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Research Article Soft Rough Approximation Operators and Related Results Zhaowen Li, Bin Qin,

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Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 241485, 15 pages
http://dx.doi.org/10.1155/2013/241485
Research Article
Soft Rough Approximation Operators and Related Results
Zhaowen Li,1 Bin Qin,2 and Zhangyong Cai3
1
School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
School of Information and Statistics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China
3
Department of Mathematics, Guangxi Teachers College, Nanning, Guangxi 530023, China
2
Correspondence should be addressed to Zhaowen Li; lizhaowen8846@126.com
Received 2 August 2012; Revised 21 January 2013; Accepted 22 January 2013
Academic Editor: Hui-Shen Shen
Copyright © 2013 Zhaowen Li et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 Section 2, 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
2
Journal of Applied Mathematics
rough sets. In Section 6, we prove that there exists a one-toone 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, π‘ˆ denotes initial universe, 𝐸
denotes the set of all possible parameters, and 2π‘ˆ denotes the
family of all subsets of π‘ˆ. We only consider the case where
both π‘ˆ 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]).
Let 𝑅 be an equivalence relation on π‘ˆ. The pair (π‘ˆ, 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:
𝑅∗ (𝑋) = {π‘₯ ∈ π‘ˆ : [π‘₯]𝑅 ⊆ 𝑋} ,
𝑅∗ (𝑋) = {π‘₯ ∈ π‘ˆ : [π‘₯]𝑅 ∩ 𝑋 =ΜΈ 0} .
(1)
∗
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 of 𝑋 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) =ΜΈ 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]). Let 𝑅 be a binary relation on π‘ˆ. The
pair (π‘ˆ, 𝑅) is called an approximation space. Based on the
approximation space (π‘ˆ, 𝑅), we define a pair of operations 𝑅,
𝑅 : 2π‘ˆ → 2π‘ˆ as follows:
𝑅 (𝑋) = {π‘₯ ∈ π‘ˆ : 𝑅 (π‘₯) ⊆ 𝑋} ,
𝑅 (𝑋) = {π‘₯ ∈ π‘ˆ : 𝑅 (π‘₯) ∩ 𝑋 =ΜΈ 0} ,
where 𝑋 ∈ 2π‘ˆ and 𝑅(π‘₯) = {𝑦 ∈ π‘ˆ : π‘₯𝑅𝑦}.
(2)
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) =ΜΈ 𝑅(𝑋).
2.2. Soft Sets
Definition 2 (see [2]). Let 𝐴 be a nonempty subset of 𝐸. A
pair (𝑓, 𝐴) is called a soft set over π‘ˆ, if 𝑓 is a mapping given
by 𝑓 : 𝐴 → 2π‘ˆ. We denote (𝑓, 𝐴) by 𝑓𝐴.
In other words, a soft set over π‘ˆ is a parametrized family
of subsets of the universe π‘ˆ. For 𝑒 ∈ 𝐴, 𝑓(𝑒) may be
considered as the set of 𝑒-approximate elements of the soft
set 𝑓𝐴.
Definition 3 (see [3]). Let 𝑓𝐴 and 𝑔𝐡 be two soft sets over π‘ˆ.
(1) 𝑓𝐴 is called a soft subset of 𝑔𝐡 , if 𝐴 ⊆ 𝐡 and 𝑓(𝑒) =
Μƒ 𝑔𝐡 .
𝑔(𝑒) for each 𝑒 ∈ 𝐴. We denote it by 𝑓𝐴⊂
Μƒ 𝑓𝐴. We denote
(2) 𝑓𝐴 is called a soft super set of 𝑔𝐡 , if 𝑔𝐡 ⊂
Μƒ 𝑔𝐡 .
it by 𝑓𝐴⊃
Definition 4 (see [3]). Let 𝑓𝐴 and 𝑔𝐡 be two soft sets over π‘ˆ.
𝑓𝐴 and 𝑔𝐡 are called soft equal, if 𝐴 = 𝐡 and 𝑓(𝑒) = 𝑔(𝑒) for
each 𝑒 ∈ 𝐴. We denote it by 𝑓𝐴 = 𝑔𝐡 .
Μƒ 𝑔𝐡 and 𝑓𝐴⊃
Μƒ 𝑔𝐡 .
Obviously, 𝑓𝐴 = 𝑔𝐡 if and only if 𝑓𝐴⊂
Definition 5 (see [10, 12]). Let 𝑓𝐴 be a soft set over π‘ˆ.
(1) 𝑓𝐴 is called full, if β‹ƒπ‘Ž∈𝐴 𝑓(π‘Ž) = π‘ˆ.
(2) 𝑓𝐴 is called partition, if {𝑓(π‘Ž) : π‘Ž ∈ 𝐴} forms a
partition of π‘ˆ.
Obviously, every partition soft set is full.
Definition 6. Let 𝑓𝐴 be a soft set over π‘ˆ.
(1) 𝑓𝐴 is called keeping intersection, if for any π‘Ž, 𝑏 ∈ 𝐴,
there exists 𝑐 ∈ 𝐴 such that 𝑓(π‘Ž) ∩ 𝑓(𝑏) = 𝑓(𝑐).
(2) 𝑓𝐴 is called keeping union, if for any π‘Ž, 𝑏 ∈ 𝐴, there
exists 𝑐 ∈ 𝐴 such that 𝑓(π‘Ž) ∪ 𝑓(𝑏) = 𝑓(𝑐).
(3) 𝑓𝐴 is called topological, if {𝑓(π‘Ž) : π‘Ž ∈ 𝐴} is a topology
on π‘ˆ.
Obviously, every topological soft set is full, keeping intersection and keeping union, and 𝑓𝐴 is keeping intersection
(resp., keeping union) if and only if for any 𝐴󸀠 ⊆ 𝐴,
there exists π‘ŽσΈ€  ∈ 𝐴 such that β‹‚π‘Ž∈𝐴󸀠 𝑓(π‘Ž) = 𝑓(π‘ŽσΈ€  ) (resp.,
β‹ƒπ‘Ž∈𝐴󸀠 𝑓(π‘Ž) = 𝑓(π‘ŽσΈ€  )).
Journal of Applied Mathematics
3
Example 7. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 }, and
let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows:
𝑓 (π‘Ž1 ) = {β„Ž1 , β„Ž2 , β„Ž5 } ,
Obviously, 𝑓𝐴 is partition. But
𝑓 (π‘Ž2 ) ∩ 𝑓 (π‘Ž3 ) = 0 =ΜΈ 𝑓 (π‘Ž)
𝑓 (π‘Ž1 ) ∪ 𝑓 (π‘Ž2 ) = {β„Ž1 , β„Ž2 , β„Ž5 } =ΜΈ 𝑓 (π‘Ž)
𝑓 (π‘Ž2 ) = 0,
(3)
𝑓 (π‘Ž3 ) = {β„Ž3 } ,
𝑓 (π‘Ž4 ) = {β„Ž3 , β„Ž4 } .
𝑓 (π‘Ž1 ) = {β„Ž1 } ,
𝑓 (π‘Ž1 ) ∩ 𝑓 (π‘Ž2 ) = 𝑓 (π‘Ž1 ) ∩ 𝑓 (π‘Ž3 ) = 𝑓 (π‘Ž1 ) ∩ 𝑓 (π‘Ž4 )
= 𝑓 (π‘Ž2 ) ∩ 𝑓 (π‘Ž3 ) = 𝑓 (π‘Ž2 ) ∩ 𝑓 (π‘Ž4 )
𝑓 (π‘Ž2 ) = {β„Ž1 , β„Ž4 } ,
(4)
𝑓 (π‘Ž3 ) = {β„Ž1 , β„Ž3 , β„Ž4 } ,
Then, 𝑓𝐴 is full and keeping intersection. But
(∀π‘Ž ∈ 𝐴) . (5)
Obviously, 𝑓𝐴 is full, keeping intersection and keeping
union. But 𝑓𝐴 is not topological.
From Examples 7, 8, 9, and 10 and 11, we have the
following relationships:
Thus, 𝑓𝐴 is not keeping union.
𝑓𝐴 is keeping intersection
Example 8. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 },
and let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows:
𝑓 (π‘Ž1 ) = {β„Ž1 } ,
𝑓𝐴 is full
(6)
𝑓𝐴 is keeping union
𝑓 (π‘Ž4 ) = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 } .
Then, 𝑓𝐴 is keeping intersection and keeping union. But
𝑓𝐴 is not full.
Example 9. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 },
and let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows:
𝑓 (π‘Ž1 ) = {β„Ž1 } ,
𝑓 (π‘Ž2 ) = {β„Ž2 } ,
𝑓 (π‘Ž3 ) = {β„Ž1 , β„Ž2 } ,
𝑓𝐴 is full and
keeping union
𝑓𝐴 is partition
Example 10. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 },
and let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows:
𝑓 (π‘Ž1 ) = {β„Ž1 , β„Ž2 } ,
𝑓 (π‘Ž4 ) = {β„Ž4 } .
𝑓𝐴 is full, keeping intersection and keeping union
(7)
Then, 𝑓𝐴 is full and keeping union. But 𝑓𝐴 is neither
keeping intersection nor partition.
𝑓 (π‘Ž3 ) = {β„Ž3 } ,
𝑓𝐴 is topological
𝑓𝐴 is full and
keeping intersection
𝑓 (π‘Ž4 ) = π‘ˆ.
𝑓 (π‘Ž2 ) = {β„Ž5 } ,
(10)
𝑓 (π‘Ž4 ) = 𝑋.
= 0 = 𝑓 (π‘Ž2 ) .
𝑓 (π‘Ž3 ) = {β„Ž1 , β„Ž2 , β„Ž3 } ,
(9)
Thus, 𝑓𝐴 is neither keeping intersection nor keeping
union.
𝑓 (π‘Ž3 ) ∩ 𝑓 (π‘Ž4 ) = {β„Ž3 } = 𝑓 (π‘Ž3 ) ,
𝑓 (π‘Ž2 ) = {β„Ž1 , β„Ž2 } ,
(∀π‘Ž ∈ 𝐴) .
Example 11. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 }
and let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows
Obviously, 𝑓𝐴 is not partition. We have
𝑓 (π‘Ž1 ) ∪ 𝑓 (π‘Ž3 ) = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž5 } =ΜΈ 𝑓 (π‘Ž)
(∀π‘Ž ∈ 𝐴) ,
(8)
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
4
Journal of Applied Mathematics
and giving their properties. Four sorts of soft rough sets
are defined by using four pairs of soft rough approximation
operators.
(1) If 𝑓𝐴 is full, then
(a) apr (𝑋) ⊆ 𝑋 ⊆ apr𝑃 (𝑋) for any 𝑋 ∈ 2π‘ˆ;
𝑃
(b) apr (π‘ˆ) = apr𝑃 (π‘ˆ) = π‘ˆ.
