# Soft Rough Approximation Operators and Related Results

Authors: Zhaowen Li; Bin Qin; Zhangyong Cai
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Soft set theory is a newly emerging tool to deal with uncertain problems. Based on soft sets, soft rough approximation operators are introduced, and soft rough sets are defined by using soft rough approximation operators. Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model. This paper is devoted to investigating soft rough approximation operations and relationships among soft sets, soft rough sets, and topologies. We consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are investigated, and their related properties are given. We show that Pawlak's rough set model can be viewed as a special case of soft rough sets, obtain the structure of soft rough sets, give the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space.


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## Body

## 1. Introduction

Most of traditional methods for formal modeling, reasoning, and computing are crisp, deterministic, and precise in character. However, many practical problems within fields such as economics, engineering, environmental science, medical science, and social sciences involve data that contain uncertainties. We cannot use traditional methods because of various types of uncertainties present in these problems.There are several theories: probability theory, fuzzy set theory, theory of interval mathematics, and rough set theory [1], which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties (see [2]). For example, theory of probabilities can deal only with stochastically stable phenomena. To overcome these kinds of difficulties, Molodtsov [2] proposed a completely new approach, which is called soft set theory, for modeling uncertainty.Presently, works on soft set theory are progressing rapidly. Maji et al. [3–5] further studied soft set theory, used this theory to solve some decision making problems, and devoted fuzzy soft sets combining soft sets with fuzzy sets. Roy et al. [6] presented a fuzzy soft set theoretic approach towards decision making problems. Jiang et al. [7] extended soft sets with description logics. Aktaş et al. [8] defined soft groups. Feng et al. [9, 10] investigated relationships among soft sets, rough sets, and fuzzy sets. Shabir et al. [11] investigated soft topological spaces. Ge et al. [12] discussed relationships between soft sets and topological spaces.The purpose of this paper is to investigate soft rough approximation operators and relationships among soft sets, soft rough sets, and topologies.The remaining part of this paper is organized as follows. In Section2, we recall some basic concepts of rough sets and soft sets. In Section 3, we consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are introduced or investigated, and the fact that Pawlak’s rough set model can be viewed as a special case of soft rough sets is proved. In Section 4, we investigate the relationships between soft sets and topologies, obtain the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space. In Section 5, we give the related properties of soft rough sets and obtain the structure of soft rough sets. In Section 6, we prove that there exists a one-to-one correspondence between the set of all soft sets and the set of all formal contexts. Conclusion is in Section 7.


## 2. Overview of Rough Sets and Soft Sets

In this section, we recall some basic concepts about rough sets and soft sets.Throughout this paper,U denotes initial universe, E denotes the set of all possible parameters, and 2U denotes the family of all subsets of U. We only consider the case where both U and E are nonempty finite sets.


### 2.1. Rough Sets

Rough set theory was initiated by [1] for dealing with vagueness and granularity in information systems. This theory handles the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems, and many other fields (see [1, 13]).LetR be an equivalence relation on U. The pair (U,R) is called a Pawlak approximation space. The equivalence relation R is often called an indiscernibility relation. Using the indiscernibility relation R, one can define the following two rough approximations:
(1)R*(X)={x∈U:[x]R⊆X},R*(X)={x∈U:[x]R∩X≠∅}.R*(X) and R*(X) are called the Pawlak lower approximation and the Pawlak upper approximation of X, respectively. In general, we refer to R* and R* as Pawlak rough approximation operators and R*(X) and R*(X) as Pawlak rough approximations of X.The Pawlak boundary region ofX is defined by the difference between these Pawlak rough approximations; that is, BndR(X)=R*(X)-R*(X). It can easily be seen that R*(X)⊆X⊆R*(X).A set is Pawlak rough if its boundary region is not empty; otherwise, the set is crisp. Thus,X is Pawlak rough if R*(X)≠R*(X).We may relax equivalence relations so that rough set theory is able to solve more complicated problems in practice. The classical rough set theory based on equivalence relations has been extended to binary relations [14].Definition 1 (see [14]).
LetR be a binary relation on U. The pair (U,R) is called an approximation space. Based on the approximation space (U,R), we define a pair of operations R_,  R¯:2U→2U as follows:
(2)R_(X)={x∈U:R(x)⊆X},R¯(X)={x∈U:R(x)∩X≠∅},
where  X∈2U and R(x)={y∈U:xRy}.
R
_
(
X
) and R¯(X) are called the lower approximation and the upper approximation of X, respectively. In general, we refer to R_ and R¯ as rough approximation operators and R_(X) and R¯(X) as rough approximations of X.
X is called a definable set if R_(X)=R¯(X);  X is called a rough set if R_(X)≠R¯(X).


### 2.2. Soft Sets

Definition 2 (see [2]).
LetA be a nonempty subset of E. A pair (f,A) is called a soft set over U, if f is a mapping given by f:A→2U. We denote (f,A) by fA.In other words, a soft set overU is a parametrized family of subsets of the universe U. For e∈A, f(e) may be considered as the set of e-approximate elements of the soft set fA.Definition 3 (see [3]).
LetfA and gB be two soft sets over U.(1)
fA is called a soft subset of gB, if A⊆B and f(e)=g(e) for each e∈A. We denote it by  fA⊂~gB.(2)
fA is called a soft super set of gB, if  gB⊂~fA. We denote it by fA⊃~gB.Definition 4 (see [3]).
LetfA and gB be two soft sets over U. fA and gB are called soft equal, if A=B and f(e)=g(e) for each e∈A. We denote it by  fA=gB.Obviously,fA=gB if and only if fA⊂~gB and fA⊃~gB.Definition 5 (see [10, 12]).
LetfA be a soft set over U.(1)
fA is called full, if ⋃a∈Af(a)=U.(2)
fA is called partition, if {f(a):a∈A} forms a partition of U.Obviously, every partition soft set is full.Definition 6.
LetfA be a soft set over U.(1)
fA is called keeping intersection, if for any a,b∈A, there exists c∈A such that f(a)∩f(b)=f(c).(2)
fA is called keeping union, if for any a,b∈A, there exists c∈A such that f(a)∪f(b)=f(c).(3)
fA is called topological, if {f(a):a∈A} is a topology on U.Obviously, every topological soft set is full, keeping intersection and keeping union, andfA is keeping intersection (resp., keeping union) if and only if for any A′⊆A, there exists a′∈A such that ⋂a∈A′f(a)=f(a′) (resp., ⋃a∈A′f(a)=f(a′)).Example 7.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(3)f(a1)={h1,h2,h5},f(a2)=∅,f(a3)={h3},f(a4)={h3,h4}.
Obviously,fA is not partition. We have
(4)f(a3)∩f(a4)={h3}=f(a3),f(a1)∩f(a2)=f(a1)∩f(a3)=f(a1)∩f(a4)aa=f(a2)∩f(a3)=f(a2)∩f(a4)aa=∅=f(a2).
Then,fA is full and keeping intersection. But
(5)f(a1)∪f(a3)={h1,h2,h3,h5}≠f(a)(∀a∈A).
Thus,fA is not keeping union.Example 8.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(6)f(a1)={h1},f(a2)={h1,h2},f(a3)={h1,h2,h3},f(a4)={h1,h2,h3,h4}.
Then,fA is keeping intersection and keeping union. But fA is not full.Example 9.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(7)f(a1)={h1},f(a2)={h2},f(a3)={h1,h2},f(a4)=U.
Then,fA is full and keeping union. But fA is neither keeping intersection nor partition.Example 10.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(8)f(a1)={h1,h2},f(a2)={h5},f(a3)={h3},f(a4)={h4}.
Obviously,fA is partition. But
(9)f(a2)∩f(a3)=∅≠f(a)(∀a∈A),f(a1)∪f(a2)={h1,h2,h5}≠f(a)(∀a∈A).
Thus,fA is neither keeping intersection nor keeping union.Example 11.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4} and let fA be a soft set over U, defined as follows
(10)f(a1)={h1},f(a2)={h1,h4},f(a3)={h1,h3,h4},f(a4)=X.
Obviously,fA is full, keeping intersection and keeping union. But fA is not topological.From Examples7, 8, 9, and 10 and 11, we have the following relationships:


## 2.1. Rough Sets

Rough set theory was initiated by [1] for dealing with vagueness and granularity in information systems. This theory handles the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems, and many other fields (see [1, 13]).LetR be an equivalence relation on U. The pair (U,R) is called a Pawlak approximation space. The equivalence relation R is often called an indiscernibility relation. Using the indiscernibility relation R, one can define the following two rough approximations:
(1)R*(X)={x∈U:[x]R⊆X},R*(X)={x∈U:[x]R∩X≠∅}.R*(X) and R*(X) are called the Pawlak lower approximation and the Pawlak upper approximation of X, respectively. In general, we refer to R* and R* as Pawlak rough approximation operators and R*(X) and R*(X) as Pawlak rough approximations of X.The Pawlak boundary region ofX is defined by the difference between these Pawlak rough approximations; that is, BndR(X)=R*(X)-R*(X). It can easily be seen that R*(X)⊆X⊆R*(X).A set is Pawlak rough if its boundary region is not empty; otherwise, the set is crisp. Thus,X is Pawlak rough if R*(X)≠R*(X).We may relax equivalence relations so that rough set theory is able to solve more complicated problems in practice. The classical rough set theory based on equivalence relations has been extended to binary relations [14].Definition 1 (see [14]).
LetR be a binary relation on U. The pair (U,R) is called an approximation space. Based on the approximation space (U,R), we define a pair of operations R_,  R¯:2U→2U as follows:
(2)R_(X)={x∈U:R(x)⊆X},R¯(X)={x∈U:R(x)∩X≠∅},
where  X∈2U and R(x)={y∈U:xRy}.
R
_
(
X
) and R¯(X) are called the lower approximation and the upper approximation of X, respectively. In general, we refer to R_ and R¯ as rough approximation operators and R_(X) and R¯(X) as rough approximations of X.
X is called a definable set if R_(X)=R¯(X);  X is called a rough set if R_(X)≠R¯(X).


