and X�� PU: By Theorem 5, we

have aprPðYÞ � aprPðXÞ ¼ aprPðUÞ: Also, it is clear

that aprPðYÞ � aprPðUÞ since Y � U: Therefore,

aprPðYÞ ¼ aprPðUÞ; and so Y �� PU as required.

Definition 17 Let S ¼ ðf ;AÞ be a soft set over U: If for

any a1; a2 2 A; there exists a3 2 A such that f ða3Þ ¼f ða1Þ \ f ða2Þ whenever f ða1Þ \ f ða2Þ 6¼ ;; then S is called

an intersection complete soft set.

Proposition 2 Let S ¼ ðf ;AÞ be an intersection complete

soft set over U and P ¼ ðU;SÞ be a soft approximation

space. Then, we have

aprPðX \ YÞ ¼ apr

PðXÞ \ apr

PðYÞ

for all X; Y � U:

Proof Note first that by Theorem 5,

aprPðX \ YÞ � apr

PðXÞ \ apr

PðYÞ

holds for every soft set S (needless to be intersection

complete). Therefore, it suffices to show the reverse

inclusion

aprPðX \ YÞ � apr

PðXÞ \ apr

PðYÞ:

In fact, let u 2 aprPðXÞ \ apr

PðYÞ: Then there exist

a1; a2 2 A such that u 2 f ða1Þ � X and u 2 f ða2Þ � Y:

Table 3 Tabular representation of the soft set T

u1 u2 u3 u4 u5 u6 u7 u8

e1 0 0 0 0 0 0 1 1

e2 1 0 0 0 1 1 0 0

e3 1 1 0 1 0 0 0 0

e4 0 0 0 1 0 0 1 0


123

Since by hypothesis S is an intersection complete soft set,

we deduce that there exists a3 2 A such that

u 2 f ða3Þ ¼ f ða1Þ \ f ða2Þ � X \ Y :

Hence, u 2 aprPðX \ YÞ as required.

Using the above assertion, one can verify the following

result on lower soft rough equal relations.

Theorem 7 Let S ¼ ðf ;AÞ be an intersection complete

soft set over U and P ¼ ðU;SÞ be a soft approximation

space. Then, we have

1. X�� P

Y , X�� PðX \ YÞ�

� PY;

2. X�� P

X1; Y �� PY1 ) ðX \ YÞ�

� PðX1 \ Y1Þ;

3. X�� P

Y ) X \ ðU � YÞ�� P;;

4. X � Y; Y �� P; ) X�

� P;;

5. X � Y;X�� P

U ) Y �� P

U;

where X; Y ;X1; Y1 � U:

Proof This can be obtained from Proposition 2 using

similar techniques as in the proof of Theorem 6.

7 Soft–rough fuzzy sets

In this section, we shall consider lower and upper soft

rough approximations of fuzzy sets in a soft approximation

space, and obtain a new hybrid model called soft–rough

fuzzy sets, which can be seen as an extension of Dubois

and Prade’s rough fuzzy sets.

Definition 18 A soft set S ¼ ðf ;AÞ over U is called a full

soft set ifS

a2A FðaÞ ¼ U:

Definition 19 A full soft set S ¼ ðf ;AÞ over U is called a

covering soft set if FðaÞ 6¼ ;; 8a 2 A:

Definition 20 Let S ¼ ðf ;AÞ be a full soft set over U and

S ¼ ðU;SÞ be a soft approximation space. For a fuzzy set

l 2FðUÞ; the lower and upper soft rough approximations

of l with respect to S are denoted by sapSðlÞ and sapSðlÞ;

respectively, which are fuzzy sets in U given by

sapSðlÞðxÞ ¼

^flðyÞ : 9a 2 A½fx; yg � FðaÞ�g;

and

sapSðlÞðxÞ ¼_flðyÞ : 9a 2 A½fx; yg � FðaÞ�g;

for all x 2 U: The operators sapS

and sapS are called the

lower and upper soft rough approximation operators on

fuzzy sets. If sapSðlÞ ¼ sapSðlÞ; l is said to be soft

definable; otherwise l is called a soft–rough fuzzy set.