𝑃
3.1. Soft Rough Approximation Operators π‘Žπ‘π‘Ÿ and π‘Žπ‘π‘Ÿπ‘ƒ
𝑃
Definition 12 (see [9, 10]). Let 𝑓𝐴 be a soft set over π‘ˆ. Then,
the pair 𝑃 = (π‘ˆ, 𝑓𝐴) is called a soft approximation space. We
define a pair of operators apr , apr𝑃 : 2π‘ˆ → 2π‘ˆ as follows:
(2) If 𝑓𝐴 is keeping union, then
(a) for any 𝑋
∈ 2π‘ˆ, there exists π‘Ž
such that apr (𝑋) = 𝑓(π‘Ž);
𝑃
𝑃
𝑃
(11)
s.t. 𝑒 ∈ 𝑓 (π‘Ž) , 𝑓 (π‘Ž) ∩ 𝑋 =ΜΈ 0} .
(3) If 𝑓𝐴 is keeping intersection, then
apr (𝑋 ∩ π‘Œ) = apr (𝑋) ∩ apr (π‘Œ)
𝑃
apr and apr𝑃 are called the soft 𝑃-lower approximation
𝑃
operator on π‘ˆ and the soft 𝑃-upper approximation operator
on π‘ˆ, respectively. In general, we refer to apr and apr𝑃 as soft
𝑃
𝑃-rough approximations operator on π‘ˆ.
apr (𝑋) and apr𝑃 (𝑋) are called the soft 𝑃-lower approx𝑃
imation and the soft 𝑃-upper approximation of 𝑋, respectively. In general, we refer to apr (𝑋) and apr𝑃 (𝑋) as soft
𝑃
rough approximations of 𝑋 with respect to 𝑃.
𝑋 is called a soft 𝑃-definable set if apr (𝑋) = apr𝑃 (𝑋); 𝑋
𝑃
is called a soft 𝑃-rough set if apr (𝑋) =ΜΈ apr𝑃 (𝑋).
𝑃
Moreover, the sets
Pos𝑃 (𝑋) = apr (𝑋) ,
𝑃
Neg𝑃 (𝑋) = π‘ˆ − apr𝑃 (𝑋) ,
𝑃
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]). Let 𝑓𝐴 be a soft set over π‘ˆ, and let
𝑃 = (π‘ˆ, 𝑓𝐴) be a soft approximation space. Then, the following
properties hold for any 𝑋, π‘Œ ∈ 2π‘ˆ.
(1) apr (𝑋) = ⋃{𝑓(π‘Ž) : π‘Ž ∈ 𝐴 and 𝑓(π‘Ž) ⊆ 𝑋} ⊆
𝑃
𝑋; apr𝑃 (𝑋) = ⋃{𝑓(π‘Ž) : π‘Ž ∈ 𝐴 and 𝑓(π‘Ž) ∩ 𝑋 =ΜΈ 0}.
(2) apr (0) = apr𝑃 (0) = 0; apr (π‘ˆ) = apr𝑃 (π‘ˆ) =
𝑃
𝑃
β‹ƒπ‘Ž∈𝐴 𝑓(π‘Ž).
(3) 𝑋 ⊆ π‘Œ ⇒ apr (𝑋) ⊆ apr (π‘Œ); 𝑋 ⊆ π‘Œ ⇒ apr𝑃 (𝑋) ⊆
𝑃
𝑃
apr𝑃 (π‘Œ).
(4) apr𝑃 (𝑋 ∪ π‘Œ) = apr𝑃 (𝑋) ∪ apr𝑃 (π‘Œ).
=
apr (𝑋); apr (apr𝑃 (𝑋))
𝑃
𝑃
𝑃
=
Proposition 14. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then, the following properties
hold.
𝑃
for any 𝑋, π‘Œ ∈ 2π‘ˆ.
(13)
(4) If 𝑓𝐴 is partition, then
apr (𝑋 ∩ π‘Œ) = apr (𝑋) ∩ apr (π‘Œ)
𝑃
𝑃
𝑃
for any 𝑋, π‘Œ ∈ 2π‘ˆ.
(14)
(5) If 𝑓𝐴 is full and keeping union, then
apr𝑃 (𝑋) = π‘ˆ
for any 𝑋 ∈ 2π‘ˆ \ {0} .
(15)
Proof. (1)(a) By Proposition 13, apr (𝑋) ⊆ 𝑋. Suppose that
𝑃
𝑋 − apr𝑃 (𝑋) =ΜΈ 0. Pick
π‘₯ ∈ 𝑋 − apr𝑃 (𝑋) =ΜΈ 0.
(12)
Bnd𝑃 (𝑋) = apr𝑃 (𝑋) − apr (𝑋)
(5) apr (apr (𝑋))
𝑃
𝑃
apr𝑃 (𝑋).
𝐴
(b) for any 𝑋 ∈ 2π‘ˆ, there exists π‘Ž ∈ 𝐴 such that
apr𝑃 (𝑋) = 𝑓(π‘Ž).
apr (𝑋) = {𝑒 ∈ π‘ˆ : ∃π‘Ž ∈ 𝐴, s.t. 𝑒 ∈ 𝑓 (π‘Ž) ⊆ 𝑋} ,
apr𝑃 (𝑋) = {𝑒 ∈ π‘ˆ : ∃π‘Ž ∈ 𝐴,
∈
(16)
Since 𝑓𝐴 is full, π‘ˆ = β‹ƒπ‘Ž∈𝐴 𝑓(π‘Ž). So, π‘₯ ∈ 𝑓(π‘Ž) for some
π‘Ž ∈ 𝐴. π‘₯ ∈ 𝑋 implies 𝑓(π‘Ž) ∩ 𝑋 =ΜΈ 0. Thus, π‘₯ ∈ apr𝑃 (𝑋) =ΜΈ 0,
contradiction. Hence,
𝑋 ⊆ apr𝑃 (𝑋) .
(17)
(1)(b) This holds by (1) and Proposition 13.
(2) This holds by Proposition 13.
(3) By Proposition 13,
apr (𝑋 ∩ π‘Œ) ⊆ apr (𝑋) ∩ apr (π‘Œ) .
𝑃
𝑃
𝑃
(18)
Suppose that apr (𝑋) ∩ apr (π‘Œ) − apr (𝑋 ∩ π‘Œ) =ΜΈ 0. Pick
𝑃
𝑃
𝑃
π‘₯ ∈ apr (𝑋) ∩ apr (π‘Œ) − apr (𝑋 ∩ π‘Œ) .
𝑃
𝑃
𝑃
(19)
Then, there exist π‘Ž, 𝑏 ∈ 𝐴 such that π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋 and
π‘₯ ∈ 𝑓(𝑏) ⊆ π‘Œ. Since 𝑓𝐴 is keeping intersection, then 𝑓(π‘Ž) ∩
𝑓(𝑏) = 𝑓(𝑐) for some 𝑐 ∈ 𝐴. This implies π‘₯ ∈ 𝑓(𝑐) ⊆ 𝑋 ∩ π‘Œ.
Thus, π‘₯ ∈ apr (𝑋 ∩ π‘Œ), contradiction. Hence,
𝑃
apr (𝑋 ∩ π‘Œ) ⊇ apr (𝑋) ∩ apr (π‘Œ) .
𝑃
𝑃
𝑃
(20)
Therefore,
apr (𝑋 ∩ π‘Œ) = apr (𝑋) ∩ apr (π‘Œ) .
𝑃
𝑃
𝑃
(21)
Journal of Applied Mathematics
5
(4) Suppose that apr (𝑋)∩apr (π‘Œ)−apr (𝑋∩π‘Œ) =ΜΈ 0. Pick
𝑃
𝑃
𝑃
π‘₯ ∈ apr (𝑋) ∩ apr (π‘Œ) − apr (𝑋 ∩ π‘Œ) .
𝑃
𝑃
𝑃
(22)
Then, there exist π‘Ž, 𝑏 ∈ 𝐴 such that π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋 and
π‘₯ ∈ 𝑓(𝑏) ⊆ π‘Œ. Since 𝑓𝐴 is partition, then 𝑓(π‘Ž) = 𝑓(𝑏). This
implies π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋 ∩ π‘Œ. So, π‘₯ ∈ apr (𝑋 ∩ π‘Œ), contradiction.
𝑃
Thus,
apr (𝑋 ∩ π‘Œ) ⊇ apr (𝑋) ∩ apr (π‘Œ) .
𝑃
𝑃
𝑃
(23)
Proof. (1) This is obvious.
(2) Suppose that 𝑦 ∈ (𝑅𝑓 )𝑠 (π‘₯). Then, π‘₯𝑅𝑓 𝑦, and so
{π‘₯, 𝑦} ⊆ 𝑓(𝑏) for some 𝑏 ∈ 𝐴. Since 𝑓𝐴 is partition and
𝑓(π‘Ž) ∩ 𝑓(𝑏) =ΜΈ 0, then 𝑓(π‘Ž) = 𝑓(𝑏). Thus, 𝑦 ∈ 𝑓(π‘Ž). This
implies 𝑓(π‘Ž) ⊇ (𝑅𝑓 )𝑠 (π‘₯). By (1),
𝑓 (π‘Ž) = (𝑅𝑓 )𝑠 (π‘₯) .
(3) Suppose that 𝑦 ∈ (𝑅𝑓 )𝑠 (π‘₯). Then, π‘₯𝑅𝑓 𝑦, and so
{π‘₯, 𝑦} ⊆ 𝑓(π‘Žπ‘¦ ) for some π‘Žπ‘¦ ∈ 𝐴. By (1), 𝑓(π‘Žπ‘¦ ) ⊆ (𝑅𝑓 )𝑠 (π‘₯).
Thus, {𝑦} ⊆ 𝑓(π‘Žπ‘¦ ) ⊆ (𝑅𝑓 )𝑠 (π‘₯). This implies
Therefore,
apr (𝑋 ∩ π‘Œ) = apr (𝑋) ∩ apr (π‘Œ) .
𝑃
𝑃
𝑃
(𝑅𝑓 )𝑠 (π‘₯) = ⋃ {𝑓 (π‘Žπ‘¦ ) : 𝑦 ∈ (𝑅𝑓 )𝑠 (π‘₯)} .
(24)
(5) Since 𝑓𝐴 is full and keeping union, then π‘ˆ =
β‹ƒπ‘Ž∈𝐴 𝑓(π‘Ž) = 𝑓(π‘Ž∗ ) for some π‘Ž∗ ∈ 𝐴. For each 𝑋 ∈ 2π‘ˆ \ {0}
and each 𝑒 ∈ π‘ˆ, 𝑒 ∈ 𝑓(π‘Ž∗ ) and 𝑓(π‘Ž∗ ) ∩ 𝑋 = 𝑋 =ΜΈ 0, and then
apr𝑃 (𝑋) = π‘ˆ.