## 2.2. Soft Sets

Definition 2 (see [2]).
LetA be a nonempty subset of E. A pair (f,A) is called a soft set over U, if f is a mapping given by f:A→2U. We denote (f,A) by fA.In other words, a soft set overU is a parametrized family of subsets of the universe U. For e∈A, f(e) may be considered as the set of e-approximate elements of the soft set fA.Definition 3 (see [3]).
LetfA and gB be two soft sets over U.(1)
fA is called a soft subset of gB, if A⊆B and f(e)=g(e) for each e∈A. We denote it by  fA⊂~gB.(2)
fA is called a soft super set of gB, if  gB⊂~fA. We denote it by fA⊃~gB.Definition 4 (see [3]).
LetfA and gB be two soft sets over U. fA and gB are called soft equal, if A=B and f(e)=g(e) for each e∈A. We denote it by  fA=gB.Obviously,fA=gB if and only if fA⊂~gB and fA⊃~gB.Definition 5 (see [10, 12]).
LetfA be a soft set over U.(1)
fA is called full, if ⋃a∈Af(a)=U.(2)
fA is called partition, if {f(a):a∈A} forms a partition of U.Obviously, every partition soft set is full.Definition 6.
LetfA be a soft set over U.(1)
fA is called keeping intersection, if for any a,b∈A, there exists c∈A such that f(a)∩f(b)=f(c).(2)
fA is called keeping union, if for any a,b∈A, there exists c∈A such that f(a)∪f(b)=f(c).(3)
fA is called topological, if {f(a):a∈A} is a topology on U.Obviously, every topological soft set is full, keeping intersection and keeping union, andfA is keeping intersection (resp., keeping union) if and only if for any A′⊆A, there exists a′∈A such that ⋂a∈A′f(a)=f(a′) (resp., ⋃a∈A′f(a)=f(a′)).Example 7.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(3)f(a1)={h1,h2,h5},f(a2)=∅,f(a3)={h3},f(a4)={h3,h4}.
Obviously,fA is not partition. We have
(4)f(a3)∩f(a4)={h3}=f(a3),f(a1)∩f(a2)=f(a1)∩f(a3)=f(a1)∩f(a4)aa=f(a2)∩f(a3)=f(a2)∩f(a4)aa=∅=f(a2).
Then,fA is full and keeping intersection. But
(5)f(a1)∪f(a3)={h1,h2,h3,h5}≠f(a)(∀a∈A).
Thus,fA is not keeping union.Example 8.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(6)f(a1)={h1},f(a2)={h1,h2},f(a3)={h1,h2,h3},f(a4)={h1,h2,h3,h4}.
Then,fA is keeping intersection and keeping union. But fA is not full.Example 9.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(7)f(a1)={h1},f(a2)={h2},f(a3)={h1,h2},f(a4)=U.
Then,fA is full and keeping union. But fA is neither keeping intersection nor partition.Example 10.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(8)f(a1)={h1,h2},f(a2)={h5},f(a3)={h3},f(a4)={h4}.
Obviously,fA is partition. But
(9)f(a2)∩f(a3)=∅≠f(a)(∀a∈A),f(a1)∪f(a2)={h1,h2,h5}≠f(a)(∀a∈A).
Thus,fA is neither keeping intersection nor keeping union.Example 11.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4} and let fA be a soft set over U, defined as follows
(10)f(a1)={h1},f(a2)={h1,h4},f(a3)={h1,h3,h4},f(a4)=X.
Obviously,fA is full, keeping intersection and keeping union. But fA is not topological.From Examples7, 8, 9, and 10 and 11, we have the following relationships:


## 3. Soft Rough Approximation Operators and Soft Rough Sets

Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model (see Example 4.6 in [10]), and defining soft rough sets and some related concepts needs using soft rough approximation operators based on soft sets. Thus, soft rough approximation operators deserve further research.In this section, we consider a pair of soft rough approximation operators which are presented by Feng et al. in [9, 10], proposing three pairs of soft rough approximation operators and giving their properties. Four sorts of soft rough sets are defined by using four pairs of soft rough approximation operators.


### 3.1. Soft Rough Approximation Operatorsapr_P and apr¯P

Definition 12 (see [9, 10]).
LetfA be a soft set over U. Then, the pair P=(U,fA) is called a soft approximation space. We define a pair of operators apr_P, apr¯P:2U→2U as follows:
(11)apr_P(X)={u∈U:∃a∈A,s.t.u∈f(a)⊆X},apr¯P(X)={u∈U:∃a∈A,s.t.u∈f(a),f(a)∩X≠∅}.
apr
_
P and apr¯P are called the soft P-lower approximation operator on U and the soft P-upper approximation operator on U, respectively. In general, we refer to apr_P and apr¯P as soft P-rough approximations operator on U.
apr
_
P
(
X
) and apr¯P(X) are called the soft P-lower approximation and the soft P-upper approximation of X, respectively. In general, we refer to apr_P(X) and apr¯P(X) as soft rough approximations of X with respect to P.
X is called a soft P-definable set if apr_P(X)=apr¯P(X);  X is called a soft P-rough set if apr_P(X)≠apr¯P(X).
Moreover, the sets(12)PosP(X)=apr_P(X),NegP(X)=U-apr¯P(X),BndP(X)=apr¯P(X)-apr_P(X)
are called the soft P-positive region, the soft P-negative region, and the soft P-boundary region of A, respectively.Proposition 13 (see [9, 10]).
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X,Y∈2U.  (1)
apr_P(X)=⋃‍{f(a):a∈Aandf(a)⊆X}⊆X;apr¯P(X)=⋃‍{f(a):a∈Aandf(a)∩X≠∅}.(2)
apr_P(∅)=apr¯P(∅)=∅;apr_P(U)=apr¯P(U)=⋃a∈Af(a).(3)
X⊆Y⇒apr_P(X)⊆apr_P(Y);X⊆Y⇒apr¯P(X)⊆apr¯P(Y).(4)
apr¯P(X∪Y)=apr¯P(X)∪apr¯P(Y).(5)
apr_P(apr_P(X))=apr_P(X);apr_P(apr¯P(X))=apr¯P(X).Proposition 14.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold.(1)
IffAis full, then
(a)
apr_P(X)⊆X⊆apr¯P(X)for  anyX∈2U;(b)
apr_P(U)=apr¯P(U)=U.(2)
IffAis keeping union, then
(a)
for  anyX∈2U, there  existsa∈Asuch  thatapr_P(X)=f(a);(b)
for  anyX∈2U, there  existsa∈Asuch  thatapr¯P(X)=f(a).(3)
IffAis keeping intersection, then(13)apr_P(X∩Y)=apr_P(X)∩apr_P(Y)foranyX,Y∈2U.(4)
IffAis partition, then(14)apr_P(X∩Y)=apr_P(X)∩apr_P(Y)foranyX,Y∈2U.(5)
IffAis full and keeping union, then(15)apr¯P(X)=UforanyX∈2U∖{∅}.Proof.
(1)(a) By Proposition13,  apr_P(X)⊆X. Suppose that X-apr¯P(X)≠∅. Pick
(16)x∈X-apr¯P(X)≠∅.
Since fA is full, U=⋃a∈Af(a). So, x∈f(a) for some a∈A. x∈X implies f(a)∩X≠∅. Thus, x∈apr¯P(X)≠∅, contradiction. Hence,
(17)X⊆apr¯P(X).
(1)(b) This holds by (1) and Proposition13.
(2)  This holds by Proposition13.
(3)  By Proposition13,
(18)apr_P(X∩Y)⊆apr_P(X)∩apr_P(Y).
Suppose thatapr_P(X)∩apr_P(Y)-apr_P(X∩Y)≠∅. Pick
(19)x∈apr_P(X)∩apr_P(Y)-apr_P(X∩Y).
Then, there exista,b∈A such that x∈f(a)⊆X and x∈f(b)⊆Y. Since fA is keeping intersection, then f(a)∩f(b)=f(c) for some c∈A. This implies x∈f(c)⊆X∩Y. Thus, x∈apr_P(X∩Y), contradiction. Hence,
(20)apr_P(X∩Y)⊇apr_P(X)∩apr_P(Y).
Therefore,(21)apr_P(X∩Y)=apr_P(X)∩apr_P(Y).
(4) Suppose thatapr_P(X)∩apr_P(Y)-apr_P(X∩Y)≠∅. Pick
(22)x∈apr_P(X)∩apr_P(Y)-apr_P(X∩Y).
Then, there exista,b∈A such that x∈f(a)⊆X and x∈f(b)⊆Y. Since fA is partition, then f(a)=f(b). This implies x∈f(a)⊆X∩Y. So, x∈apr_P(X∩Y), contradiction. Thus,
(23)apr_P(X∩Y)⊇apr_P(X)∩apr_P(Y).
Therefore,(24)apr_P(X∩Y)=apr_P(X)∩apr_P(Y).
(5)  SincefA is full and keeping union, then U=⋃a∈Af(a)=f(a*) for some a*∈A. For each X∈2U∖{∅} and each u∈U, u∈f(a*) and f(a*)∩X=X≠∅, and then apr¯P(X)=U.