Theorem 8 Let S ¼ ðf ;AÞ be a full soft set over U; S ¼ðU;SÞ be a soft approximation space and l; m 2FðUÞ:Then we have

1. sapSðlÞ � l � sapSðlÞ;

2. sapSð;Þ ¼ sapSð;Þ ¼ ;;

3. sapSðUÞ ¼ sapSðUÞ ¼ U;

4. ðsapSðlÞÞc ¼ sapSðlcÞ;

5. ðsapSðlÞÞc ¼ sapSðlcÞ;

6. sapSðl \ mÞ ¼ sap

SðlÞ \ sap

SðmÞ;

7. sapSðl [ mÞ � sap

SðlÞ [ sap

SðmÞ;

8. sapSðl [ mÞ ¼ sapSðlÞ [ sapSðmÞ;9. sapSðl \ mÞ � sapSðlÞ \ sapSðmÞ;

10. l � m) sapSðlÞ � sap

SðmÞ;

11. l � m) sapSðlÞ � sapSðmÞ:

Proof

(1) Let l 2FðUÞ and x 2 U: Since S ¼ ðf ;AÞ is a full

soft set over U; there exists some a0 2 A such that

x 2 Fða0Þ: By definition, we have

sapSðlÞðxÞ ¼

^flðyÞ : 9a 2 A½fx; yg � FðaÞ�g;

and

sapSðlÞðxÞ ¼_flðyÞ : 9a 2 A½fx; yg � FðaÞ�g:

Hence, it follows that sapSðlÞðxÞ� lðxÞ� sapS

ðlÞðxÞ: This shows that sapSðlÞ � l � sapSðlÞ:



(4) Let l 2FðUÞ; x 2 U and let

NðxÞ ¼ fy : 9a 2 A½fx; yg � FðaÞ�g:

Note that sapSðlÞðxÞ ¼WflðyÞ : y 2 NðxÞg: Hence,

we have that lðyÞ� sapSðlÞðxÞ for all y 2 NðxÞ:Now it follows that

ðsapSðlÞÞcðxÞ ¼ 1� sapSðlÞðxÞ� 1� lðyÞ¼ lcðyÞ;

for all y 2 NðxÞ: Thus we deduce that

ðsapSðlÞÞcðxÞ�^flcðyÞ : y 2 NðxÞg

¼ sapSðlcÞðxÞ:

This shows that ðsapSðlÞÞc � sap

SðlcÞ:

Next, we only need to prove the reverse inclusion

ðsapSðlÞÞc � sapSðlcÞ . To see this, note first that by

virtue of similar methods as used above, we can

prove that ðsapSðmÞÞc � sapSðmcÞ hold for all fuzzy

set m 2FðUÞ . Taking m ¼ lc , it follows that

ðsapSðlcÞÞc � sapSðlÞ . Consequently we have

sapSðlcÞ � ðsapSðlÞÞ

cas required.


(6) Let l; m 2FðUÞ; x 2 U and let


At first, note that

908 F. Feng et al.

123

sapSðl \ mÞðxÞ ¼

^flðyÞ ^ mðyÞ : y 2 NðxÞg:

Hence, sapSðl \ mÞðxÞ� lðyÞ ^ mðyÞ� lðyÞ for all

y 2 NðxÞ: Since sapSðlÞðxÞ ¼

VflðyÞ : y 2 NðxÞg;

it follows that sapSðl \ mÞðxÞ� sap

SðlÞðxÞ:

Similarly, we obtain sapSðl \ mÞðxÞ� sap

SðmÞðxÞ:

Therefore,

sapSðl \ mÞðxÞ� sap

SðlÞðxÞ ^ sap

SðmÞðxÞ:

This says that sapSðl \ mÞ � sap

SðlÞ \ sap

SðmÞ:

Now it remains to show the reverse inclusion. To

prove this, notefirst that

ðsapSðlÞ \ sap

SðmÞÞðxÞ ¼ sap

SðlÞðxÞ ^ sap

SðmÞðxÞ

� sapSðlÞðxÞ� lðyÞ;

for all y 2 NðxÞ . In a similar way, we have

ðsapSðlÞ \ sap

SðmÞÞðxÞ� sap

SðmÞðxÞ� mðyÞ

for all y 2 NðxÞ . Thus

ðsapSðlÞ \ sap

SðmÞÞðxÞ� lðyÞ ^ mðyÞ

for all y 2 NðxÞ . Now it follows that

ðsapSðlÞ \ sap

SðmÞÞðxÞ�

^flðyÞ ^ mðyÞ : y 2 NðxÞg

¼ sapSðl \ mÞðxÞ:

Thus, sapSðl \ mÞ � sap

SðlÞ \ sap

SðmÞ as required.