σΈ€ 
3.2. Soft Rough Approximation Operators π‘Žπ‘π‘ŸσΈ€  and π‘Žπ‘π‘Ÿπ‘ƒ ,
σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
π‘Žπ‘π‘ŸσΈ€ σΈ€  and π‘Žπ‘π‘Ÿπ‘ƒ , and π‘Žπ‘π‘ŸσΈ€ σΈ€ σΈ€  and π‘Žπ‘π‘Ÿπ‘ƒ
𝑃
𝑃
𝑃
Since 𝑓𝐴 is keeping union, then ⋃{𝑓(π‘Žπ‘¦ )
(𝑅𝑓 )𝑠 (π‘₯)} = 𝑓(π‘Ž) for some π‘Ž ∈ 𝐴. Thus,
𝑃
apr󸀠𝑃 (𝑋) = {π‘₯ ∈ π‘ˆ : (𝑅𝑓 )𝑠 (π‘₯) ∩ 𝑋 =ΜΈ 0} .
(25)
(1) 𝑅𝑓 is a symmetric relation.
(2) If 𝑓𝐴 is full, then 𝑅𝑓 is a reflexive relation.
(3) If 𝑓𝐴 is partition, then 𝑅𝑓 is an equivalence relation.
Proposition 17. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑅𝑓 be
the binary relation induced by 𝑓𝐴 on π‘ˆ. Then, the following
properties hold.
(1) If π‘₯ ∈ 𝑓(π‘Ž) with π‘Ž ∈ 𝐴, then 𝑓(π‘Ž) ⊆ (𝑅𝑓 )𝑠 (π‘₯).
(2) If 𝑓𝐴 is partition and π‘₯ ∈ 𝑓(π‘Ž) with π‘Ž ∈ 𝐴, then 𝑓(π‘Ž) =
(𝑅𝑓 )𝑠 (π‘₯).
(3) If 𝑓𝐴 is keeping union, then for each π‘₯ ∈ π‘ˆ, there exists
π‘Ž ∈ 𝐴 such that (𝑅𝑓 )𝑠 (π‘₯) = 𝑓(π‘Ž).
(30)
𝑃
is called a soft 𝑃󸀠 -rough set if aprσΈ€  (𝑋) =ΜΈ apr󸀠𝑃 (𝑋). The sets
𝑃
Pos󸀠𝑃 (𝑋)
= aprσΈ€  (𝑋) ,
𝑃
Neg󸀠𝑃 (𝑋) = π‘ˆ − apr󸀠𝑃 (𝑋) ,
(31)
Bnd󸀠𝑃 (𝑋) = apr󸀠𝑃 (𝑋) − aprσΈ€  (𝑋)
(26)
Proposition 16. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑅𝑓 be
the binary relation induced by 𝑓𝐴 on π‘ˆ. Then, the following
properties hold.
∈
𝑋 is called a soft 𝑃󸀠 -definable set if apr (𝑋) = apr󸀠𝑃 (𝑋). 𝑋
(2) For each π‘₯ ∈ π‘ˆ, define a successor neighborhood
𝑅𝑓𝑠 (π‘₯) of π‘₯ in π‘ˆ by
Since the following Proposition 16 is clear, we omit its
proof.
𝑦
(29)
aprσΈ€  (𝑋) = {π‘₯ ∈ π‘ˆ : (𝑅𝑓 )𝑠 (π‘₯) ⊆ 𝑋} ,
for each π‘₯, 𝑦 ∈ π‘ˆ. Then, 𝑅𝑓 is called the binary
relation induced by 𝑓𝐴 on π‘ˆ.
(𝑅𝑓 )𝑠 (π‘₯) = {𝑦 ∈ π‘ˆ : π‘₯𝑅𝑓 𝑦} .
:
Definition 18. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. We define three pairs of soft
rough approximation operations: 2π‘ˆ → 2π‘ˆ as follows:
(1)
(1) Define a binary relation 𝑅𝑓 on π‘ˆ by
{π‘₯, 𝑦} ⊆ 𝑓 (π‘Ž)
(28)
(𝑅𝑓 )𝑠 (π‘₯) = 𝑓 (π‘Ž) .
Definition 15. Let 𝑓𝐴 be a soft set over π‘ˆ.
π‘₯𝑅𝑓 𝑦 ⇐⇒ ∃π‘Ž ∈ 𝐴,
(27)
𝑃
σΈ€ 
are called the soft 𝑃 -positive region, the soft 𝑃󸀠 -negative
region, and the soft 𝑃󸀠 -boundary region of 𝑋, respectively.
Consider,
(2)
aprσΈ€ σΈ€  (𝑋) = ⋃ {(𝑅𝑓 )𝑠 (π‘₯) : π‘₯ ∈ π‘ˆ, (𝑅𝑓 )𝑠 (π‘₯) ⊆ 𝑋} ,
𝑃
apr󸀠󸀠𝑃 (𝑋) = π‘ˆ − aprσΈ€ σΈ€  (π‘ˆ − 𝑋) .
(32)
𝑃
𝑋 is called a soft 𝑃 -definable set if aprσΈ€ σΈ€  (𝑋) = apr󸀠󸀠𝑃 (𝑋).
σΈ€ σΈ€ 
𝑃
𝑋 is called a soft 𝑃󸀠󸀠 -rough set if aprσΈ€ σΈ€  (𝑋) =ΜΈ apr󸀠󸀠𝑃 (𝑋). The sets
𝑃
Pos󸀠󸀠𝑃 (𝑋) = aprσΈ€ σΈ€  (𝑋) ,
𝑃
Neg󸀠󸀠𝑃 (𝑋) = π‘ˆ − apr󸀠󸀠𝑃 (𝑋) ,
(33)
Bnd󸀠󸀠𝑃 (𝑋) = apr󸀠󸀠𝑃 (𝑋) − aprσΈ€ σΈ€  (𝑋)
𝑃
are called the soft 𝑃󸀠󸀠 -positive region, the soft 𝑃󸀠󸀠 -negative
region, and the soft 𝑃󸀠󸀠 -boundary region of 𝑋, respectively.
Consider,
6
Journal of Applied Mathematics
σΈ€ σΈ€ 
(2) aprσΈ€ σΈ€  (0) = 0; apr𝑃 (π‘ˆ) = π‘ˆ. If 𝑓𝐴 is full, then
(3)
𝑃
aprσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) = ∪ {(𝑅𝑓 )𝑠 (π‘₯) : π‘₯ ∈ π‘ˆ, (𝑅𝑓 )𝑠 (π‘₯) ∩ 𝑋 =ΜΈ 0} ,
aprσΈ€ σΈ€ σΈ€ 
𝑃
(𝑋) = π‘ˆ −
aprσΈ€ σΈ€ σΈ€ 
𝑃
σΈ€ 
aprσΈ€  (0) = apr𝑃 (0) = 0;
𝑃
(π‘ˆ − 𝑋) .
(34)
𝑋 is called a soft 𝑃 -definable set if aprσΈ€ σΈ€ σΈ€  (𝑋) = aprσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋).
𝑃
σΈ€ σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
is called a soft 𝑃 -rough set if apr (𝑋) =ΜΈ apr𝑃 (𝑋). The
𝑃
(39)
σΈ€ 
aprσΈ€  (π‘ˆ) = apr𝑃 (π‘ˆ) = π‘ˆ.
𝑃
σΈ€ σΈ€ σΈ€ 
𝑋
sets
σΈ€ σΈ€ σΈ€ 
PosσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) = apr (𝑋) ,
σΈ€ σΈ€ 
𝑃
aprσΈ€ σΈ€  (𝑋).
𝑃
(4) aprσΈ€ σΈ€  (aprσΈ€ σΈ€  (𝑋))
𝑃
σΈ€ σΈ€ σΈ€ 
NegσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) = π‘ˆ − apr𝑃 (𝑋) ,
𝑃
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 to aprσΈ€  (𝑋) and apr󸀠𝑃 (𝑋), aprσΈ€ σΈ€  (𝑋)
𝑃
σΈ€ σΈ€ σΈ€ 
aprσΈ€ σΈ€ σΈ€  (𝑋) ⊆ 𝑋 ⊆ apr𝑃 (𝑋) .
σΈ€ 
(36)
𝑃
σΈ€ σΈ€ σΈ€ 
(2) apr𝑃 (0) = 0; aprσΈ€ σΈ€ σΈ€  (π‘ˆ) = π‘ˆ. If 𝑓𝐴 is full, then
𝑃
σΈ€ σΈ€ σΈ€ 
aprσΈ€ σΈ€ σΈ€  (0) = apr𝑃 (0) = 0;
𝑃
σΈ€ σΈ€ σΈ€ 
𝑃
(37)
σΈ€ 
𝑃
σΈ€ 
(3) 𝑋 ⊆ π‘Œ ⇒ aprσΈ€  (𝑋) ⊆ aprσΈ€  (π‘Œ); 𝑋 ⊆ π‘Œ ⇒ apr𝑃 (𝑋) ⊆
𝑃
(7)
(5)
σΈ€ σΈ€ σΈ€ 
apr𝑃 (𝑋
𝑃
𝑃
∪ π‘Œ) =
∪
σΈ€ σΈ€ σΈ€ 
apr𝑃 (π‘Œ).
σΈ€ σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
𝑃
aprσΈ€ σΈ€ σΈ€  (𝑋).
𝑃
𝑃
𝑃
σΈ€ σΈ€ σΈ€ 
apr𝑃 (𝑋).
− 𝑋) = π‘ˆ −
𝑃
σΈ€ σΈ€ σΈ€ 
apr𝑃 (𝑋)
(6) aprσΈ€ σΈ€ σΈ€  (π‘ˆ − 𝑋) = π‘ˆ − apr𝑃 (𝑋); apr𝑃 (π‘ˆ − 𝑋) = π‘ˆ −
(7) aprσΈ€ σΈ€ σΈ€  (aprσΈ€ σΈ€ σΈ€  (𝑋))
(4) aprσΈ€  (𝑋 ∩ π‘Œ) = aprσΈ€  (𝑋) ∩ aprσΈ€  (π‘Œ).
(6)
𝑃
(4) aprσΈ€ σΈ€ σΈ€  (𝑋 ∩ π‘Œ) = aprσΈ€ σΈ€ σΈ€  (𝑋) ∩ aprσΈ€ σΈ€ σΈ€  (π‘Œ).
σΈ€ 
= apr𝑃 (0) = 0;
aprσΈ€  (π‘ˆ) = apr𝑃 (π‘ˆ) = π‘ˆ.
(5)
σΈ€ σΈ€ σΈ€ 
apr𝑃 (𝑋) ⊆ apr𝑃 (π‘Œ).
𝑃
𝑃
𝑃
𝑃
σΈ€ 
σΈ€ 
σΈ€ 
apr𝑃 (𝑋 ∪ π‘Œ) = apr𝑃 (𝑋) ∪ apr𝑃 (π‘Œ).
σΈ€ 
σΈ€ 
aprσΈ€  (π‘ˆ − 𝑋) = π‘ˆ − apr𝑃 (𝑋); apr𝑃 (π‘ˆ
𝑃
aprσΈ€  (𝑋).
𝑃
σΈ€ 
σΈ€ 
apr𝑃 (aprσΈ€  (𝑋)) ⊆ 𝑋 ⊆ aprσΈ€  (apr𝑃 (𝑋)).