### 3.2. Soft Rough Approximation Operatorsapr_P′ and apr¯P′, apr_P′′ and apr¯P′′, and apr_P′′′ and apr¯P′′′

Definition 15.
LetfA be a soft set over U.(1)
Define a binary relationRf on U by
(25)xRfy⟺∃a∈A,{x,y}⊆f(a)
for each x,y∈U. Then, Rf is called the binary relation induced by fA on U.(2)
For eachx∈U, define a successor neighborhood Rfs(x) of x in U by
(26)(Rf)s(x)={y∈U:xRfy}.Since the following Proposition16 is clear, we omit its proof.Proposition 16.
LetfA be a soft set over U, and let Rf be the binary relation induced by fA on U. Then, the following properties hold.(1)
Rfis a symmetric relation.(2)
IffAis full, thenRfis a reflexive relation.(3)
IffAis partition, thenRfis an equivalence relation.Proposition 17.
LetfA be a soft set over U, and let Rf be the binary relation induced by fA on U. Then, the following properties hold.(1)
Ifx∈f(a)witha∈A, thenf(a)⊆(Rf)s(x).(2)
IffAis partition andx∈f(a)witha∈A, thenf(a)=(Rf)s(x).(3)
IffAis keeping union, then for eachx∈U, there existsa∈Asuch that(Rf)s(x)=f(a).Proof.
(1) This is obvious.
(2) Suppose thaty∈(Rf)s(x). Then, xRfy, and so {x,y}⊆f(b) for some b∈A. Since fA is partition and f(a)∩f(b)≠∅, then f(a)=f(b). Thus, y∈f(a). This implies f(a)⊇(Rf)s(x). By (1),
(27)f(a)=(Rf)s(x).
(3) Suppose thaty∈(Rf)s(x). Then, xRfy, and so {x,y}⊆f(ay) for some ay∈A. By (1), f(ay)⊆(Rf)s(x). Thus, {y}⊆f(ay)⊆(Rf)s(x). This implies
(28)(Rf)s(x)=⋃{f(ay):y∈(Rf)s(x)}.
SincefA is keeping union, then ⋃‍{f(ay):y∈(Rf)s(x)}=f(a) for some a∈A. Thus,
(29)(Rf)s(x)=f(a).Definition 18.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. We define three pairs of soft rough approximation operations: 2U→2U as follows:
(1)(30)apr_P′(X)={x∈U:(Rf)s(x)⊆X},apr¯P′(X)={x∈U:(Rf)s(x)∩X≠∅}.
X is called a soft P′-definable set if apr_P(X)=apr¯P′(X). X is called a soft P′-rough set if  apr_P′(X)≠apr¯P′(X). The sets
(31)PosP′(X)=apr_P′(X),NegP′(X)=U-apr¯P′(X),BndP′(X)=apr¯P′(X)-apr_P′(X)
are called the soft P′-positive region, the soft P′-negative region, and the soft P′-boundary region of X, respectively. Consider,
(2)(32)apr_P′′(X)=⋃{(Rf)s(x):x∈U,(Rf)s(x)⊆X},apr¯P′′(X)=U-apr_P′′(U-X).
X is called a soft P′′-definable set if apr_P′′(X)=apr¯P′′(X). X is called a soft P′′-rough set if apr_P′′(X)≠apr¯P′′(X). The sets
(33)PosP′′(X)=apr_P′′(X),NegP′′(X)=U-apr¯P′′(X),BndP′′(X)=apr¯P′′(X)-apr_P′′(X)
are called the soft P′′-positive region, the soft P′′-negative region, and the soft P′′-boundary region of X, respectively. Consider,
(3)(34)apr¯P′′′(X)=∪{(Rf)s(x):x∈U,(Rf)s(x)∩X≠∅},apr_P′′′(X)=U-apr¯P′′′(U-X).
X is called a soft P′′′-definable set if apr_P′′′(X)=apr¯P′′′(X). X is called a soft P′′′-rough set if apr_P′′′(X)≠apr¯P′′′(X). The sets
(35)PosP′′′(X)=apr_P′′′(X),NegP′′′(X)=U-apr¯P′′′(X),BndP′′′(X)=apr¯P′′′(X)-apr_P′′′(X)
are called the soft P′′′-positive region, the soft P′′′-negative region, and the soft P′′′-boundary region of X, respectively.In general, we also refer toapr_P′(X) and apr¯P′(X),  apr_P′′(X) and apr¯P′′(X), and apr_P′′′(X) and apr¯P′′′(X) as soft rough approximations of X with respect to P′, P′′, P′′′, respectively.It is not very difficult to prove Propositions19, 20, and 21 (see [15]).Proposition 19.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X,Y∈2U.  (1)
apr_P′(X)⊆X. IffAis full, then(36)apr_P′(X)⊆X⊆apr¯P′(X).(2)
apr¯P′(∅)=∅;apr_P′(U)=U.   IffAis full, then(37)apr_P′(∅)=apr¯P′(∅)=∅;apr_P′(U)=apr¯P′(U)=U.(3)
X⊆Y⇒apr_P′(X)⊆apr_P′(Y);X⊆Y⇒apr¯P′(X)⊆apr¯P′(Y).(4)
apr_P′(X∩Y)=apr_P′(X)∩apr_P′(Y).(5)
apr¯P′(X∪Y)=apr¯P′(X)∪apr¯P′(Y).(6)
apr_P′(U-X)=U-apr¯P′(X);apr¯P′(U-X)=U-apr_P′(X).(7)
apr¯P′(apr_P′(X))⊆X⊆apr_P′(apr¯P′(X)).Proposition 20.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.  (1)
(38)apr_P′′(X)⊆X⊆apr¯P′′(X).(2)
apr_P′′(∅)=∅;apr¯P′′(U)=U.   IffAis full, then(39)apr_P′(∅)=apr¯P′(∅)=∅;apr_P′(U)=apr¯P′(U)=U.(3)
apr_P′′(U-X)=U-apr¯P′′(X);apr¯P′′(U-X)=U-apr_P′′(X).(4)
apr_P′′(apr_P′′(X))=apr_P′′(X);apr¯P′′(apr¯P′′(X))=apr¯P′′(X).Proposition 21.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X,Y∈2U.  (1)
IffAis full, then(40)apr_P′′′(X)⊆X⊆apr¯P′′′(X).(2)
apr¯P′′′(∅)=∅;apr_P′′′(U)=U. IffAis full, then(41)apr_P′′′(∅)=apr¯P′′′(∅)=∅;apr_P′′′(U)=apr¯P′(U)=U.(3)
X⊆Y⇒apr_P′′′(X)⊆apr_P′′′(Y);X⊆Y⇒apr¯P′′′(X)⊆apr¯P′′′(Y).(4)
apr_P′′′(X∩Y)=apr_P′′′(X)∩apr_P′′′(Y).(5)
apr¯P′′′(X∪Y)=apr¯P′′′(X)∪apr¯P′′′(Y).(6)
apr_P′′′(U-X)=U-apr¯P′′′(X);apr¯P′′′(U-X)=U-apr_P′′′(X).(7)
apr_P′′′(apr_P′′′(X))⊆apr_P′′′(X);apr¯P′′′(apr¯P′′′(X))⊇apr¯P′′′(X).(8)
apr¯P′′′(apr_P′′′(X))⊆X⊆apr_P′′′(apr¯P′′′(X)).Example 22.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(42)f(a1)={h2},f(a2)={h1,h4},f(a3)={h3},f(a4)={h1,h3}.
Obviously,fA is not full. We have
(43)(Rf)s(h1)={h1,h3,h4},(Rf)s(h2)={h2},(Rf)s(h3)={h1,h3},(Rf)s(h4)={h1,h4},(Rf)s(h5)=∅.
LetX={h3,h5}, Y={h1,h2,h3}, and Z={h2,h4,h5}.(1)
We have(44)apr¯P′(X)={h1,h3}.
Thus,
(45)X⊈apr¯P′(X).(2)
We have(46)apr_P′(X)={h5},apr_P′(Y)={h2,h3,h5}.
Thus,
(47)apr_P′(X)⊆apr_P′(Y)⇏X⊆Y.(3)
We have(48)apr¯P′(X)={h1,h3},apr¯P′(Y)={h1,h2,h3,h4}.
Thus,
(49)apr¯P′(X)⊆apr¯P′(Y)⇏X⊆Y.(4)
We have(50)apr_P′(Y)={h2,h3,h5},apr_P′(Z)={h2,h5},apr_P′(Y∪Z)=U.
Thus,
(51)apr_P′(Y∪Z)≠apr_P′(y)∪apr_P′(Z).(5)
We have(52)apr¯P′(Y)={h1,h2,h3,h4},apr¯P′(Z)={h1,h2,h4},apr¯P′(Y∩Z)={h2}.
Thus,
(53)apr¯P′(Y∩Z)≠apr¯P′(Y)∩apr_P′(Z).


### 3.3. The Relationships among Four Pairs of Soft Rough Approximation Operators

Lemma 23.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
IffA is full, then
(54)apr_P(X)⊇apr_P′(X).(2)
IffA is full and keeping union, then
(55)apr¯P(X)⊇apr¯P′(X).(3)
IffA is partition, then
(a)
apr_P(X)=apr_P′(X);(b)
apr¯P(X)=apr¯P′(X).(c)
apr_P′(X)=apr_P′′(X)=apr_P′′′(X);(d)
apr¯P′=apr¯P′′(X)=apr¯P′′′(X).Proof.
(1) Suppose thatx∈apr_P′(X). Then, (Rf)s(x)⊆X. Since fA is full, then x∈f(a) for some a∈A. By Proposition 17, f(a)⊆(Rf)s(x). Thus, x∈f(a)⊆X. This implies x∈apr_P(X). Thus,
(56)apr_P(X)⊇apr_P′(X).
(2) IfX=∅, then apr¯P(X)=∅=apr¯P′(X). If X≠∅, by Proposition 14, apr¯P(X)=U.
Hence,(57)apr¯P(X)⊇apr¯P′(X).
(3)(a) Suppose thatx∈apr_P(X). Then, x∈f(a)⊆X for some a∈A. Since fA is partition and x∈f(a), then f(a)=(Rf)s(x) by Proposition 17. This implies x∈apr_P′(X). Thus,
(58)apr_P(X)⊆apr_P′(X).
By (1),
(59)apr_P(X)=apr_P′(X).
(3)(b) This is similar to the proof of (3) (a).
Suppose thatx∈apr¯P′(X). Then, (Rf)s(x)∩X≠∅. Since fA is full, then x∈f(a) for some a∈A. Since fA is partition and x∈f(a), then f(a)=(Rf)s(x) by Proposition 17. This implies x∈apr¯P(X). Thus,
(60)apr¯P(X)⊇apr¯P′(X).
Hence,apr¯P(X)=apr¯P′(X).By Propositions13 and 17, we have Lemma 24.Lemma 24.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space.  If fA is keeping union, then for any X∈2U,
(61)apr_P(X)⊇apr_P′′(X),apr¯P(X)⊇apr¯P′′′(X).Lemma 25.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
IffAis full, then(62)apr_P′′′(X)⊆apr_P′(X)⊆apr_P′′(X)⊆X⊆apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X).(2)
IffAis partition, then
(a)
apr_P′(X)=apr_P′′(X)=apr_P′′′(X);(b)
apr¯P′=apr¯P′′(X)=apr¯P′′′(X).Proof.
(1) Suppose thatx∈apr_P′(X). Then, (Rf)s(x)⊆X. Since fA is full, then x∈(Rf)s(x)⊆X by Proposition 17. This implies x∈apr_P′′(X). Thus,
(63)apr_P′(X)⊆apr_P′′(X).
Suppose thatapr_P′′′(X)-apr_P′(X)≠∅. Pick
(64)x∈apr_P′′′(X)-apr_P′(X).
x
∉
apr
_
P
′
(
X
) implies (Rf)s(x)⊈X. So, (Rf)s(x)∩(U-X)≠∅. Since fA is full, then x∈(Rf)s(x) by Proposition 17. This implies x∈apr¯P′′(U-X). Thus, x∉U-apr¯P′′(U-X)=apr_P′′′(X), contradiction.
Hence,apr_P′′′(X)⊆apr_P′(X).
By Proposition20,
(65)apr_P′′(X)⊆X⊆apr¯P′′(X).
Since(66)apr_P′′′(U-X)⊆apr_P′(U-X)⊆apr_P′′(U-X),
then
(67)U-apr_P′′′(U-X)⊇U-apr_P′(U-X)⊇U-apr_P′′(U-X).
By Propositions19, 20, and 21,
(68)apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X).
(2)(a)  Suppose thatx∈apr_P′′(X). Then, there exists y∈U such that x∈(Rf)s(y)⊆X. Since fA is partition, then Rf is an equivalence relation by Proposition 16. Thus, x∈(Rf)s(y) follows (Rf)s(x)=(Rf)s(y). So, (Rf)s(x)⊆X. This implies x∈apr_P′(X). Hence,
(69)apr_P′(X)⊇apr_P′′(X).
By (1),(70)apr_P′(X)=apr_P′′(X).
Suppose thatapr_P′(X)∩apr¯P′′′(U-X)≠∅. Pick
(71)x∈apr_P′(X)∩apr¯P′′′(U-X).
x
∈
apr
_
P
′
(
X
) implies (Rf)s(x)⊆X. x∈apr¯P′′′(U-X) implies that there exists y∈U such that x∈(Rf)s(y) and (Rf)s(y)∩(U-X)≠∅. So, (Rf)s(y)⊈X. Note that Rf is an equivalence relation. Then, (Rf)s(x)=(Rf)s(y). Thus, (Rf)s(x)⊈X, contradiction.
Hence,apr_P′(X)∩apr¯P′′′(U-X)=∅.
This proves that(72)apr_P′(X)⊆U-apr¯P′′′(U-X)=apr_P′′′(X).
By (1),(73)apr_P′(X)=apr_P′′′(X).
(2)(b) By (2)(a),(74)apr_P′(U-X)=apr_P′′(U-X)=apr_P′′′(U-X).
Then,(75)U-apr_P′(U-X)=U-apr_P′′(U-X)=U-apr_P′′′(U-X).
By Propositions19, 20, and 21,
(76)apr¯P′(X)=apr¯P′′(X)=apr¯P′′′(X).Example 26.
LetU={h1,h2,h3,h4,h5,h6}, A={a1,a2,a3,a4,a5}, and let fA be a soft set over U, defined as follows:
(77)f(a1)={h1},f(a2)={h6},f(a3)={h2,h5},f(a4)={h2,h4},f(a5)={h1,h3,h5}.
Obviously,fA is full. We have
(78)(Rf)s(h1)={h1,h3,h5},(Rf)s(h2)={h2,h4,h5},(Rf)s(h3)={h1,h3,h5},(Rf)s(h4)={h2,h4},(Rf)s(h5)={h1,h2,h3,h5}.
LetX={h2,h4,h6}. We have
(79)apr_P(X)={h2,h4,h6},apr_P′(X)={h4},apr_P′′(X)={h2,h4},apr¯P(X)={h2,h4,h5,h6},apr¯P′(X)={h2,h4,h5},apr¯P′′(X)={h1,h2,h3,h4,h5}.
Thus,(80)apr_P(X)≠apr_P′(X),apr_P′(X)≠apr_P′′(X),apr¯P(X)≠apr¯P′(X),apr¯P′(X)≠apr¯P′′(X).By Proposition16 and Lemmas 23, 24, and 25, we have Theorem 27.Theorem 27.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
IffAis full, then(81)apr_P′′′(X)⊆apr_P′(X)⊆apr_P′′(X)⊆X⊆apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X).(2)
IffAis full and keeping union, then(82)apr_P′′′(X)⊆apr_P′(X)⊆apr_P′′(X)⊆apr_P(X)⊆X⊆apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X)⊆apr¯P(X).