Then it is clear that

sapSðlÞðxÞ ¼

^flðyÞ : y 2 NðxÞg� lðyÞ

� lðyÞ _ mðyÞ;

for all y 2 NðxÞ: Thus we have

sapSðl [ mÞðxÞ ¼

^flðyÞ _ mðyÞ : y 2 NðxÞg

sapSðlÞðxÞ:

Similarly, we obtain that sapSðl [ mÞðxÞ sap

SðmÞðxÞ: Hence, it follows that

sapSðl [ mÞðxÞ sap

SðlÞðxÞ _ sap

SðmÞðxÞ

¼ ðsapSðlÞ [ sap

SðmÞÞðxÞ:

Thus, we conclude that sapSðl [ mÞ � sap

SðlÞ[

sapSðmÞ:





If l � m; then it is easy to see that

sapSðlÞðxÞ ¼

^flðyÞ : y 2 NðxÞg� lðyÞ� mðyÞ;

for all y 2 NðxÞ:(11) This is similar to the proof of (10).

To illustrate soft–rough approximations of fuzzy sets, let

us consider the following example which is a continuation

to Example 1.

Example 4 Let U; E and the soft set S ¼ ðF;EÞ over U

be the same as in Example 1. Let S ¼ ðU;SÞ be a soft

approximation space. Then for fuzzy set

l ¼ f0:8=h1; 0:5=h2; 0:7=h3; 0:2=h4; 0:3=h5g;

by definition we compute

sapSðlÞ ¼ f0:2=h1; 0:5=h2; 0:3=h3; 0:2=h4; 0:3=h5g;

and

sapSðlÞ ¼ f0:8=h1; 0:8=h2; 0:8=h3; 0:8=h4; 0:7=h5g:

Similarly for fuzzy set

m ¼ f0:1=h1; 0:3=h2; 0:6=h3; 0:8=h4; 0:5=h5g;

we have

sapSðmÞ ¼ f0:1=h1; 0:1=h2; 0:1=h3; 0:1=h4; 0:5=h5g;

and

sapSðmÞ ¼ f0:8=h1; 0:6=h2; 0:6=h3; 0:8=h4; 0:6=h5g:

It is easy to see that

l [ m ¼ f0:8=h1; 0:5=h2; 0:7=h3; 0:8=h4; 0:5=h5g;

and

l \ m ¼ f0:1=h1; 0:3=h2; 0:6=h3; 0:2=h4; 0:3=h5g:

By computation we obtain

sapSðl [ mÞ ¼ f0:5=h1; 0:5=h2; 0:5=h3; 0:8=h4; 0:5=h5g;

and

sapSðl \ mÞ ¼ f0:6=h1; 0:6=h2; 0:6=h3; 0:2=h4; 0:6=h5g:

Furthermore, we have

sapSðlÞ[sap

SðmÞ¼f0:2=h1;0:5=h2;0:3=h3;0:2=h4;0:5=h5g;

and

sapSðlÞ\sapSðmÞ¼f0:8=h1;0:6=h2;0:6=h3;0:8=h4;0:6=h5g:

This shows that the inclusions in Theorem 8 may hold

strictly.

Definition 21 A soft set S ¼ ðf ;AÞ over U is called a

partition soft set if fFðaÞ : a 2 Ag forms a partition of U:


123

By definition, every partition soft set is a covering soft

set. The following example shows that every quotient set

may be considered a partition soft set.

Example 5 Let R be an equivalence relation on U: Then

the set-valued mapping fR : U ! PðUÞ in Theorem 2

coincides with the natural mapping of the equivalence

relation R: That is, fRðxÞ ¼ ½x�R for all x 2 U: Then the

canonical soft set SR of the equivalence relation R can be

identified with the quotient set U=R: Moreover, we claim

that SR ¼ ðfR;UÞ is a partition soft set since ffRðxÞ : x 2Ug ¼ U=R is a partition of U:

Theorem 9 Let R be an equivalence relation on U: Let

SR ¼ ðfR;UÞ be the canonical soft set of R and S ¼ðU;SRÞ be the soft approximation space. Then,

sapSðlÞðxÞ ¼

^flðyÞ : y 2 ½x�Rg;

and

sapSðlÞðxÞ ¼_flðyÞ : y 2 ½x�Rg;

where l 2FðUÞ; x 2 U: Thus in this case, l is a rough

fuzzy set with respect to the approximation space ðU;RÞ if

and only if l is a soft–rough fuzzy set with respect to the

soft approximation space S ¼ ðU;SRÞ:

Proof Let l 2FðUÞ and x 2 U: Since SR ¼ ðfR;UÞ is

the canonical soft set of R and S ¼ ðU;SRÞ; by definition

we have

sapSðlÞðxÞ ¼

^flðyÞ : 9a 2 U; fx; yg � fRðaÞg

¼^flðyÞ : 9a 2 U; fx; yg � ½a�Rg

¼^flðyÞ : y 2 ½x�Rg:

The second assertion

sapSðlÞðxÞ ¼_flðyÞ : y 2 ½x�Rg

can be proved in a similar way.