𝑃
𝑃
aprσΈ€ σΈ€ σΈ€  (𝑋) ⊆ aprσΈ€ σΈ€ σΈ€  (π‘Œ); 𝑋 ⊆ π‘Œ ⇒
(3) 𝑋 ⊆ π‘Œ ⇒
σΈ€ 
𝑃
⊆
σΈ€ σΈ€ σΈ€ 
𝑃
σΈ€ σΈ€ σΈ€ 
σΈ€ σΈ€ 
⊇
σΈ€ σΈ€ σΈ€ 
(8) apr𝑃 (aprσΈ€ σΈ€ σΈ€  (𝑋)) ⊆ 𝑋 ⊆ aprσΈ€ σΈ€ σΈ€  (apr𝑃 (𝑋)).
𝑃
𝑃
𝑓 (π‘Ž1 ) = {β„Ž2 } ,
𝑓 (π‘Ž2 ) = {β„Ž1 , β„Ž4 } ,
𝑓 (π‘Ž3 ) = {β„Ž3 } ,
(1)
𝑃
σΈ€ σΈ€ σΈ€ 
aprσΈ€ σΈ€ σΈ€  (𝑋); apr𝑃 (apr𝑃 (𝑋))
Example 22. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 , π‘Ž4 },
and let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows:
Proposition 20. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 =
(π‘ˆ, 𝑓𝐴) be a soft approximation space. Then, the following
properties hold for any 𝑋 ∈ 2π‘ˆ.
aprσΈ€ σΈ€  (𝑋) ⊆ 𝑋 ⊆ apr𝑃 (𝑋) .
(41)
σΈ€ 
aprσΈ€ σΈ€ σΈ€  (π‘ˆ) = apr𝑃 (π‘ˆ) = π‘ˆ.
(2) apr𝑃 (0) = 0; aprσΈ€  (π‘ˆ) = π‘ˆ. If 𝑓𝐴 is full, then
σΈ€ 
(40)
𝑃
𝑃
aprσΈ€  (𝑋) ⊆ 𝑋 ⊆ apr𝑃 (𝑋) .
apr𝑃 (π‘Œ).
=
(1) If 𝑓𝐴 is full, then
⊆ 𝑋. If 𝑓𝐴 is full, then
aprσΈ€  (0)
𝑃
σΈ€ σΈ€ 
𝑃
Proposition 21. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then, the following properties
hold for any 𝑋, π‘Œ ∈ 2π‘ˆ.
Proposition 19. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then, the following properties
hold for any 𝑋, π‘Œ ∈ 2π‘ˆ.
(1)
σΈ€ σΈ€ 
aprσΈ€ σΈ€  (𝑋); apr𝑃 (apr𝑃 (𝑋))
𝑃
and apr󸀠󸀠𝑃 (𝑋), and aprσΈ€ σΈ€ σΈ€  (𝑋) and aprσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) as soft rough
𝑃
approximations of 𝑋 with respect to 𝑃󸀠 , 𝑃󸀠󸀠 , 𝑃󸀠󸀠󸀠 , respectively.
It is not very difficult to prove Propositions 19, 20, and 21
(see [15]).
aprσΈ€  (𝑋)
𝑃
=
𝑃
𝑃
σΈ€ σΈ€ 
apr𝑃 (𝑋).
(35)
σΈ€ σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
BndσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) = apr𝑃 (𝑋) − apr (𝑋)
σΈ€ σΈ€ 
(3) aprσΈ€ σΈ€  (π‘ˆ − 𝑋) = π‘ˆ − apr𝑃 (𝑋); apr𝑃 (π‘ˆ − 𝑋) = π‘ˆ −
(38)
𝑓 (π‘Ž4 ) = {β„Ž1 , β„Ž3 } .
(42)
Journal of Applied Mathematics
7
Obviously, 𝑓𝐴 is not full. We have
3.3. The Relationships among Four Pairs of Soft Rough Approximation Operators
(𝑅𝑓 )𝑠 (β„Ž1 ) = {β„Ž1 , β„Ž3 , β„Ž4 } ,
(𝑅𝑓 )𝑠 (β„Ž2 ) = {β„Ž2 } ,
(𝑅𝑓 )𝑠 (β„Ž3 ) = {β„Ž1 , β„Ž3 } ,
(43)
Lemma 23. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴) be
a soft approximation space. Then, the following properties hold
for any 𝑋 ∈ 2π‘ˆ.
(1) If 𝑓𝐴 is full, then
(𝑅𝑓 )𝑠 (β„Ž4 ) = {β„Ž1 , β„Ž4 } ,
apr (𝑋) ⊇ aprσΈ€  (𝑋) .
(𝑅𝑓 )𝑠 (β„Ž5 ) = 0.
Let 𝑋 = {β„Ž3 , β„Ž5 }, π‘Œ = {β„Ž1 , β„Ž2 , β„Ž3 }, and 𝑍 = {β„Ž2 , β„Ž4 , β„Ž5 }.
(2) If 𝑓𝐴 is full and keeping union, then
σΈ€ 
apr𝑃 (𝑋) ⊇ apr𝑃 (𝑋) .
(1) We have
apr󸀠𝑃 (𝑋) = {β„Ž1 , β„Ž3 } .
(44)
(a) apr (𝑋) = aprσΈ€  (𝑋);
𝑃
σΈ€ 
𝑃
apr󸀠𝑃 (𝑋) .
(b) apr𝑃 (𝑋) = apr𝑃 (𝑋).
(45)
(2) We have
aprσΈ€  (𝑋) = {β„Ž5 } ,
𝑃
(46)
aprσΈ€  (π‘Œ) = {β„Ž2 , β„Ž3 , β„Ž5 } .
Proof. (1) Suppose that π‘₯ ∈ aprσΈ€  (𝑋). Then, (𝑅𝑓 )𝑠 (π‘₯) ⊆
𝑃
𝑋. Since 𝑓𝐴 is full, then π‘₯ ∈ 𝑓(π‘Ž) for some π‘Ž ∈ 𝐴. By
Proposition 17, 𝑓(π‘Ž) ⊆ (𝑅𝑓 )𝑠 (π‘₯). Thus, π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋. This
implies π‘₯ ∈ apr (𝑋). Thus,
𝑃
𝑃
apr (𝑋) ⊇ aprσΈ€  (𝑋) .
𝑃
Thus,
aprσΈ€  (𝑋) ⊆ aprσΈ€  (π‘Œ) 󴁁󴁙󴀑 𝑋 ⊆ π‘Œ.
𝑃
𝑃
(47)
(3) We have
apr󸀠𝑃 (π‘Œ) = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 } .
(48)
Thus,
apr󸀠𝑃 (𝑋) ⊆ apr󸀠𝑃 (π‘Œ) 󴁁󴁙󴀑 𝑋 ⊆ π‘Œ.
(2) If 𝑋 = 0, then apr𝑃 (𝑋) = 0 = apr󸀠𝑃 (𝑋). If 𝑋 =ΜΈ 0, by
Proposition 14, apr𝑃 (𝑋) = π‘ˆ.
Hence,
(49)
apr (𝑋) ⊆ aprσΈ€  (𝑋) .
(58)
apr (𝑋) = aprσΈ€  (𝑋) .
(59)
𝑃
𝑃
By (1),
aprσΈ€  (π‘Œ) = {β„Ž2 , β„Ž3 , β„Ž5 } ,
𝑃
= {β„Ž2 , β„Ž5 } ,
𝑃
(50)
aprσΈ€  (π‘Œ ∪ 𝑍) = π‘ˆ.
𝑃
Thus,
aprσΈ€  (π‘Œ ∪ 𝑍) =ΜΈ aprσΈ€  (𝑦) ∪ aprσΈ€  (𝑍) .
𝑃
(57)
(3)(a) Suppose that π‘₯ ∈ apr (𝑋). Then, π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋
𝑃
for some π‘Ž ∈ 𝐴. Since 𝑓𝐴 is partition and π‘₯ ∈ 𝑓(π‘Ž), then
𝑓(π‘Ž) = (𝑅𝑓 )𝑠 (π‘₯) by Proposition 17. This implies π‘₯ ∈ aprσΈ€  (𝑋).
𝑃
Thus,
(4) We have
𝑃
(56)
𝑃
apr𝑃 (𝑋) ⊇ apr󸀠𝑃 (𝑋) .
apr󸀠𝑃 (𝑋) = {β„Ž1 , β„Ž3 } ,
aprσΈ€  (𝑍)
𝑃
(55)
(3) If 𝑓𝐴 is partition, then
Thus,
𝑋 ⊆ΜΈ
(54)
𝑃
𝑃
𝑃
(51)
(3)(b) This is similar to the proof of (3) (a).
Suppose that π‘₯ ∈ apr󸀠𝑃 (𝑋). Then, (𝑅𝑓 )𝑠 (π‘₯) ∩ 𝑋 =ΜΈ 0. Since
𝑓𝐴 is full, then π‘₯ ∈ 𝑓(π‘Ž) for some π‘Ž ∈ 𝐴. Since 𝑓𝐴 is partition
and π‘₯ ∈ 𝑓(π‘Ž), then 𝑓(π‘Ž) = (𝑅𝑓 )𝑠 (π‘₯) by Proposition 17. This
implies π‘₯ ∈ apr𝑃 (𝑋). Thus,
apr𝑃 (𝑋) ⊇ apr󸀠𝑃 (𝑋) .
(5) We have
(60)
Hence, apr𝑃 (𝑋) = apr󸀠𝑃 (𝑋).
apr󸀠𝑃 (π‘Œ) = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 } ,
apr󸀠𝑃 (𝑍) = {β„Ž1 , β„Ž2 , β„Ž4 } ,
𝑃
(52)
By Propositions 13 and 17, we have Lemma 24.
Lemma 24. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴) be
a soft approximation space. If 𝑓𝐴 is keeping union, then for any
𝑋 ∈ 2π‘ˆ,
apr󸀠𝑃 (π‘Œ ∩ 𝑍) = {β„Ž2 } .
Thus,
apr󸀠𝑃 (π‘Œ ∩ 𝑍) =ΜΈ apr󸀠𝑃 (π‘Œ) ∩ aprσΈ€  (𝑍) .
𝑃
(53)
apr (𝑋) ⊇ aprσΈ€ σΈ€  (𝑋) ,
𝑃
𝑃
σΈ€ σΈ€ σΈ€ 
apr𝑃 (𝑋) ⊇ apr𝑃 (𝑋) .
(61)
8
Journal of Applied Mathematics
Lemma 25. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴) be
a soft approximation space. Then, the following properties hold
for any 𝑋 ∈ 2π‘ˆ.
(1) If 𝑓𝐴 is full, then
aprσΈ€ σΈ€ σΈ€ 
𝑃
aprσΈ€  (𝑋)
𝑃
(𝑋) ⊆
σΈ€ σΈ€ 
apr𝑃
⊆
⊆
aprσΈ€ σΈ€ 
𝑃
(𝑋) ⊆ 𝑋
σΈ€ 
apr𝑃 (𝑋)
(𝑋) ⊆
⊆
σΈ€ σΈ€ σΈ€ 
apr𝑃
(62)
(𝑋) .