### 3.4. The Relationship between Soft Rough Approximation Operators and Pawlak Rough Approximation Operators

In this section, we shall explore the relationship between soft rough approximation operators and Pawlak rough approximation operators.Definition 28.
LetR be an equivalence relation on U. Define a mapping fR:A→2U by
(83)fR(a)=[a]R
for each a∈A, where A=U. Then, (fR)A is called the soft set induced by R on U.Theorem 29 (see [10]).
LetR be an equivalence relation on U, let (fR)A be the soft set induced by R on U, and let P=(U,(fR)A) be a soft approximation space. Then, for each X∈2U,
(84)R_(X)=apr_P(X),R¯(X)=apr¯P(X).
Thus, in this case,X∈2U is a Pawlak rough set if and only if X is a soft P-rough set.By Proposition16 and Lemmas 23 and 25, we have Theorem 30.Theorem 30.
LetfA be a partition soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
Rf_(X)=apr_P(X)=apr_P′(X)=apr_P′′(X)=apr_P′′′(X);(2)
Rf¯(X)=apr¯P(X)=apr¯P′(X)=apr¯P′′(X)=apr¯P′′′(X),where Rf_(X) and Rf¯(X) are the Pawlak rough approximations of X.Corollary 31.
LetfA be a full soft set over U, and let P=(U,fA) be a soft approximation space. Then,(1)
every softP′′′-definable set is a soft P′-definable set.(2)
every softP′-definable set is a soft P′′-definable set.Remark 32.
Theorems29 and 30 illustrate that Pawlak’s rough set models can be viewed as a special case of soft rough sets.Remark 33.
Example 4.6 in [10] illustrates that a soft rough approximation is a worth considering alternative to the rough approximation. Soft rough sets could provide a better approximation than rough sets do.


## 3.1. Soft Rough Approximation Operatorsapr_P and apr¯P

Definition 12 (see [9, 10]).
LetfA be a soft set over U. Then, the pair P=(U,fA) is called a soft approximation space. We define a pair of operators apr_P, apr¯P:2U→2U as follows:
(11)apr_P(X)={u∈U:∃a∈A,s.t.u∈f(a)⊆X},apr¯P(X)={u∈U:∃a∈A,s.t.u∈f(a),f(a)∩X≠∅}.
apr
_
P and apr¯P are called the soft P-lower approximation operator on U and the soft P-upper approximation operator on U, respectively. In general, we refer to apr_P and apr¯P as soft P-rough approximations operator on U.
apr
_
P
(
X
) and apr¯P(X) are called the soft P-lower approximation and the soft P-upper approximation of X, respectively. In general, we refer to apr_P(X) and apr¯P(X) as soft rough approximations of X with respect to P.
X is called a soft P-definable set if apr_P(X)=apr¯P(X);  X is called a soft P-rough set if apr_P(X)≠apr¯P(X).
Moreover, the sets(12)PosP(X)=apr_P(X),NegP(X)=U-apr¯P(X),BndP(X)=apr¯P(X)-apr_P(X)
are called the soft P-positive region, the soft P-negative region, and the soft P-boundary region of A, respectively.Proposition 13 (see [9, 10]).
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X,Y∈2U.  (1)
apr_P(X)=⋃‍{f(a):a∈Aandf(a)⊆X}⊆X;apr¯P(X)=⋃‍{f(a):a∈Aandf(a)∩X≠∅}.(2)
apr_P(∅)=apr¯P(∅)=∅;apr_P(U)=apr¯P(U)=⋃a∈Af(a).(3)
X⊆Y⇒apr_P(X)⊆apr_P(Y);X⊆Y⇒apr¯P(X)⊆apr¯P(Y).(4)
apr¯P(X∪Y)=apr¯P(X)∪apr¯P(Y).(5)
apr_P(apr_P(X))=apr_P(X);apr_P(apr¯P(X))=apr¯P(X).Proposition 14.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold.(1)
IffAis full, then
(a)
apr_P(X)⊆X⊆apr¯P(X)for  anyX∈2U;(b)
apr_P(U)=apr¯P(U)=U.(2)
IffAis keeping union, then
(a)
for  anyX∈2U, there  existsa∈Asuch  thatapr_P(X)=f(a);(b)
for  anyX∈2U, there  existsa∈Asuch  thatapr¯P(X)=f(a).(3)
IffAis keeping intersection, then(13)apr_P(X∩Y)=apr_P(X)∩apr_P(Y)foranyX,Y∈2U.(4)
IffAis partition, then(14)apr_P(X∩Y)=apr_P(X)∩apr_P(Y)foranyX,Y∈2U.(5)
IffAis full and keeping union, then(15)apr¯P(X)=UforanyX∈2U∖{∅}.Proof.
(1)(a) By Proposition13,  apr_P(X)⊆X. Suppose that X-apr¯P(X)≠∅. Pick
(16)x∈X-apr¯P(X)≠∅.
Since fA is full, U=⋃a∈Af(a). So, x∈f(a) for some a∈A. x∈X implies f(a)∩X≠∅. Thus, x∈apr¯P(X)≠∅, contradiction. Hence,
(17)X⊆apr¯P(X).
(1)(b) This holds by (1) and Proposition13.
(2)  This holds by Proposition13.
(3)  By Proposition13,
(18)apr_P(X∩Y)⊆apr_P(X)∩apr_P(Y).
Suppose thatapr_P(X)∩apr_P(Y)-apr_P(X∩Y)≠∅. Pick
(19)x∈apr_P(X)∩apr_P(Y)-apr_P(X∩Y).
Then, there exista,b∈A such that x∈f(a)⊆X and x∈f(b)⊆Y. Since fA is keeping intersection, then f(a)∩f(b)=f(c) for some c∈A. This implies x∈f(c)⊆X∩Y. Thus, x∈apr_P(X∩Y), contradiction. Hence,
(20)apr_P(X∩Y)⊇apr_P(X)∩apr_P(Y).
Therefore,(21)apr_P(X∩Y)=apr_P(X)∩apr_P(Y).
(4) Suppose thatapr_P(X)∩apr_P(Y)-apr_P(X∩Y)≠∅. Pick
(22)x∈apr_P(X)∩apr_P(Y)-apr_P(X∩Y).
Then, there exista,b∈A such that x∈f(a)⊆X and x∈f(b)⊆Y. Since fA is partition, then f(a)=f(b). This implies x∈f(a)⊆X∩Y. So, x∈apr_P(X∩Y), contradiction. Thus,
(23)apr_P(X∩Y)⊇apr_P(X)∩apr_P(Y).
Therefore,(24)apr_P(X∩Y)=apr_P(X)∩apr_P(Y).
(5)  SincefA is full and keeping union, then U=⋃a∈Af(a)=f(a*) for some a*∈A. For each X∈2U∖{∅} and each u∈U, u∈f(a*) and f(a*)∩X=X≠∅, and then apr¯P(X)=U.