Theorem 10 Let S ¼ ðf ;AÞ be a partition soft set over U

and S ¼ ðU;SÞ be the soft approximation space. Define a

binary relation R on U by

ðx; yÞ 2 R, 9a 2 A; fx; yg � FðaÞ;

where x; y 2 U: Then R is an equivalence relation on U

such that

sapSðlÞðxÞ ¼

^flðyÞ : y 2 ½x�Rg;

and

sapSðlÞðxÞ ¼_flðyÞ : y 2 ½x�Rg;

where l 2FðUÞ; x 2 U: Thus in this case, l is a soft–

rough fuzzy set with respect to the soft approximation

space S ¼ ðU;SÞ if and only if l is a rough fuzzy set with

respect to the approximation space ðU;RÞ:

Proof First, we show that the relation R induced by the

partition soft set S ¼ ðf ;AÞ is an equivalence relation on

U: For any x 2 U; since S ¼ ðf ;AÞ is clearly a full soft set,

there exists a 2 A such that x 2 FðaÞ; and so ðx; xÞ 2 R;

which shows that R is reflexive. If ðx; yÞ 2 R; then there

exists a 2 A such that fx; yg ¼ fy; xg � FðaÞ: Thus, we

deduce that ðy; xÞ 2 R; indicating that R is symmetric.

Assume that ðx; yÞ 2 R and ðy; zÞ 2 R: Then there exist

a; b 2 A such that fx; yg � FðaÞ and fy; zg � FðbÞ: Hence,

FðaÞ \ FðbÞ 6¼ ;: But fFðaÞ : a 2 Ag is a partition of U for

S ¼ ðf ;AÞ is a partition soft set over U: It follows that

FðaÞ ¼ FðbÞ and so ðx; zÞ 2 R: This shows that R is tran-

sitive as required.

Now let l 2FðUÞ and x 2 U: By definition,

sapSðlÞðxÞ ¼

^flðyÞ : 9a 2 A; fx; yg � FðaÞg

¼^flðyÞ : ðx; yÞ 2 Rg

¼^flðyÞ : y 2 ½x�Rg:

The second assertion

sapSðlÞðxÞ ¼_flðyÞ : y 2 ½x�Rg

can be proved in a similar way.

From the above results, one easily sees that (classical)

rough fuzzy sets can be identified with soft–rough fuzzy

sets when the underlying soft set in the soft approximation

space is a partition soft set. Consequently, every rough

fuzzy set may be considered a soft–rough fuzzy set.

However, we claim that the reverse statement is generally

not true, since a (full) soft set is not necessarily a partition

soft set. In this sense, the notion of soft–rough fuzzy sets

can be seen as a natural generalization of rough fuzzy sets

by applying soft set theory.

8 Conclusions

We have investigated in this paper the problem of combing

soft sets with fuzzy sets and rough sets. In general, three

different types of hybrid models are presented, which are

called rough soft sets, soft rough sets and soft–rough fuzzy

sets, respectively. A rough soft set is the approximation of

a soft set in a Pawlak approximation space, whereas a soft

rough set is based on soft rough approximations in a soft

approximation space. The approximation of a fuzzy set in a

soft approximation space is also investigated to obtain

soft–rough fuzzy sets which extend Dubois and Prade’s

rough fuzzy sets in a natural way. In addition, we have

investigated some basic properties of these new

910 F. Feng et al.

123

hybridizations with illustrating examples. Further study

will be needed to establish whether the notions put forth in

this paper may lead to a fruitful theory.

Acknowledgments We are highly grateful to the anonymous ref-

erees for their helpful comments and suggestions for improving the

paper. We are indebted to Dr. Brunella Gerla and Dr. Vincenzo Marra

for their kindly help. This work is supported by a grant (No. 08JK432)

from the Education Department of Shaanxi Province of China, and by

the Shaanxi Provincial Research and Development Plan of Science

and Technology under Grant No. 2008K0133.

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Soft sets combined with fuzzy sets and rough sets: a tentative approach