(2) If 𝑓𝐴 is partition, then
(a) aprσΈ€  (𝑋) = aprσΈ€ σΈ€  (𝑋) = aprσΈ€ σΈ€ σΈ€  (𝑋);
𝑃
σΈ€ 
𝑃
σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
𝑃
(b) apr𝑃 = apr𝑃 (𝑋) = apr𝑃 (𝑋).
∈ aprσΈ€  (𝑋).
𝑃
(𝑅𝑓 )𝑠 (π‘₯) ⊆
Proof. (1) Suppose that π‘₯
Since 𝑓𝐴 is full, then π‘₯ ∈
This implies π‘₯ ∈ aprσΈ€ σΈ€  (𝑋). Thus,
Suppose that aprσΈ€  (𝑋) ∩ aprσΈ€ σΈ€ σΈ€ 
𝑃 (π‘ˆ − 𝑋) =ΜΈ 0. Pick
𝑃
π‘₯ ∈ aprσΈ€  (𝑋) ∩ aprσΈ€ σΈ€ σΈ€ 
𝑃 (π‘ˆ − 𝑋) .
𝑃
π‘₯ ∈ aprσΈ€  (𝑋) implies (𝑅𝑓 )𝑠 (π‘₯) ⊆ 𝑋. π‘₯ ∈ aprσΈ€ σΈ€ σΈ€ 
𝑃 (π‘ˆ − 𝑋)
𝑃
implies that there exists 𝑦 ∈ π‘ˆ such that π‘₯ ∈ (𝑅𝑓 )𝑠 (𝑦) and
(𝑅𝑓 )𝑠 (𝑦) ∩ (π‘ˆ − 𝑋) =ΜΈ 0. So, (𝑅𝑓 )𝑠 (𝑦) ⊆ΜΈ 𝑋. Note that 𝑅𝑓 is
an equivalence relation. Then, (𝑅𝑓 )𝑠 (π‘₯) = (𝑅𝑓 )𝑠 (𝑦). Thus,
(𝑅𝑓 )𝑠 (π‘₯) ⊆ΜΈ 𝑋, contradiction.
Hence, aprσΈ€  (𝑋) ∩ aprσΈ€ σΈ€ σΈ€ 
𝑃 (π‘ˆ − 𝑋) = 0.
𝑃
This proves that
σΈ€ σΈ€ σΈ€ 
aprσΈ€  (𝑋) ⊆ π‘ˆ − aprσΈ€ σΈ€ σΈ€ 
𝑃 (π‘ˆ − 𝑋) = apr (𝑋) .
Then, (𝑅𝑓 )𝑠 (π‘₯) ⊆ 𝑋.
𝑋 by Proposition 17.
𝑃
Suppose that
aprσΈ€ σΈ€ σΈ€  (𝑋)
𝑃
π‘₯∈
−
aprσΈ€ σΈ€ σΈ€ 
𝑃
⊆
aprσΈ€  (𝑋) = aprσΈ€ σΈ€ σΈ€  (𝑋) .
aprσΈ€ σΈ€ 
𝑃
𝑃
(𝑋) .
aprσΈ€  (𝑋) =ΜΈ 0.
𝑃
(𝑋) −
(63)
Pick
aprσΈ€  (𝑋) .
𝑃
aprσΈ€ σΈ€  (𝑋) ⊆ 𝑋 ⊆ apr󸀠󸀠𝑃 (𝑋) .
𝑃
𝑃
𝑃
𝑃
𝑃
(74)
Then,
π‘ˆ − aprσΈ€  (π‘ˆ − 𝑋) = π‘ˆ − aprσΈ€ σΈ€  (π‘ˆ − 𝑋) = π‘ˆ − aprσΈ€ σΈ€ σΈ€  (π‘ˆ − 𝑋) .
𝑃
𝑃
𝑃
(75)
By Propositions 19, 20, and 21,
apr󸀠𝑃 (𝑋) = apr󸀠󸀠𝑃 (𝑋) = aprσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) .
(76)
(65)
Example 26. Let π‘ˆ = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 , β„Ž6 }, 𝐴 = {π‘Ž1 , π‘Ž2 , π‘Ž3 ,
π‘Ž4 , π‘Ž5 }, and let 𝑓𝐴 be a soft set over π‘ˆ, defined as follows:
(66)
π‘ˆ − aprσΈ€ σΈ€ σΈ€  (π‘ˆ − 𝑋) ⊇ π‘ˆ − aprσΈ€  (π‘ˆ − 𝑋) ⊇ π‘ˆ − aprσΈ€ σΈ€  (π‘ˆ − 𝑋) .
𝑃
𝑃
𝑃
(67)
𝑓 (π‘Ž1 ) = {β„Ž1 } ,
𝑓 (π‘Ž2 ) = {β„Ž6 } ,
𝑓 (π‘Ž3 ) = {β„Ž2 , β„Ž5 } ,
(77)
𝑓 (π‘Ž4 ) = {β„Ž2 , β„Ž4 } ,
By Propositions 19, 20, and 21,
apr󸀠󸀠𝑃 (𝑋) ⊆ apr󸀠𝑃 (𝑋) ⊆ aprσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) .
(68)
(2)(a) Suppose that π‘₯ ∈ aprσΈ€ σΈ€  (𝑋). Then, there exists 𝑦 ∈ π‘ˆ
𝑃
such that π‘₯ ∈ (𝑅𝑓 )𝑠 (𝑦) ⊆ 𝑋. Since 𝑓𝐴 is partition, then 𝑅𝑓 is
an equivalence relation by Proposition 16. Thus, π‘₯ ∈ (𝑅𝑓 )𝑠 (𝑦)
follows (𝑅𝑓 )𝑠 (π‘₯) = (𝑅𝑓 )𝑠 (𝑦). So, (𝑅𝑓 )𝑠 (π‘₯) ⊆ 𝑋. This implies
π‘₯ ∈ aprσΈ€  (𝑋). Hence,
𝑃
aprσΈ€  (𝑋) ⊇ aprσΈ€ σΈ€  (𝑋) .
𝑃
(69)
𝑓 (π‘Ž5 ) = {β„Ž1 , β„Ž3 , β„Ž5 } .
Obviously, 𝑓𝐴 is full. We have
(𝑅𝑓 )𝑠 (β„Ž1 ) = {β„Ž1 , β„Ž3 , β„Ž5 } ,
(𝑅𝑓 )𝑠 (β„Ž2 ) = {β„Ž2 , β„Ž4 , β„Ž5 } ,
(𝑅𝑓 )𝑠 (β„Ž3 ) = {β„Ž1 , β„Ž3 , β„Ž5 } ,
(𝑅𝑓 )𝑠 (β„Ž4 ) = {β„Ž2 , β„Ž4 } ,
By (1),
aprσΈ€  (𝑋) = aprσΈ€ σΈ€  (𝑋) .
𝑃
(73)
(64)
then
𝑃
(72)
(2)(b) By (2)(a),
𝑃
Since
aprσΈ€ σΈ€ σΈ€  (π‘ˆ − 𝑋) ⊆ aprσΈ€  (π‘ˆ − 𝑋) ⊆ aprσΈ€ σΈ€  (π‘ˆ − 𝑋) ,
𝑃
aprσΈ€  (π‘ˆ − 𝑋) = aprσΈ€ σΈ€  (π‘ˆ − 𝑋) = aprσΈ€ σΈ€ σΈ€  (π‘ˆ − 𝑋) .
π‘₯ ∉ aprσΈ€  (𝑋) implies (𝑅𝑓 )𝑠 (π‘₯) ⊆ΜΈ 𝑋. So, (𝑅𝑓 )𝑠 (π‘₯) ∩ (π‘ˆ −
𝑃
𝑋) =ΜΈ 0. Since 𝑓𝐴 is full, then π‘₯ ∈ (𝑅𝑓 )𝑠 (π‘₯) by Proposition 17.
This implies π‘₯ ∈ apr󸀠󸀠𝑃 (π‘ˆ − 𝑋). Thus, π‘₯ ∉ π‘ˆ − apr󸀠󸀠𝑃 (π‘ˆ − 𝑋) =
aprσΈ€ σΈ€ σΈ€  (𝑋), contradiction.
𝑃
Hence, aprσΈ€ σΈ€ σΈ€  (𝑋) ⊆ aprσΈ€  (𝑋).
𝑃
𝑃
By Proposition 20,
𝑃
𝑃
By (1),
𝑃
aprσΈ€  (𝑋)
𝑃
(71)
𝑃
(70)
(𝑅𝑓 )𝑠 (β„Ž5 ) = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž5 } .
(78)
Journal of Applied Mathematics
9
Let 𝑋 = {β„Ž2 , β„Ž4 , β„Ž6 }. We have
Theorem 29 (see [10]). Let 𝑅 be an equivalence relation on
π‘ˆ, let (𝑓𝑅 )𝐴 be the soft set induced by 𝑅 on π‘ˆ, and let 𝑃 =
(π‘ˆ, (𝑓𝑅 )𝐴 ) be a soft approximation space. Then, for each 𝑋 ∈
2π‘ˆ,
apr (𝑋) = {β„Ž2 , β„Ž4 , β„Ž6 } ,
𝑃
aprσΈ€  (𝑋) = {β„Ž4 } ,
𝑅 (𝑋) = apr (𝑋) ,
𝑃
aprσΈ€ σΈ€ 
𝑃
(𝑋) = {β„Ž2 , β„Ž4 } ,
(79)
apr𝑃 (𝑋) = {β„Ž2 , β„Ž4 , β„Ž5 , β„Ž6 } ,
apr󸀠󸀠𝑃 (𝑋) = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 , β„Ž5 } .
apr (𝑋) =ΜΈ aprσΈ€  (𝑋) ,
𝑃
(1) 𝑅𝑓 (𝑋) = apr (𝑋) = aprσΈ€  (𝑋) = aprσΈ€ σΈ€  (𝑋) = aprσΈ€ σΈ€ σΈ€  (𝑋);
aprσΈ€  (𝑋)
𝑃
=ΜΈ aprσΈ€ σΈ€ 
𝑃
(2) 𝑅𝑓 (𝑋) = apr𝑃 (𝑋) = apr𝑃 (𝑋) = apr𝑃 (𝑋) = apr𝑃 (𝑋),
𝑃
(𝑋) ,
(80)
apr󸀠𝑃 (𝑋) =ΜΈ apr󸀠󸀠𝑃 (𝑋) .
By Proposition 16 and Lemmas 23, 24, and 25, we have
Theorem 27.
Theorem 27. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then, the following properties
hold for any 𝑋 ∈ 2π‘ˆ.