## 3.2. Soft Rough Approximation Operatorsapr_P′ and apr¯P′, apr_P′′ and apr¯P′′, and apr_P′′′ and apr¯P′′′

Definition 15.
LetfA be a soft set over U.(1)
Define a binary relationRf on U by
(25)xRfy⟺∃a∈A,{x,y}⊆f(a)
for each x,y∈U. Then, Rf is called the binary relation induced by fA on U.(2)
For eachx∈U, define a successor neighborhood Rfs(x) of x in U by
(26)(Rf)s(x)={y∈U:xRfy}.Since the following Proposition16 is clear, we omit its proof.Proposition 16.
LetfA be a soft set over U, and let Rf be the binary relation induced by fA on U. Then, the following properties hold.(1)
Rfis a symmetric relation.(2)
IffAis full, thenRfis a reflexive relation.(3)
IffAis partition, thenRfis an equivalence relation.Proposition 17.
LetfA be a soft set over U, and let Rf be the binary relation induced by fA on U. Then, the following properties hold.(1)
Ifx∈f(a)witha∈A, thenf(a)⊆(Rf)s(x).(2)
IffAis partition andx∈f(a)witha∈A, thenf(a)=(Rf)s(x).(3)
IffAis keeping union, then for eachx∈U, there existsa∈Asuch that(Rf)s(x)=f(a).Proof.
(1) This is obvious.
(2) Suppose thaty∈(Rf)s(x). Then, xRfy, and so {x,y}⊆f(b) for some b∈A. Since fA is partition and f(a)∩f(b)≠∅, then f(a)=f(b). Thus, y∈f(a). This implies f(a)⊇(Rf)s(x). By (1),
(27)f(a)=(Rf)s(x).
(3) Suppose thaty∈(Rf)s(x). Then, xRfy, and so {x,y}⊆f(ay) for some ay∈A. By (1), f(ay)⊆(Rf)s(x). Thus, {y}⊆f(ay)⊆(Rf)s(x). This implies
(28)(Rf)s(x)=⋃{f(ay):y∈(Rf)s(x)}.
SincefA is keeping union, then ⋃‍{f(ay):y∈(Rf)s(x)}=f(a) for some a∈A. Thus,
(29)(Rf)s(x)=f(a).Definition 18.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. We define three pairs of soft rough approximation operations: 2U→2U as follows:
(1)(30)apr_P′(X)={x∈U:(Rf)s(x)⊆X},apr¯P′(X)={x∈U:(Rf)s(x)∩X≠∅}.
X is called a soft P′-definable set if apr_P(X)=apr¯P′(X). X is called a soft P′-rough set if  apr_P′(X)≠apr¯P′(X). The sets
(31)PosP′(X)=apr_P′(X),NegP′(X)=U-apr¯P′(X),BndP′(X)=apr¯P′(X)-apr_P′(X)
are called the soft P′-positive region, the soft P′-negative region, and the soft P′-boundary region of X, respectively. Consider,
(2)(32)apr_P′′(X)=⋃{(Rf)s(x):x∈U,(Rf)s(x)⊆X},apr¯P′′(X)=U-apr_P′′(U-X).
X is called a soft P′′-definable set if apr_P′′(X)=apr¯P′′(X). X is called a soft P′′-rough set if apr_P′′(X)≠apr¯P′′(X). The sets
(33)PosP′′(X)=apr_P′′(X),NegP′′(X)=U-apr¯P′′(X),BndP′′(X)=apr¯P′′(X)-apr_P′′(X)
are called the soft P′′-positive region, the soft P′′-negative region, and the soft P′′-boundary region of X, respectively. Consider,
(3)(34)apr¯P′′′(X)=∪{(Rf)s(x):x∈U,(Rf)s(x)∩X≠∅},apr_P′′′(X)=U-apr¯P′′′(U-X).
X is called a soft P′′′-definable set if apr_P′′′(X)=apr¯P′′′(X). X is called a soft P′′′-rough set if apr_P′′′(X)≠apr¯P′′′(X). The sets
(35)PosP′′′(X)=apr_P′′′(X),NegP′′′(X)=U-apr¯P′′′(X),BndP′′′(X)=apr¯P′′′(X)-apr_P′′′(X)
are called the soft P′′′-positive region, the soft P′′′-negative region, and the soft P′′′-boundary region of X, respectively.In general, we also refer toapr_P′(X) and apr¯P′(X),  apr_P′′(X) and apr¯P′′(X), and apr_P′′′(X) and apr¯P′′′(X) as soft rough approximations of X with respect to P′, P′′, P′′′, respectively.It is not very difficult to prove Propositions19, 20, and 21 (see [15]).Proposition 19.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X,Y∈2U.  (1)
apr_P′(X)⊆X. IffAis full, then(36)apr_P′(X)⊆X⊆apr¯P′(X).(2)
apr¯P′(∅)=∅;apr_P′(U)=U.   IffAis full, then(37)apr_P′(∅)=apr¯P′(∅)=∅;apr_P′(U)=apr¯P′(U)=U.(3)
X⊆Y⇒apr_P′(X)⊆apr_P′(Y);X⊆Y⇒apr¯P′(X)⊆apr¯P′(Y).(4)
apr_P′(X∩Y)=apr_P′(X)∩apr_P′(Y).(5)
apr¯P′(X∪Y)=apr¯P′(X)∪apr¯P′(Y).(6)
apr_P′(U-X)=U-apr¯P′(X);apr¯P′(U-X)=U-apr_P′(X).(7)
apr¯P′(apr_P′(X))⊆X⊆apr_P′(apr¯P′(X)).Proposition 20.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.  (1)
(38)apr_P′′(X)⊆X⊆apr¯P′′(X).(2)
apr_P′′(∅)=∅;apr¯P′′(U)=U.   IffAis full, then(39)apr_P′(∅)=apr¯P′(∅)=∅;apr_P′(U)=apr¯P′(U)=U.(3)
apr_P′′(U-X)=U-apr¯P′′(X);apr¯P′′(U-X)=U-apr_P′′(X).(4)
apr_P′′(apr_P′′(X))=apr_P′′(X);apr¯P′′(apr¯P′′(X))=apr¯P′′(X).Proposition 21.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X,Y∈2U.  (1)
IffAis full, then(40)apr_P′′′(X)⊆X⊆apr¯P′′′(X).(2)
apr¯P′′′(∅)=∅;apr_P′′′(U)=U. IffAis full, then(41)apr_P′′′(∅)=apr¯P′′′(∅)=∅;apr_P′′′(U)=apr¯P′(U)=U.(3)
X⊆Y⇒apr_P′′′(X)⊆apr_P′′′(Y);X⊆Y⇒apr¯P′′′(X)⊆apr¯P′′′(Y).(4)
apr_P′′′(X∩Y)=apr_P′′′(X)∩apr_P′′′(Y).(5)
apr¯P′′′(X∪Y)=apr¯P′′′(X)∪apr¯P′′′(Y).(6)
apr_P′′′(U-X)=U-apr¯P′′′(X);apr¯P′′′(U-X)=U-apr_P′′′(X).(7)
apr_P′′′(apr_P′′′(X))⊆apr_P′′′(X);apr¯P′′′(apr¯P′′′(X))⊇apr¯P′′′(X).(8)
apr¯P′′′(apr_P′′′(X))⊆X⊆apr_P′′′(apr¯P′′′(X)).Example 22.
LetU={h1,h2,h3,h4,h5}, A={a1,a2,a3,a4}, and let fA be a soft set over U, defined as follows:
(42)f(a1)={h2},f(a2)={h1,h4},f(a3)={h3},f(a4)={h1,h3}.
Obviously,fA is not full. We have
(43)(Rf)s(h1)={h1,h3,h4},(Rf)s(h2)={h2},(Rf)s(h3)={h1,h3},(Rf)s(h4)={h1,h4},(Rf)s(h5)=∅.
LetX={h3,h5}, Y={h1,h2,h3}, and Z={h2,h4,h5}.(1)
We have(44)apr¯P′(X)={h1,h3}.
Thus,
(45)X⊈apr¯P′(X).(2)
We have(46)apr_P′(X)={h5},apr_P′(Y)={h2,h3,h5}.
Thus,
(47)apr_P′(X)⊆apr_P′(Y)⇏X⊆Y.(3)
We have(48)apr¯P′(X)={h1,h3},apr¯P′(Y)={h1,h2,h3,h4}.
Thus,
(49)apr¯P′(X)⊆apr¯P′(Y)⇏X⊆Y.(4)
We have(50)apr_P′(Y)={h2,h3,h5},apr_P′(Z)={h2,h5},apr_P′(Y∪Z)=U.
Thus,
(51)apr_P′(Y∪Z)≠apr_P′(y)∪apr_P′(Z).(5)
We have(52)apr¯P′(Y)={h1,h2,h3,h4},apr¯P′(Z)={h1,h2,h4},apr¯P′(Y∩Z)={h2}.
Thus,
(53)apr¯P′(Y∩Z)≠apr¯P′(Y)∩apr_P′(Z).


## 3.3. The Relationships among Four Pairs of Soft Rough Approximation Operators

Lemma 23.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
IffA is full, then
(54)apr_P(X)⊇apr_P′(X).(2)
IffA is full and keeping union, then
(55)apr¯P(X)⊇apr¯P′(X).(3)
IffA is partition, then
(a)
apr_P(X)=apr_P′(X);(b)
apr¯P(X)=apr¯P′(X).(c)
apr_P′(X)=apr_P′′(X)=apr_P′′′(X);(d)
apr¯P′=apr¯P′′(X)=apr¯P′′′(X).Proof.
(1) Suppose thatx∈apr_P′(X). Then, (Rf)s(x)⊆X. Since fA is full, then x∈f(a) for some a∈A. By Proposition 17, f(a)⊆(Rf)s(x). Thus, x∈f(a)⊆X. This implies x∈apr_P(X). Thus,
(56)apr_P(X)⊇apr_P′(X).
(2) IfX=∅, then apr¯P(X)=∅=apr¯P′(X). If X≠∅, by Proposition 14, apr¯P(X)=U.
Hence,(57)apr¯P(X)⊇apr¯P′(X).
(3)(a) Suppose thatx∈apr_P(X). Then, x∈f(a)⊆X for some a∈A. Since fA is partition and x∈f(a), then f(a)=(Rf)s(x) by Proposition 17. This implies x∈apr_P′(X). Thus,
(58)apr_P(X)⊆apr_P′(X).
By (1),
(59)apr_P(X)=apr_P′(X).
(3)(b) This is similar to the proof of (3) (a).
Suppose thatx∈apr¯P′(X). Then, (Rf)s(x)∩X≠∅. Since fA is full, then x∈f(a) for some a∈A. Since fA is partition and x∈f(a), then f(a)=(Rf)s(x) by Proposition 17. This implies x∈apr¯P(X). Thus,
(60)apr¯P(X)⊇apr¯P′(X).
Hence,apr¯P(X)=apr¯P′(X).By Propositions13 and 17, we have Lemma 24.Lemma 24.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space.  If fA is keeping union, then for any X∈2U,
(61)apr_P(X)⊇apr_P′′(X),apr¯P(X)⊇apr¯P′′′(X).Lemma 25.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
IffAis full, then(62)apr_P′′′(X)⊆apr_P′(X)⊆apr_P′′(X)⊆X⊆apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X).(2)
IffAis partition, then
(a)
apr_P′(X)=apr_P′′(X)=apr_P′′′(X);(b)
apr¯P′=apr¯P′′(X)=apr¯P′′′(X).Proof.
(1) Suppose thatx∈apr_P′(X). Then, (Rf)s(x)⊆X. Since fA is full, then x∈(Rf)s(x)⊆X by Proposition 17. This implies x∈apr_P′′(X). Thus,
(63)apr_P′(X)⊆apr_P′′(X).
Suppose thatapr_P′′′(X)-apr_P′(X)≠∅. Pick
(64)x∈apr_P′′′(X)-apr_P′(X).
x
∉
apr
_
P
′
(
X
) implies (Rf)s(x)⊈X. So, (Rf)s(x)∩(U-X)≠∅. Since fA is full, then x∈(Rf)s(x) by Proposition 17. This implies x∈apr¯P′′(U-X). Thus, x∉U-apr¯P′′(U-X)=apr_P′′′(X), contradiction.
Hence,apr_P′′′(X)⊆apr_P′(X).
By Proposition20,
(65)apr_P′′(X)⊆X⊆apr¯P′′(X).
Since(66)apr_P′′′(U-X)⊆apr_P′(U-X)⊆apr_P′′(U-X),
then
(67)U-apr_P′′′(U-X)⊇U-apr_P′(U-X)⊇U-apr_P′′(U-X).
By Propositions19, 20, and 21,
(68)apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X).
(2)(a)  Suppose thatx∈apr_P′′(X). Then, there exists y∈U such that x∈(Rf)s(y)⊆X. Since fA is partition, then Rf is an equivalence relation by Proposition 16. Thus, x∈(Rf)s(y) follows (Rf)s(x)=(Rf)s(y). So, (Rf)s(x)⊆X. This implies x∈apr_P′(X). Hence,
(69)apr_P′(X)⊇apr_P′′(X).
By (1),(70)apr_P′(X)=apr_P′′(X).
Suppose thatapr_P′(X)∩apr¯P′′′(U-X)≠∅. Pick
(71)x∈apr_P′(X)∩apr¯P′′′(U-X).
x
∈
apr
_
P
′
(
X
) implies (Rf)s(x)⊆X. x∈apr¯P′′′(U-X) implies that there exists y∈U such that x∈(Rf)s(y) and (Rf)s(y)∩(U-X)≠∅. So, (Rf)s(y)⊈X. Note that Rf is an equivalence relation. Then, (Rf)s(x)=(Rf)s(y). Thus, (Rf)s(x)⊈X, contradiction.
Hence,apr_P′(X)∩apr¯P′′′(U-X)=∅.
This proves that(72)apr_P′(X)⊆U-apr¯P′′′(U-X)=apr_P′′′(X).
By (1),(73)apr_P′(X)=apr_P′′′(X).
(2)(b) By (2)(a),(74)apr_P′(U-X)=apr_P′′(U-X)=apr_P′′′(U-X).
Then,(75)U-apr_P′(U-X)=U-apr_P′′(U-X)=U-apr_P′′′(U-X).
By Propositions19, 20, and 21,
(76)apr¯P′(X)=apr¯P′′(X)=apr¯P′′′(X).Example 26.
LetU={h1,h2,h3,h4,h5,h6}, A={a1,a2,a3,a4,a5}, and let fA be a soft set over U, defined as follows:
(77)f(a1)={h1},f(a2)={h6},f(a3)={h2,h5},f(a4)={h2,h4},f(a5)={h1,h3,h5}.
Obviously,fA is full. We have
(78)(Rf)s(h1)={h1,h3,h5},(Rf)s(h2)={h2,h4,h5},(Rf)s(h3)={h1,h3,h5},(Rf)s(h4)={h2,h4},(Rf)s(h5)={h1,h2,h3,h5}.
LetX={h2,h4,h6}. We have
(79)apr_P(X)={h2,h4,h6},apr_P′(X)={h4},apr_P′′(X)={h2,h4},apr¯P(X)={h2,h4,h5,h6},apr¯P′(X)={h2,h4,h5},apr¯P′′(X)={h1,h2,h3,h4,h5}.
Thus,(80)apr_P(X)≠apr_P′(X),apr_P′(X)≠apr_P′′(X),apr¯P(X)≠apr¯P′(X),apr¯P′(X)≠apr¯P′′(X).By Proposition16 and Lemmas 23, 24, and 25, we have Theorem 27.Theorem 27.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
IffAis full, then(81)apr_P′′′(X)⊆apr_P′(X)⊆apr_P′′(X)⊆X⊆apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X).(2)
IffAis full and keeping union, then(82)apr_P′′′(X)⊆apr_P′(X)⊆apr_P′′(X)⊆apr_P(X)⊆X⊆apr¯P′′(X)⊆apr¯P′(X)⊆apr¯P′′′(X)⊆apr¯P(X).


## 3.4. The Relationship between Soft Rough Approximation Operators and Pawlak Rough Approximation Operators

In this section, we shall explore the relationship between soft rough approximation operators and Pawlak rough approximation operators.Definition 28.
LetR be an equivalence relation on U. Define a mapping fR:A→2U by
(83)fR(a)=[a]R
for each a∈A, where A=U. Then, (fR)A is called the soft set induced by R on U.Theorem 29 (see [10]).
LetR be an equivalence relation on U, let (fR)A be the soft set induced by R on U, and let P=(U,(fR)A) be a soft approximation space. Then, for each X∈2U,
(84)R_(X)=apr_P(X),R¯(X)=apr¯P(X).
Thus, in this case,X∈2U is a Pawlak rough set if and only if X is a soft P-rough set.By Proposition16 and Lemmas 23 and 25, we have Theorem 30.Theorem 30.
LetfA be a partition soft set over U, and let P=(U,fA) be a soft approximation space. Then, the following properties hold for any X∈2U.(1)
Rf_(X)=apr_P(X)=apr_P′(X)=apr_P′′(X)=apr_P′′′(X);(2)
Rf¯(X)=apr¯P(X)=apr¯P′(X)=apr¯P′′(X)=apr¯P′′′(X),where Rf_(X) and Rf¯(X) are the Pawlak rough approximations of X.Corollary 31.
LetfA be a full soft set over U, and let P=(U,fA) be a soft approximation space. Then,(1)
every softP′′′-definable set is a soft P′-definable set.(2)
every softP′-definable set is a soft P′′-definable set.Remark 32.
Theorems29 and 30 illustrate that Pawlak’s rough set models can be viewed as a special case of soft rough sets.Remark 33.
Example 4.6 in [10] illustrates that a soft rough approximation is a worth considering alternative to the rough approximation. Soft rough sets could provide a better approximation than rough sets do.


## 4. The Relationships between Soft Sets and Topologies

LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Denote
(85)τf={X∈2U:X=apr_P(X)},σf={X∈2U:X=apr¯P(X)};τf′={X∈2U:X=apr_P′(X)},σf′={X∈2U:X=apr¯P′(X)};τf′′={X∈2U:X=apr_P′′(X)},σf′′={X∈2U:X=apr¯P′′(X)};τf′′′={X∈2U:X=apr_P′′′(X)},σf′′′={X∈2U:X=apr¯P′′′(X)}.


### 4.1. The First Sort of Topologies Induced by a Soft Set and Related Results

By Propositions13 and 14, we have Theorem 34.Theorem 34.
LetfA be a full and keeping intersection or a partition soft set over U and let P=(U,fA) be a soft approximation space. Then τf is a topology on U.Remark 35.
LetfA be a full and keeping union soft set over U, and let P=(U,fA) be an soft approximation space. Then, by Proposition 14(5),  σf={∅,U} is a indiscrete topology on U.The following theorem gives the structure of the first sort of topologies induced by a soft set.Theorem 36.
LetfA be a full and keeping intersection soft set over U, let P=(U,fA) be a soft approximation space, and let τf be the topology induced by fA on U. Then,(1)
(86){apr¯P(X):X∈2U}⊆τf={apr_P(X):X∈2U};(2)
(87)τf⊇{f(a):a∈A};(3)
iffA is topological, then
(88)τf={f(a):a∈A};(4)
apr_Pis an interior operator ofτf.Proof.
(1) By Proposition13, we have
(89){apr¯P(X):X∈2U}⊆τf.
Obviously,(90)τf⊆{apr_P(X):X∈2U}.
LetY∈{apr_P(X):X∈2U}. Then, Y=apr_P(X) for some X∈2U. By Proposition 13,  apr_P(apr_P(X))=apr_P(X). This implies Y∈τf. Thus,
(91)τf⊇{apr_P(X):X∈2U}.
Hence,(92){apr¯P(X):X∈2U}⊆τf={apr_P(X):X∈2U}.
(2) For eacha∈A, by Proposition 13,
(93)apr_P(f(a))=⋃{f(a′):a′∈A,f(a′)⊆f(a)}⊆f(a).
Then,f(a)=apr_P(f(a)). So f(a)∈τf. Thus,
(94){f(a):a∈A}⊆τf.
(3) Suppose thatX∈τf. If X=∅, by fA is topological, there exists a∈A such that X=f(a). If X≠∅, for each x∈X, X=apr_P(X), there exists ax∈A such that x∈f(ax)⊆X. Then,
(95)X=⋃x∈X{x}⊆⋃x∈Xf(ax)⊆X.
So, X=⋃x∈Xf(ax). Since fA is keeping union, then
(96)⋃x∈Xf(ax)=f(a)forsomea∈A.
This impliesX∈{f(a):a∈A}. Thus,  τf⊆{f(a):a∈A}.
By (1),τf⊇{f(a):a∈A}.
Hence,(97)τf={f(a):a∈A}.
(4) It suffices to show that(98)apr_P(X)=int(X)foreachX∈2U.
By (1),apr_P(X)∈τf. By Proposition 13, apr_P(X)⊆X. Thus
(99)apr_P(X)⊆int(X).
Conversely, for eachY∈τf with Y⊆X, we have Y=apr_P(Y)⊆apr_P(X) by Proposition 13. Thus,
(100)int(X)=∪{Y:Y∈τf,Y⊆X}⊆apr_P(X).
Hence,(101)apr_P(X)=int(X).Definition 37.
Letτ be a topology on U. Put τ={Ua:a∈A}, where A is the set of indexes. Define a mapping fτ:A→2U by fτ(a)=Ua for each a∈A. Then, the soft set (fτ)A over U is called the soft set induced by τ on U.Definition 38.
Let(U,μ) be a topological space. If there exists a full and keeping intersection or a partition soft set fA over U such that τf=μ, then (U,μ) is called a soft approximating space.The following proposition can easily be proved.Proposition 39.
(1) Letτ be a topology on U, and let (fτ)A be the soft set induced by τ on U. Then, (fτ)A is a full, keeping intersection, and keeping union soft set over U.
(2) Letτ1 and τ2 be two topologies on U, and let (fτ1)A1 and (fτ2)A2 be two soft sets induced, respectively, by τ1 and τ2 on U. If  τ1⊆τ2, then
(102)(fτ1)A1⊂~(fτ2)A2.Theorem 40.
Letτ be a topology on U, let (fτ)A be the soft set induced by τ on U, and let τfτ be the topology induced by (fτ)A on U. Then, τ=τfτ.Proof.
Putτ={Ua:a∈A}; then, fτ:A→2U is a mapping, where fτ(a)=Ua for each a∈A. By Proposition 39, (fτ)A is full, keeping intersection, and keeping union.
By Theorem36, τfτ={fτ(a):a∈A}.
Hence,τfτ=τ.Corollary 41.
Every topological space on the initial universe is a soft approximating space.Theorem 42.
Let(U,τ) be a topological space. Then, there exists a full, keeping intersection, and keeping union soft set fA over U such that
(103)apr_P(X)=int(X)foreachX∈2U,
where P=(U,fA) is a soft approximation space.Proof.
Putτ={Ua:a∈A}, where A is the set of indexes. Define a mapping f:A→2U by
(104)f(a)=Uaforeacha∈A.
By Proposition 39, fA is full, keeping intersection, and keeping union.
LetX∈2U. For each x∈apr_P(X), x∈f(a)⊆X for some a∈A. Then, x∈Ua⊆X with Ua∈τ. This implies x∈int(X).
Conversely, for eachx∈int(X), there exists an open neighborhood W of x in U such that W⊆X. So, W=Ua for some a∈A. This implies x∈f(a)⊆X. Thus, x∈apr_P(X).
Hence,apr_P(X)=int(X).Theorem 43.
LetfA be a full and keeping intersection soft set over U, let  τf be the topology induced by fA on U, and let (fτf)B be the soft set induced by τf on U. Then,(1)
(105)fA⊂~(fτf)B.(2)
IffA is topological, then
(106)fA=(fτf)B.Proof.
(1) By Theorem36, τf⊇{f(a):a∈A}. Denote
(107)τf={Ua:a∈B},whereA⊆B,Ua=f(a)foreacha∈A.
Thus fτf is a mapping given by fτf:B→2U, where fτf(a)=Ua for each a∈B.
Hence,fA⊂~(fτf)B.
(2) SincefA is topological, then by Theorem 36,  A=B.
Hence,(108)fA=(fτf)B.