(1) If 𝑓𝐴 is full, then
aprσΈ€  (𝑋)
𝑃
⊆
aprσΈ€ σΈ€ 
𝑃
σΈ€ σΈ€ 
(𝑋) ⊆ 𝑋
σΈ€ 
(81)
σΈ€ σΈ€ σΈ€ 
⊆ apr𝑃 (𝑋) ⊆ apr𝑃 (𝑋) ⊆ apr𝑃 (𝑋) .
aprσΈ€ σΈ€ σΈ€  (𝑋) ⊆ aprσΈ€  (𝑋) ⊆ aprσΈ€ σΈ€  (𝑋) ⊆ apr (𝑋)
𝑃
σΈ€ σΈ€ 
𝑃
σΈ€ 
⊆ 𝑋 ⊆ apr𝑃 (𝑋) ⊆ apr𝑃 (𝑋)
(82)
σΈ€ σΈ€ σΈ€ 
𝑃
σΈ€ σΈ€ σΈ€ 
where 𝑅𝑓 (𝑋) and 𝑅𝑓 (𝑋) are the Pawlak rough approximations
of 𝑋.
Corollary 31. Let 𝑓𝐴 be a full soft set over π‘ˆ, and let 𝑃 =
(π‘ˆ, 𝑓𝐴) be a soft approximation space. Then,
(1) every soft 𝑃󸀠󸀠󸀠 -definable set is a soft 𝑃󸀠 -definable set.
(2) every soft 𝑃󸀠 -definable set is a soft 𝑃󸀠󸀠 -definable set.
Remark 32. Theorems 29 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.
Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴) be a soft
approximation space. Denote
πœπ‘“ = {𝑋 ∈ 2π‘ˆ : 𝑋 = apr (𝑋)} ,
𝑃
⊆ apr𝑃 (𝑋) ⊆ apr𝑃 (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. Let 𝑅 be an equivalence relation on π‘ˆ. Define
a mapping 𝑓𝑅 : 𝐴 → 2π‘ˆ by
𝑓𝑅 (π‘Ž) = [π‘Ž]𝑅
𝑃
σΈ€ σΈ€ 
4. The Relationships between Soft
Sets and Topologies
(2) If 𝑓𝐴 is full and keeping union, then
𝑃
𝑃
σΈ€ 
𝑃
apr𝑃 (𝑋) =ΜΈ apr󸀠𝑃 (𝑋) ,
𝑃
Thus, in this case, 𝑋 ∈ 2π‘ˆ is a Pawlak rough set if and only
if X is a soft 𝑃-rough set.
Theorem 30. Let 𝑓𝐴 be a partition soft set over π‘ˆ, and let
𝑃 = (π‘ˆ, 𝑓𝐴) be a soft approximation space. Then, the following
properties hold for any 𝑋 ∈ 2π‘ˆ.
Thus,
(𝑋) ⊆
(84)
By Proposition 16 and Lemmas 23 and 25, we have
Theorem 30.
apr󸀠𝑃 (𝑋) = {β„Ž2 , β„Ž4 , β„Ž5 } ,
aprσΈ€ σΈ€ σΈ€ 
𝑃
𝑅 (𝑋) = apr𝑃 (𝑋) .
𝑃
(83)
for each π‘Ž ∈ 𝐴, where 𝐴 = π‘ˆ. Then, (𝑓𝑅 )𝐴 is called the soft
set induced by 𝑅 on π‘ˆ.
πœŽπ‘“ = {𝑋 ∈ 2π‘ˆ : 𝑋 = apr𝑃 (𝑋)} ;
πœπ‘“σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 = aprσΈ€  (𝑋)} ,
𝑃
πœŽπ‘“σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 = apr󸀠𝑃 (𝑋)} ;
πœπ‘“σΈ€ σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 = aprσΈ€ σΈ€  (𝑋)} ,
𝑃
πœŽπ‘“σΈ€ σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 = apr󸀠󸀠𝑃 (𝑋)} ;
πœπ‘“σΈ€ σΈ€ σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 = aprσΈ€ σΈ€ σΈ€  (𝑋)} ,
𝑃
πœŽπ‘“σΈ€ σΈ€ σΈ€ 
π‘ˆ
= {𝑋 ∈ 2 : 𝑋 = aprσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋)} .
(85)
10
Journal of Applied Mathematics
Then, 𝑓(π‘Ž) = apr (𝑓(π‘Ž)). So 𝑓(π‘Ž) ∈ πœπ‘“ . Thus,
4.1. The First Sort of Topologies Induced by a Soft Set
and Related Results. By Propositions 13 and 14, we have
Theorem 34.
Theorem 34. Let 𝑓𝐴 be a full and keeping intersection or
a partition soft set over π‘ˆ and let 𝑃 = (π‘ˆ, 𝑓𝐴) be a soft
approximation space. Then πœπ‘“ is a topology on π‘ˆ.
Remark 35. Let 𝑓𝐴 be a full and keeping union soft set over
π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴) be an soft approximation space. Then,
by Proposition 14(5), πœŽπ‘“ = {0, π‘ˆ} is a indiscrete topology on
π‘ˆ.
𝑃
{𝑓 (π‘Ž) : π‘Ž ∈ 𝐴} ⊆ πœπ‘“ .
(3) Suppose that 𝑋 ∈ πœπ‘“ . If 𝑋 = 0, by 𝑓𝐴 is topological,
there exists π‘Ž ∈ 𝐴 such that 𝑋 = 𝑓(π‘Ž). If 𝑋 =ΜΈ 0, for each π‘₯ ∈ 𝑋,
𝑋 = apr (𝑋), there exists π‘Žπ‘₯ ∈ 𝐴 such that π‘₯ ∈ 𝑓(π‘Žπ‘₯ ) ⊆ 𝑋.
𝑃
Then,
𝑋 = ⋃ {π‘₯} ⊆ ⋃ 𝑓 (π‘Žπ‘₯ ) ⊆ 𝑋.
π‘₯∈𝑋
π‘₯∈𝑋
⋃ 𝑓 (π‘Žπ‘₯ ) = 𝑓 (π‘Ž)
for some π‘Ž ∈ 𝐴.
π‘₯∈𝑋
𝐴}.
By (1), πœπ‘“ ⊇ {𝑓(π‘Ž) : π‘Ž ∈ 𝐴}.
Hence,
πœπ‘“ = {𝑓 (π‘Ž) : π‘Ž ∈ 𝐴} .
π‘ˆ
{apr𝑃 (𝑋) : 𝑋 ∈ 2 } ⊆ πœπ‘“ = {apr (𝑋) : 𝑋 ∈ 2 } ;
𝑃
(86)
apr (𝑋) = int (𝑋)
𝑃
(87)
(97)
(4) It suffices to show that
(2)
πœπ‘“ ⊇ {𝑓 (π‘Ž) : π‘Ž ∈ 𝐴} ;
for each 𝑋 ∈ 2π‘ˆ.
apr (𝑋) ⊆ int (𝑋) .
(99)
𝑃
(88)
Conversely, for each π‘Œ ∈ πœπ‘“ with π‘Œ ⊆ 𝑋, we have π‘Œ =
apr (π‘Œ) ⊆ apr (𝑋) by Proposition 13. Thus,
𝑃
(4) apr is an interior operator of πœπ‘“ .
𝑃
int (𝑋) = ∪ {π‘Œ : π‘Œ ∈ πœπ‘“ , π‘Œ ⊆ 𝑋} ⊆ apr (𝑋) .
𝑃
𝑃
Proof. (1) By Proposition 13, we have
(98)
By (1), apr (𝑋) ∈ πœπ‘“ . By Proposition 13, apr (𝑋) ⊆ 𝑋.
𝑃
𝑃
Thus
(3) if 𝑓𝐴 is topological, then
πœπ‘“ = {𝑓 (π‘Ž) : π‘Ž ∈ 𝐴} ;
(96)
This implies 𝑋 ∈ {𝑓(π‘Ž) : π‘Ž ∈ 𝐴}. Thus, πœπ‘“ ⊆ {𝑓(π‘Ž) : π‘Ž ∈
(1)
π‘ˆ
(95)
So, 𝑋 = ⋃π‘₯∈𝑋 𝑓(π‘Žπ‘₯ ). Since 𝑓𝐴 is keeping union, then
The following theorem gives the structure of the first sort
of topologies induced by a soft set.
Theorem 36. Let 𝑓𝐴 be a full and keeping intersection soft set
over π‘ˆ, let 𝑃 = (π‘ˆ, 𝑓𝐴) be a soft approximation space, and let
πœπ‘“ be the topology induced by 𝑓𝐴 on π‘ˆ. Then,
(94)
(100)
Hence,
π‘ˆ
{apr𝑃 (𝑋) : 𝑋 ∈ 2 } ⊆ πœπ‘“ .
(89)
apr (𝑋) = int (𝑋) .
(90)
Definition 37. Let 𝜏 be a topology on π‘ˆ. Put 𝜏 = {π‘ˆπ‘Ž : π‘Ž ∈ 𝐴},
where 𝐴 is the set of indexes. Define a mapping π‘“πœ : 𝐴 → 2π‘ˆ
by π‘“πœ (π‘Ž) = π‘ˆπ‘Ž for each π‘Ž ∈ 𝐴. Then, the soft set (π‘“πœ )𝐴 over π‘ˆ
is called the soft set induced by 𝜏 on π‘ˆ.
Obviously,
π‘ˆ
πœπ‘“ ⊆ {apr (𝑋) : 𝑋 ∈ 2 } .
𝑃
Let π‘Œ ∈ {apr (𝑋) : 𝑋 ∈ 2π‘ˆ}. Then, π‘Œ = apr (𝑋) for some
𝑃
𝑃
𝑋 ∈ 2π‘ˆ. By Proposition 13, apr (apr (𝑋)) = apr (𝑋). This
𝑃
𝑃
𝑃
implies π‘Œ ∈ πœπ‘“ . Thus,
πœπ‘“ ⊇ {apr (𝑋) : 𝑋 ∈ 2π‘ˆ} .
𝑃
(91)
𝑃
(101)
Definition 38. Let (π‘ˆ, πœ‡) be a topological space. If there exists
a full and keeping intersection or a partition soft set 𝑓𝐴 over
π‘ˆ such that πœπ‘“ = πœ‡, then (π‘ˆ, πœ‡) is called a soft approximating
space.
The following proposition can easily be proved.
Hence,
{apr𝑃 (𝑋) : 𝑋 ∈ 2π‘ˆ} ⊆ πœπ‘“ = {apr (𝑋) : 𝑋 ∈ 2π‘ˆ} .
𝑃
(92)
(2) For each π‘Ž ∈ 𝐴, by Proposition 13,
apr (𝑓 (π‘Ž))
𝑃
= ⋃ {𝑓 (π‘ŽσΈ€  ) : π‘ŽσΈ€  ∈ 𝐴, 𝑓 (π‘ŽσΈ€  ) ⊆ 𝑓 (π‘Ž)} ⊆ 𝑓 (π‘Ž) .