### 4.2. The Second Sort of Topologies Induced by a Soft Set

By Proposition19, we have Theorem 44.Theorem 44.
LetfA be a full soft set over U, and let P=(X,fA) be a soft approximation space. Then, τf′ is a topology on U.Definition 45.
Letτ be a topology on U.(1)
τ is called an Alexandrov topology on U, if τ is closed for arbitrary intersections.(2)
(U,τ) is called an Alexandrov space, if τ is an Alexandrov topology on U.(3)
τ is called a pseudodiscrete topology on U, if A∈τ if and only if U-A∈τ.Obviously, every pseudodiscrete topology is an Alexandrov topology.The following theorem gives the structure of the second sort of topologies induced by a soft set.Theorem 46.
LetfA be a full soft set over U, and let τf′ be the topology induced by fA on U. Then, (U,τf′) is an Alexandrov space.Proof.
By Proposition16, Rf is reflexive and symmetric. Then, by Proposition 5 in [16], τf′ is a pseudodiscrete topology on U.
Thus,(109)(U,τf′)isanAlexandrovspace.


### 4.3. The Third Sort of Topologies Induced by a Soft Set

Example 47.
LetfA be a full soft set over U and let P=(U,fA) be a soft approximation space in Example 26.
LetY={h2,h4}, and Z={h1,h2,h3,h5}. We have
(110)apr_P′′(Y)=Y,apr_P′′(Z)=Z,apr_P′′(Y∩Z)=∅≠Y∩Z.
Thus,τf′′ is not a topology on U.By Proposition21, we have Theorem 48.Theorem 48.
LetfA be a full soft set over U, and let P=(U,fA) be a soft approximation space. Then, τf′′′ is a topology on U.


### 4.4. The Relationships among Three Sorts of Topologies Induced by a Soft Set and Related Results

By Theorem27, we have Theorem 49, which illustrates relationships among three sorts of topologies induced by a soft set.Theorem 49.
(1) IffA is a full soft set over U, then
(111)τf′′′⊆τf′.
(2) IffA is a full and keeping intersection soft set over U, then
(112)τf′′′⊆τf′⊆τf.
(3) IffA is a partition soft set over U, then
(113)τf=τf′=τf′′′.By Theorem36, Proposition 39 and Theorem 49, we have Theorem 50.Theorem 50.
Letτ be a topology on U, let (fτ)A be the soft set induced by τ on U, and let  τfτ, τfτ′ and τfτ′′′ be the topology induced, respectively, by (fτ)A on U. Then
(114)τ=τfτ⊇τfτ′⊇τfτ′′′.By Proposition39 and Theorems 43 and 49, we have Theorem 51.Theorem 51.
(1) IffA is a full soft set over U, let τf′ and τf′′′ be the topologies induced, respectively, by fA on U, and let (fτf′)C  (resp., (fτf′′′)D) be the soft set induced by τf′  (resp., τf′′′) on U. Then,
(115)(fτf′)C⊃~(fτf′′′)D.
(2) LetfA be a full and keeping intersection soft set over U, let  τf, τf′, and τf′′′ be the topologies induced respectively by fA on U and let (fτf)B  (resp., (fτf′)C, (fτf′′′)D) be the soft set induced by τf  (resp., τf′, τf′′′) on U. Then,
(a)(116)(fτf)B⊃~fA,(fτf)B⊃~(fτf′)C⊃~(fτf′′′)D.
(b) IffA is keeping union, then
(117)fA=(fτf)B⊃~(fτf′)C⊃~(fτf′′′)D.
(3) LetfA be a partition soft set over U, let  τfτ, τfτ′ and τfτ′′′ be the topologies induced, respectively, by fA on U and let (fτf)B  (resp., (fτf′)C, (fτf′′′)D) be the soft set induced by τf  (resp., τf′, τf′′′) on U. Then,
(118)(fτf)B⊃~fA,(fτf)B=(fτf′)C=(fτf′′′)D.Example 52.
LetfA be a partition soft set over U in Example 11, let τf be the topology induced by fA on U, and let (fτf)B be the soft set induced by τf on U. We have
(119)τf={Ua:a∈B},whereB={a1,a2,…,a16},Ua1=f(a1)={h1,h2},Ua2=f(a2)={h5},Ua3=f(a3)={h3},Ua4=f(a4)={h4},Ua5={h3,h4},Ua6={h3,h5},Ua7={h4,h5},Ua8={h1,h2,h3},Ua9={h1,h2,h4},Ua10={h1,h2,h5},Ua11={h3,h4,h5},Ua12={h1,h2,h3,h4},Ua13={h1,h2,h3,h5},Ua14={h1,h2,h4,h5},Ua15=∅,Ua16=U.
Obviously,(120)fA⊂~(fτf)B,fA⊉~(fτf)B.
Thus,(121)fA≠(fτf)B.


## 4.1. The First Sort of Topologies Induced by a Soft Set and Related Results

By Propositions13 and 14, we have Theorem 34.Theorem 34.
LetfA be a full and keeping intersection or a partition soft set over U and let P=(U,fA) be a soft approximation space. Then τf is a topology on U.Remark 35.
LetfA be a full and keeping union soft set over U, and let P=(U,fA) be an soft approximation space. Then, by Proposition 14(5),  σf={∅,U} is a indiscrete topology on U.The following theorem gives the structure of the first sort of topologies induced by a soft set.Theorem 36.
LetfA be a full and keeping intersection soft set over U, let P=(U,fA) be a soft approximation space, and let τf be the topology induced by fA on U. Then,(1)
(86){apr¯P(X):X∈2U}⊆τf={apr_P(X):X∈2U};(2)
(87)τf⊇{f(a):a∈A};(3)
iffA is topological, then
(88)τf={f(a):a∈A};(4)
apr_Pis an interior operator ofτf.Proof.
(1) By Proposition13, we have
(89){apr¯P(X):X∈2U}⊆τf.
Obviously,(90)τf⊆{apr_P(X):X∈2U}.
LetY∈{apr_P(X):X∈2U}. Then, Y=apr_P(X) for some X∈2U. By Proposition 13,  apr_P(apr_P(X))=apr_P(X). This implies Y∈τf. Thus,
(91)τf⊇{apr_P(X):X∈2U}.
Hence,(92){apr¯P(X):X∈2U}⊆τf={apr_P(X):X∈2U}.
(2) For eacha∈A, by Proposition 13,
(93)apr_P(f(a))=⋃{f(a′):a′∈A,f(a′)⊆f(a)}⊆f(a).
Then,f(a)=apr_P(f(a)). So f(a)∈τf. Thus,
(94){f(a):a∈A}⊆τf.
(3) Suppose thatX∈τf. If X=∅, by fA is topological, there exists a∈A such that X=f(a). If X≠∅, for each x∈X, X=apr_P(X), there exists ax∈A such that x∈f(ax)⊆X. Then,
(95)X=⋃x∈X{x}⊆⋃x∈Xf(ax)⊆X.
So, X=⋃x∈Xf(ax). Since fA is keeping union, then
(96)⋃x∈Xf(ax)=f(a)forsomea∈A.
This impliesX∈{f(a):a∈A}. Thus,  τf⊆{f(a):a∈A}.
By (1),τf⊇{f(a):a∈A}.
Hence,(97)τf={f(a):a∈A}.
(4) It suffices to show that(98)apr_P(X)=int(X)foreachX∈2U.
By (1),apr_P(X)∈τf. By Proposition 13, apr_P(X)⊆X. Thus
(99)apr_P(X)⊆int(X).
Conversely, for eachY∈τf with Y⊆X, we have Y=apr_P(Y)⊆apr_P(X) by Proposition 13. Thus,
(100)int(X)=∪{Y:Y∈τf,Y⊆X}⊆apr_P(X).
Hence,(101)apr_P(X)=int(X).Definition 37.
Letτ be a topology on U. Put τ={Ua:a∈A}, where A is the set of indexes. Define a mapping fτ:A→2U by fτ(a)=Ua for each a∈A. Then, the soft set (fτ)A over U is called the soft set induced by τ on U.Definition 38.
Let(U,μ) be a topological space. If there exists a full and keeping intersection or a partition soft set fA over U such that τf=μ, then (U,μ) is called a soft approximating space.The following proposition can easily be proved.Proposition 39.
(1) Letτ be a topology on U, and let (fτ)A be the soft set induced by τ on U. Then, (fτ)A is a full, keeping intersection, and keeping union soft set over U.
(2) Letτ1 and τ2 be two topologies on U, and let (fτ1)A1 and (fτ2)A2 be two soft sets induced, respectively, by τ1 and τ2 on U. If  τ1⊆τ2, then
(102)(fτ1)A1⊂~(fτ2)A2.Theorem 40.
Letτ be a topology on U, let (fτ)A be the soft set induced by τ on U, and let τfτ be the topology induced by (fτ)A on U. Then, τ=τfτ.Proof.
Putτ={Ua:a∈A}; then, fτ:A→2U is a mapping, where fτ(a)=Ua for each a∈A. By Proposition 39, (fτ)A is full, keeping intersection, and keeping union.
By Theorem36, τfτ={fτ(a):a∈A}.
Hence,τfτ=τ.Corollary 41.
Every topological space on the initial universe is a soft approximating space.Theorem 42.
Let(U,τ) be a topological space. Then, there exists a full, keeping intersection, and keeping union soft set fA over U such that
(103)apr_P(X)=int(X)foreachX∈2U,
where P=(U,fA) is a soft approximation space.Proof.
Putτ={Ua:a∈A}, where A is the set of indexes. Define a mapping f:A→2U by
(104)f(a)=Uaforeacha∈A.
By Proposition 39, fA is full, keeping intersection, and keeping union.
LetX∈2U. For each x∈apr_P(X), x∈f(a)⊆X for some a∈A. Then, x∈Ua⊆X with Ua∈τ. This implies x∈int(X).
Conversely, for eachx∈int(X), there exists an open neighborhood W of x in U such that W⊆X. So, W=Ua for some a∈A. This implies x∈f(a)⊆X. Thus, x∈apr_P(X).
Hence,apr_P(X)=int(X).Theorem 43.
LetfA be a full and keeping intersection soft set over U, let  τf be the topology induced by fA on U, and let (fτf)B be the soft set induced by τf on U. Then,(1)
(105)fA⊂~(fτf)B.(2)
IffA is topological, then
(106)fA=(fτf)B.Proof.
(1) By Theorem36, τf⊇{f(a):a∈A}. Denote
(107)τf={Ua:a∈B},whereA⊆B,Ua=f(a)foreacha∈A.
Thus fτf is a mapping given by fτf:B→2U, where fτf(a)=Ua for each a∈B.
Hence,fA⊂~(fτf)B.
(2) SincefA is topological, then by Theorem 36,  A=B.
Hence,(108)fA=(fτf)B.


## 4.2. The Second Sort of Topologies Induced by a Soft Set

By Proposition19, we have Theorem 44.Theorem 44.
LetfA be a full soft set over U, and let P=(X,fA) be a soft approximation space. Then, τf′ is a topology on U.Definition 45.
Letτ be a topology on U.(1)
τ is called an Alexandrov topology on U, if τ is closed for arbitrary intersections.(2)
(U,τ) is called an Alexandrov space, if τ is an Alexandrov topology on U.(3)
τ is called a pseudodiscrete topology on U, if A∈τ if and only if U-A∈τ.Obviously, every pseudodiscrete topology is an Alexandrov topology.The following theorem gives the structure of the second sort of topologies induced by a soft set.Theorem 46.
LetfA be a full soft set over U, and let τf′ be the topology induced by fA on U. Then, (U,τf′) is an Alexandrov space.Proof.
By Proposition16, Rf is reflexive and symmetric. Then, by Proposition 5 in [16], τf′ is a pseudodiscrete topology on U.
Thus,(109)(U,τf′)isanAlexandrovspace.