(93)
Proposition 39. (1) Let 𝜏 be a topology on π‘ˆ, and let (π‘“πœ )𝐴
be the soft set induced by 𝜏 on π‘ˆ. Then, (π‘“πœ )𝐴 is a full, keeping
intersection, and keeping union soft set over π‘ˆ.
(2) Let 𝜏1 and 𝜏2 be two topologies on π‘ˆ, and let (π‘“πœ1 )𝐴 1 and
(π‘“πœ2 )𝐴 2 be two soft sets induced, respectively, by 𝜏1 and 𝜏2 on π‘ˆ.
If 𝜏1 ⊆ 𝜏2 , then
Μƒ (π‘“πœ ) .
(π‘“πœ1 )𝐴 ⊂
2 𝐴
1
2
(102)
Journal of Applied Mathematics
11
Theorem 40. Let 𝜏 be a topology on π‘ˆ, let (π‘“πœ )𝐴 be the soft set
induced by 𝜏 on π‘ˆ, and let πœπ‘“πœ be the topology induced by (π‘“πœ )𝐴
on π‘ˆ. Then, 𝜏 = πœπ‘“πœ .
Proof. Put 𝜏 = {π‘ˆπ‘Ž : π‘Ž ∈ 𝐴}; then, π‘“πœ : 𝐴 → 2π‘ˆ is a mapping,
where π‘“πœ (π‘Ž) = π‘ˆπ‘Ž for each π‘Ž ∈ 𝐴. By Proposition 39, (π‘“πœ )𝐴 is
full, keeping intersection, and keeping union.
By Theorem 36, πœπ‘“πœ = {π‘“πœ (π‘Ž) : π‘Ž ∈ 𝐴}.
Hence, πœπ‘“πœ = 𝜏.
Corollary 41. Every topological space on the initial universe is
a soft approximating space.
Theorem 42. Let (π‘ˆ, 𝜏) be a topological space. Then, there
exists a full, keeping intersection, and keeping union soft set 𝑓𝐴
over π‘ˆ such that
apr (𝑋) = int (𝑋)
𝑃
for each 𝑋 ∈ 2π‘ˆ,
(103)
where 𝑃 = (π‘ˆ, 𝑓𝐴) is a soft approximation space.
Proof. Put 𝜏 = {π‘ˆπ‘Ž : π‘Ž ∈ 𝐴}, where 𝐴 is the set of indexes.
Define a mapping 𝑓 : 𝐴 → 2π‘ˆ by
𝑓 (π‘Ž) = π‘ˆπ‘Ž
for each π‘Ž ∈ 𝐴.
(104)
By Proposition 39, 𝑓𝐴 is full, keeping intersection, and keeping union.
Let 𝑋 ∈ 2π‘ˆ. For each π‘₯ ∈ apr (𝑋), π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋 for
𝑃
some π‘Ž ∈ 𝐴. Then, π‘₯ ∈ π‘ˆπ‘Ž ⊆ 𝑋 with π‘ˆπ‘Ž ∈ 𝜏. This implies
π‘₯ ∈ int(𝑋).
Conversely, for each π‘₯ ∈ int(𝑋), there exists an open
neighborhood π‘Š of π‘₯ in π‘ˆ such that π‘Š ⊆ 𝑋. So, π‘Š = π‘ˆπ‘Ž for
some π‘Ž ∈ 𝐴. This implies π‘₯ ∈ 𝑓(π‘Ž) ⊆ 𝑋. Thus, π‘₯ ∈ apr (𝑋).
𝑃
Hence, apr (𝑋) = int(𝑋).
4.2. The Second Sort of Topologies Induced by a Soft Set. By
Proposition 19, we have Theorem 44.
Theorem 44. Let 𝑓𝐴 be a full soft set over π‘ˆ, and let 𝑃 =
(𝑋, 𝑓𝐴) be a soft approximation space. Then, πœπ‘“σΈ€  is a topology
on π‘ˆ.
Definition 45. Let 𝜏 be a topology on π‘ˆ.
(1) 𝜏 is called an Alexandrov topology on π‘ˆ, if 𝜏 is closed
for arbitrary intersections.
(2) (π‘ˆ, 𝜏) is called an Alexandrov space, if 𝜏 is an
Alexandrov topology on π‘ˆ.
(3) 𝜏 is called a pseudodiscrete topology on π‘ˆ, if 𝐴 ∈ 𝜏 if
and only if π‘ˆ − 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. Let 𝑓𝐴 be a full soft set over π‘ˆ, and let πœπ‘“σΈ€  be the
topology induced by 𝑓𝐴 on π‘ˆ. Then, (π‘ˆ, πœπ‘“σΈ€  ) is an Alexandrov
space.
Proof. By Proposition 16, 𝑅𝑓 is reflexive and symmetric.
Then, by Proposition 5 in [16], πœπ‘“σΈ€  is a pseudodiscrete topology
on π‘ˆ.
Thus,
(π‘ˆ, πœπ‘“σΈ€  ) is an Alexandrov space.
(109)
𝑃
Theorem 43. Let 𝑓𝐴 be a full and keeping intersection soft set
over π‘ˆ, let πœπ‘“ be the topology induced by 𝑓𝐴 on π‘ˆ, and let (π‘“πœπ‘“ )𝐡
be the soft set induced by πœπ‘“ on π‘ˆ. Then,
(1)
Μƒ (π‘“πœ ) .
𝑓𝐴 ⊂
𝑓
𝐡
(105)
(2) If 𝑓𝐴 is topological, then
aprσΈ€ σΈ€  (π‘Œ) = π‘Œ,
𝑃
𝑃
𝐡
(106)
(107)
Thus π‘“πœπ‘“ is a mapping given by π‘“πœπ‘“ : 𝐡 → 2π‘ˆ, where π‘“πœπ‘“ (π‘Ž) =
π‘ˆπ‘Ž for each π‘Ž ∈ 𝐡.
Μƒ (π‘“πœ )𝐡 .
Hence, 𝑓𝐴 ⊂
𝑓
(2) Since 𝑓𝐴 is topological, then by Theorem 36, 𝐴 = 𝐡.
Hence,
𝑓𝐴 = (π‘“πœπ‘“ ) .
𝐡
aprσΈ€ σΈ€  (π‘Œ ∩ 𝑍) = 0 =ΜΈ π‘Œ ∩ 𝑍.
Thus, πœπ‘“σΈ€ σΈ€  is not a topology on π‘ˆ.
where 𝐴 ⊆ 𝐡,
for each π‘Ž ∈ 𝐴.
(110)
𝑃
Proof. (1) By Theorem 36, πœπ‘“ ⊇ {𝑓(π‘Ž) : π‘Ž ∈ 𝐴}. Denote
π‘ˆπ‘Ž = 𝑓 (π‘Ž)
Example 47. Let 𝑓𝐴 be a full soft set over π‘ˆ and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space in Example 26.
Let π‘Œ = {β„Ž2 , β„Ž4 }, and 𝑍 = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž5 }. We have
aprσΈ€ σΈ€  (𝑍) = 𝑍,
𝑓𝐴 = (π‘“πœπ‘“ ) .
πœπ‘“ = {π‘ˆπ‘Ž : π‘Ž ∈ 𝐡} ,
4.3. The Third Sort of Topologies Induced by a Soft Set
(108)
By Proposition 21, we have Theorem 48.
Theorem 48. Let 𝑓𝐴 be a full soft set over π‘ˆ, and let 𝑃 =
(π‘ˆ, 𝑓𝐴) be a soft approximation space. Then, πœπ‘“σΈ€ σΈ€ σΈ€  is a topology
on π‘ˆ.
4.4. The Relationships among Three Sorts of Topologies Induced
by a Soft Set and Related Results. By Theorem 27, we have
Theorem 49, which illustrates relationships among three sorts
of topologies induced by a soft set.
12
Journal of Applied Mathematics
Theorem 49. (1) If 𝑓𝐴 is a full soft set over π‘ˆ, then
πœπ‘“σΈ€ σΈ€ σΈ€  ⊆ πœπ‘“σΈ€  .
(111)
Example 52. Let 𝑓𝐴 be a partition soft set over π‘ˆ in
Example 11, let πœπ‘“ be the topology induced by 𝑓𝐴 on π‘ˆ, and
let (π‘“πœπ‘“ )𝐡 be the soft set induced by πœπ‘“ on π‘ˆ. We have
πœπ‘“ = {π‘ˆπ‘Ž : π‘Ž ∈ 𝐡} ,
(2) If 𝑓𝐴 is a full and keeping intersection soft set over π‘ˆ,
then
πœπ‘“σΈ€ σΈ€ σΈ€  ⊆ πœπ‘“σΈ€  ⊆ πœπ‘“ .
where 𝐡 = {π‘Ž1 , π‘Ž2 , . . . , π‘Ž16 } ,
π‘ˆπ‘Ž1 = 𝑓 (π‘Ž1 ) = {β„Ž1 , β„Ž2 } ,
(112)
π‘ˆπ‘Ž2 = 𝑓 (π‘Ž2 ) = {β„Ž5 } ,
(3) If 𝑓𝐴 is a partition soft set over π‘ˆ, then
πœπ‘“ =
πœπ‘“σΈ€ 
=
π‘ˆπ‘Ž3 = 𝑓 (π‘Ž3 ) = {β„Ž3 } ,
πœπ‘“σΈ€ σΈ€ σΈ€  .
(113)
π‘ˆπ‘Ž4 = 𝑓 (π‘Ž4 ) = {β„Ž4 } ,
π‘ˆπ‘Ž5 = {β„Ž3 , β„Ž4 } ,
By Theorem 36, Proposition 39 and Theorem 49, we have
Theorem 50.
π‘ˆπ‘Ž6 = {β„Ž3 , β„Ž5 } ,
Theorem 50. Let 𝜏 be a topology on π‘ˆ, let (π‘“πœ )𝐴 be the soft
set induced by 𝜏 on π‘ˆ, and let πœπ‘“πœ , πœπ‘“σΈ€  𝜏 and πœπ‘“σΈ€ σΈ€ σΈ€ πœ be the topology
induced, respectively, by (π‘“πœ )𝐴 on π‘ˆ. Then
𝜏 = πœπ‘“πœ ⊇ πœπ‘“σΈ€  𝜏 ⊇ πœπ‘“σΈ€ σΈ€ σΈ€ πœ .
π‘ˆπ‘Ž7 = {β„Ž4 , β„Ž5 } ,
π‘ˆπ‘Ž9 = {β„Ž1 , β„Ž2 , β„Ž4 } ,
(114)
π‘ˆπ‘Ž10 = {β„Ž1 , β„Ž2 , β„Ž5 } ,
By Proposition 39 and Theorems 43 and 49, we have
Theorem 51.