## 4.3. The Third Sort of Topologies Induced by a Soft Set

Example 47.
LetfA be a full soft set over U and let P=(U,fA) be a soft approximation space in Example 26.
LetY={h2,h4}, and Z={h1,h2,h3,h5}. We have
(110)apr_P′′(Y)=Y,apr_P′′(Z)=Z,apr_P′′(Y∩Z)=∅≠Y∩Z.
Thus,τf′′ is not a topology on U.By Proposition21, we have Theorem 48.Theorem 48.
LetfA be a full soft set over U, and let P=(U,fA) be a soft approximation space. Then, τf′′′ is a topology on U.


## 4.4. The Relationships among Three Sorts of Topologies Induced by a Soft Set and Related Results

By Theorem27, we have Theorem 49, which illustrates relationships among three sorts of topologies induced by a soft set.Theorem 49.
(1) IffA is a full soft set over U, then
(111)τf′′′⊆τf′.
(2) IffA is a full and keeping intersection soft set over U, then
(112)τf′′′⊆τf′⊆τf.
(3) IffA is a partition soft set over U, then
(113)τf=τf′=τf′′′.By Theorem36, Proposition 39 and Theorem 49, we have Theorem 50.Theorem 50.
Letτ be a topology on U, let (fτ)A be the soft set induced by τ on U, and let  τfτ, τfτ′ and τfτ′′′ be the topology induced, respectively, by (fτ)A on U. Then
(114)τ=τfτ⊇τfτ′⊇τfτ′′′.By Proposition39 and Theorems 43 and 49, we have Theorem 51.Theorem 51.
(1) IffA is a full soft set over U, let τf′ and τf′′′ be the topologies induced, respectively, by fA on U, and let (fτf′)C  (resp., (fτf′′′)D) be the soft set induced by τf′  (resp., τf′′′) on U. Then,
(115)(fτf′)C⊃~(fτf′′′)D.
(2) LetfA be a full and keeping intersection soft set over U, let  τf, τf′, and τf′′′ be the topologies induced respectively by fA on U and let (fτf)B  (resp., (fτf′)C, (fτf′′′)D) be the soft set induced by τf  (resp., τf′, τf′′′) on U. Then,
(a)(116)(fτf)B⊃~fA,(fτf)B⊃~(fτf′)C⊃~(fτf′′′)D.
(b) IffA is keeping union, then
(117)fA=(fτf)B⊃~(fτf′)C⊃~(fτf′′′)D.
(3) LetfA be a partition soft set over U, let  τfτ, τfτ′ and τfτ′′′ be the topologies induced, respectively, by fA on U and let (fτf)B  (resp., (fτf′)C, (fτf′′′)D) be the soft set induced by τf  (resp., τf′, τf′′′) on U. Then,
(118)(fτf)B⊃~fA,(fτf)B=(fτf′)C=(fτf′′′)D.Example 52.
LetfA be a partition soft set over U in Example 11, let τf be the topology induced by fA on U, and let (fτf)B be the soft set induced by τf on U. We have
(119)τf={Ua:a∈B},whereB={a1,a2,…,a16},Ua1=f(a1)={h1,h2},Ua2=f(a2)={h5},Ua3=f(a3)={h3},Ua4=f(a4)={h4},Ua5={h3,h4},Ua6={h3,h5},Ua7={h4,h5},Ua8={h1,h2,h3},Ua9={h1,h2,h4},Ua10={h1,h2,h5},Ua11={h3,h4,h5},Ua12={h1,h2,h3,h4},Ua13={h1,h2,h3,h5},Ua14={h1,h2,h4,h5},Ua15=∅,Ua16=U.
Obviously,(120)fA⊂~(fτf)B,fA⊉~(fτf)B.
Thus,(121)fA≠(fτf)B.


## 5. The Related Properties of Soft Rough Sets

In this section, four sorts of soft rough sets based on four pairs of soft rough approximations are investigated.For𝒜,ℬ⊆2U, we denote
(122)𝒜∖ℬ={X∈2U:X∈𝒜,X∉ℬ}.Lemma 53 (see [10]).
LetfA be a soft set over U and let P=(U,fA) be a soft approximation space. Then for each X∈2U,
(123)XisasoftP-roughset⟺apr¯P(X)⊆X.By Corollary31, we have the following Lemma 54.Lemma 54.
LetfA be a full soft set over U, and let P=(U,fA) be a soft approximation space. Then,(1)
every softP′-rough set is a soft P′′′-rough set.(2)
every softP′′-rough set is a soft P′-rough set.By Theorem27, we have Lemma 55.Lemma 55.
LetfA be a full and keeping union soft set over U, and let P=(U,fA) be a soft approximation space. If X∈2U is a soft P-definable set, then,
(124)X∈σf′′′⊆σf′⊆σf′′.The following theorem gives the structure of soft rough sets.Theorem 56.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Denote
(125)Ω={X∈2U:XisasoftP-roughset},Ω′={X∈2U:XisasoftP′-roughset},Ω′′={X∈2U:XisasoftP′′-roughset},Ω′′′={X∈2U:XisasoftP′′′-roughset}.(1)
IffA is full, then
(a)
(126)σf=2U∖Ω;(b)
(127)Ω′′⊆Ω′⊆Ω′′′;(c)
(128)2U∖τf′⊆Ω′,2U∖τf′′⊆Ω′′,2U∖τf′′′⊆Ω′′′;(d)
(129)τf′∩σf′=2U∖Ω′,τf′′∩σf′′=2U∖Ω′′,τf′′′∩σf′′′=2U∖Ω′′′.(2)
IffA is full and keeping union, then
(a)
2U∖σf′′⊆2U∖σf′⊆2U∖σf′′′⊆2U∖σf⊆Ω;(b)
σf={∅,U}=2U∖Ω.(3)
IffA is full and keeping intersection, then
(a)
σf=2U∖Ω⊆τf.Proof.
These hold by Proposition14 and Lemmas 53, 54, and 55.Theorem 57.
LetfA be a soft set over U and let P=(U,fA) be a soft approximation space. Then, for X∈2U, one has
(130)PosP′(U-X)=NegP′(X),PosP′′(U-X)=NegP′′(X),PosP′′′(U-X)=NegP′′′(X).Proof.
This is obtained from Propositions19, 20, and 21.By Proposition14 and Theorem 27, we have Theorem 58.Theorem 58.
LetfA be a soft set over U, and let P=(U,fA) be a soft approximation space. Then, for X∈2U, one has the following.(1)
IffA be a full, then
(131)BndP′′(X)⊆BndP′(X)⊆BndP′′′(X).(2)
IffA is full and keeping union, then
(a)
apr¯P′′(X)-X⊆apr¯P′(X)-X⊆apr¯P′′′(X)-X⊆BndP(X);(b)
NegP(X)=∅ and BndP(X)=U-PosP(X) where X≠∅.Remark 59.
Theorem58 illustrates that soft P′′-rough sets could provide a better approximation than soft P′-rough sets do and soft P′-rough sets could provide a better approximation than soft P′′′-rough sets do.


## 6. A Correspondence Relationship

In this section, we give a one-to-one correspondence relationship in order to reveal the broad application prospect of soft sets.Definition 60 (see [17]).
LetU be a finite set of objects, let A be a finite set of attributes, and let I be a binary relation on from U to A. The triple (U,A,I) is called a formal context.Let(U,A,I) be a formal context. For u∈U and a∈A, uRa, which is also written as (u,a)∈R, means that the object u possesses the attribute a.Denote(132){a}*={u∈U:uIa},{u}*={a∈A:uIa}.Definition 61.
LetFC=(U,A,I) be a formal context. Define a mapping fFC:A→2U by
(133)fFC(a)={a}*
for each a∈A. Then, (fFC)A=(fFC,A) is called the soft set over U induced by FC. We denote (fFC)A by SFC.Definition 62.
LetS=fA be a soft set over U. Define a binary relation IS∈2U×A by
(134)(u,a)∈IS⟺u∈f(a)
for each u∈U and each a∈A. Then, (U,A,IS) is called the formal context induced by S. We denote (U,A,IS) by FCS.Lemma 63.
LetFC=(U,A,I) be a formal context, let  SFC be the soft set induced by FC, and let  FCSFC be the formal context induced by  SFC. Then,
(135)FC=FCSFC.Proof.
Obviously,FCSFC=(U,A,ISFC).
For each(u,a)∈U×A,
(136)(u,a)∈ISFC⟺u∈fFC(a).
For eacha∈A,
(137)fFC(a)={a}*={u∈U:uIa}={u∈U:(u,a)∈I}.
Then,(138)u∈fFC(a)⟺(u,a)∈I.
Thus, for each(u,a)∈U×A, (u,a)∈ISFC⇔(u,a)∈I. This implies ISFC=I.
Hence,(139)FC=FCSFC.Lemma 64.
LetS=fA be a soft set over U, let  FCS be the formal context induced by S, and let
(140)SFCS
be the soft set induced by FCS. Then,
(141)S=SFCS.Proof.
Obviously,SFCS=(fFCS,A).
SinceFCS=(U,A,IS), then
(142)fFCS(a)={u∈U:(u,a)∈IS}foreacha∈A.
So, (u,a)∈IS⇔u∈f(a).
Obviously,f(a)={u∈U:u∈f(a)} for each a∈A.
Thus, for eacha∈A, fFC(a)=f(a).
Hence,(143)S=SFCS.Theorem 65.
Let(144)Γ={S=fA:SisasoftsetoverU},Σ={FC=(U,A,I):FCisaformalcontext}.
Then, there exists a one-to-one correspondence betweenΓ and Σ.Proof.
Two mappingsk:Σ→Γ and l:Γ→Σ are defined as follows:
(145)k(FC)=SFC,foranyFC=(U,A,I)∈Σ,l(S)=FCS,foranyS=fA∈Γ.
By Lemma63,(146)l∘k=iΣ,
where l∘k is the composition of k and G, and iΣ is the identity mapping on Σ.
By Lemma64,(147)k∘l=iΓ,
where l∘F is the composition of l and k, and iΓ is the identity mapping on Γ.
Hence,F and G are both a one-to-one correspondence. This prove that there exists a one-to-one correspondence between Σ and Γ.Remark 66.
Theorem65 illustrates that we can do formal concept analysis for soft sets or do soft analysis for formal contexts.


## 7. Conclusions

In this paper, we investigated soft rough approximation operators and the problems of combing soft sets with soft rough sets and topologies. Four pairs of soft rough approximation operators were considered, and their properties were given. Four sorts of soft rough sets are defined by using four pairs of soft rough approximation operators, and Pawlak’s rough set models can be viewed as a special case of soft rough sets. We researched relationships among soft sets, soft rough sets and topologies, obtained the structure of soft rough sets, and revealed that every topological space on the initial universe is a soft approximating space. We may mention that soft rough sets can be used in object evaluation and group decision making. It should be noted that the use of soft rough sets could, to some extent, automatically reduce the noise factor caused by the subjective nature of the expert's evaluation. We will investigate these problems in future papers.



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