π‘ˆπ‘Ž11 = {β„Ž3 , β„Ž4 , β„Ž5 } ,
Theorem 51. (1) If 𝑓𝐴 is a full soft set over π‘ˆ, let πœπ‘“σΈ€  and πœπ‘“σΈ€ σΈ€ σΈ€ 
be the topologies induced, respectively, by 𝑓𝐴 on π‘ˆ, and let
(π‘“πœπ‘“σΈ€  )𝐢 (resp., (π‘“πœπ‘“σΈ€ σΈ€ σΈ€  )D ) be the soft set induced by πœπ‘“σΈ€  (resp., πœπ‘“σΈ€ σΈ€ σΈ€  )
on π‘ˆ. Then,
Μƒ (π‘“πœσΈ€ σΈ€ σΈ€  ) .
(π‘“πœπ‘“σΈ€  ) ⊃
𝑓
𝐢
(2) Let 𝑓𝐴 be a full and keeping intersection soft set over
π‘ˆ, let πœπ‘“ , πœπ‘“σΈ€  , and πœπ‘“σΈ€ σΈ€ σΈ€  be the topologies induced respectively by
𝑓𝐴 on π‘ˆ and let (π‘“πœπ‘“ )𝐡 (resp., (π‘“πœπ‘“σΈ€  )𝐢, (π‘“πœπ‘“σΈ€ σΈ€ σΈ€  )D ) be the soft set
induced by πœπ‘“ (resp., πœπ‘“σΈ€  , πœπ‘“σΈ€ σΈ€ σΈ€  ) on π‘ˆ. Then,
(a)
Μƒ 𝑓𝐴,
(π‘“πœπ‘“ ) ⊃
𝐡
𝐢
𝐷
(116)
𝐢
(117)
𝐷
(3) Let 𝑓𝐴 be a partition soft set over π‘ˆ, let πœπ‘“πœ , πœπ‘“σΈ€  𝜏 and
πœπ‘“σΈ€ σΈ€ σΈ€ πœ be the topologies induced, respectively, by 𝑓𝐴 on π‘ˆ and
let (π‘“πœπ‘“ )𝐡 (resp., (π‘“πœπ‘“σΈ€  )𝐢, (π‘“πœπ‘“σΈ€ σΈ€ σΈ€  )𝐷) be the soft set induced by
πœπ‘“ (resp., πœπ‘“σΈ€  , πœπ‘“σΈ€ σΈ€ σΈ€  ) on π‘ˆ. Then,
𝐡
π‘ˆπ‘Ž16 = π‘ˆ.
Obviously,
Μƒ (π‘“πœ ) ,
𝑓𝐴 ⊂
𝑓
Μƒ ΜΈ (π‘“πœ ) .
𝑓𝐴 ⊇
𝑓
𝐡
𝑓𝐴 =ΜΈ (π‘“πœπ‘“ ) .
𝐡
(120)
(121)
In this section, four sorts of soft rough sets based on four pairs
of soft rough approximations are investigated.
For A, B ⊆ 2π‘ˆ, we denote
A \ B = {𝑋 ∈ 2π‘ˆ : 𝑋 ∈ A, 𝑋 ∉ B} .
𝐢
𝐷
(118)
(122)
Lemma 53 (see [10]). Let 𝑓𝐴 be a soft set over π‘ˆ and let 𝑃 =
(π‘ˆ, 𝑓𝐴) be a soft approximation space. Then for each 𝑋 ∈ 2π‘ˆ,
𝑋 is a soft 𝑃-rough set ⇐⇒ apr𝑃 (𝑋) ⊆ 𝑋.
(π‘“πœπ‘“ ) = (π‘“πœπ‘“σΈ€  ) = (π‘“πœπ‘“σΈ€ σΈ€ σΈ€  ) .
𝐡
π‘ˆπ‘Ž15 = 0,
5. The Related Properties of Soft Rough Sets
Μƒ (π‘“πœσΈ€  ) ⊃
Μƒ (π‘“πœσΈ€ σΈ€ σΈ€  ) .
𝑓𝐴 = (π‘“πœπ‘“ ) ⊃
𝑓
𝑓
Μƒ 𝑓𝐴,
(π‘“πœπ‘“ ) ⊃
π‘ˆπ‘Ž14 = {β„Ž1 , β„Ž2 , β„Ž4 , β„Ž5 } ,
Thus,
(b) If 𝑓𝐴 is keeping union, then
𝐡
π‘ˆπ‘Ž13 = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž5 } ,
𝐡
Μƒ (π‘“πœσΈ€  ) ⊃
Μƒ (π‘“πœσΈ€ σΈ€ σΈ€  ) .
(π‘“πœπ‘“ ) ⊃
𝑓
𝑓
𝐡
π‘ˆπ‘Ž12 = {β„Ž1 , β„Ž2 , β„Ž3 , β„Ž4 } ,
(115)
𝐷
(119)
π‘ˆπ‘Ž8 = {β„Ž1 , β„Ž2 , β„Ž3 } ,
By Corollary 31, we have the following Lemma 54.
(123)
Journal of Applied Mathematics
13
Lemma 54. Let 𝑓𝐴 be a full soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then,
(1) every soft 𝑃󸀠 -rough set is a soft 𝑃󸀠󸀠󸀠 -rough set.
Theorem 57. Let 𝑓𝐴 be a soft set over U and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then, for 𝑋 ∈ 2π‘ˆ, one has
(2) every soft 𝑃󸀠󸀠 -rough set is a soft 𝑃󸀠 -rough set.
Pos󸀠𝑃 (π‘ˆ − 𝑋) = Neg󸀠𝑃 (𝑋) ,
By Theorem 27, we have Lemma 55.
Lemma 55. Let 𝑓𝐴 be a full and keeping union soft set over π‘ˆ,
and let 𝑃 = (π‘ˆ, 𝑓𝐴) be a soft approximation space. If 𝑋 ∈ 2π‘ˆ is
a soft 𝑃-definable set, then,
𝑋 ∈ πœŽπ‘“σΈ€ σΈ€ σΈ€  ⊆ πœŽπ‘“σΈ€  ⊆ πœŽπ‘“σΈ€ σΈ€  .
(124)
The following theorem gives the structure of soft rough
sets.
Theorem 56. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Denote
(125)
(a)
πœŽπ‘“ = 2π‘ˆ \ Ω;
(126)
ΩσΈ€ σΈ€  ⊆ ΩσΈ€  ⊆ ΩσΈ€ σΈ€ σΈ€  ;
(127)
(b)
(c)
2π‘ˆ \ πœπ‘“σΈ€ σΈ€ σΈ€  ⊆ ΩσΈ€ σΈ€ σΈ€  ;
(128)
(d)
πœπ‘“σΈ€  ∩ πœŽπ‘“σΈ€  = 2π‘ˆ \ ΩσΈ€  ,
∩
πœŽπ‘“σΈ€ σΈ€ 
π‘ˆ
σΈ€ σΈ€ 
=2 \Ω ,
(129)
πœπ‘“σΈ€ σΈ€ σΈ€  ∩ πœŽπ‘“σΈ€ σΈ€ σΈ€  = 2π‘ˆ \ ΩσΈ€ σΈ€ σΈ€  .
(2) If 𝑓𝐴 is full and keeping union, then
(a) 2π‘ˆ \ πœŽπ‘“σΈ€ σΈ€  ⊆ 2π‘ˆ \ πœŽπ‘“σΈ€  ⊆ 2π‘ˆ \ πœŽπ‘“σΈ€ σΈ€ σΈ€  ⊆ 2π‘ˆ \ πœŽπ‘“ ⊆ Ω;
(b) πœŽπ‘“ = {0, π‘ˆ} = 2π‘ˆ \ Ω.
(3) If 𝑓𝐴 is full and keeping intersection, then
π‘ˆ
(a) πœŽπ‘“ = 2 \ Ω ⊆ πœπ‘“ .
Proof. This is obtained from Propositions 19, 20, and 21.
By Proposition 14 and Theorem 27, we have Theorem 58.
Theorem 58. Let 𝑓𝐴 be a soft set over π‘ˆ, and let 𝑃 = (π‘ˆ, 𝑓𝐴)
be a soft approximation space. Then, for 𝑋 ∈ 2π‘ˆ, one has the
following.
(131)
(2) If 𝑓𝐴 is full and keeping union, then
σΈ€ σΈ€ 
(1) If 𝑓𝐴 is full, then
πœπ‘“σΈ€ σΈ€ 
σΈ€ σΈ€ σΈ€ 
PosσΈ€ σΈ€ σΈ€ 
𝑃 (π‘ˆ − 𝑋) = Neg𝑃 (𝑋) .
σΈ€ 
σΈ€ σΈ€ σΈ€ 
(a) apr𝑃 (𝑋) − 𝑋 ⊆ apr𝑃 (𝑋) − 𝑋 ⊆ apr𝑃 (𝑋) − 𝑋 ⊆
Bnd𝑃 (𝑋);
(b) Neg𝑃 (𝑋) = 0 and Bnd𝑃 (𝑋) = π‘ˆ−Pos𝑃 (𝑋) where
𝑋 =ΜΈ 0.
ΩσΈ€ σΈ€ σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 is a soft 𝑃󸀠󸀠󸀠 -rough set} .
2π‘ˆ \ πœπ‘“σΈ€ σΈ€  ⊆ ΩσΈ€ σΈ€  ,
(130)
Bnd󸀠󸀠𝑃 (𝑋) ⊆ Bnd󸀠𝑃 (𝑋) ⊆ BndσΈ€ σΈ€ σΈ€ 
𝑃 (𝑋) .
ΩσΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 is a soft 𝑃󸀠 -rough set} ,
ΩσΈ€ σΈ€  = {𝑋 ∈ 2π‘ˆ : 𝑋 is a soft 𝑃󸀠󸀠 -rough set} ,
Pos󸀠󸀠𝑃 (π‘ˆ − 𝑋) = Neg󸀠󸀠𝑃 (𝑋) ,
(1) If 𝑓𝐴 be a full, then
Ω = {𝑋 ∈ 2π‘ˆ : 𝑋 is a soft 𝑃-rough set} ,
2π‘ˆ \ πœπ‘“σΈ€  ⊆ ΩσΈ€  ,
Proof. These hold by Proposition 14 and Lemmas 53, 54, and
55.
Remark 59. Theorem 58 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]). Let π‘ˆ be a finite set of objects, let 𝐴
be a finite set of attributes, and let 𝐼 be a binary relation on
from π‘ˆ to 𝐴. The triple (π‘ˆ, 𝐴, 𝐼) is called a formal context.
Let (π‘ˆ, 𝐴, 𝐼) be a formal context. For 𝑒 ∈ π‘ˆ and π‘Ž ∈ 𝐴,
π‘’π‘…π‘Ž, which is also written as (𝑒, π‘Ž) ∈ 𝑅, means that the object
𝑒 possesses the attribute π‘Ž.
Denote
{π‘Ž}∗ = {𝑒 ∈ π‘ˆ : π‘’πΌπ‘Ž} ,
{𝑒}∗ = {π‘Ž ∈ 𝐴 : π‘’πΌπ‘Ž} .
(132)
Definition 61. Let FC = (π‘ˆ, 𝐴, 𝐼) be a formal context. Define
a mapping 𝑓FC : 𝐴 → 2π‘ˆ by
𝑓FC (π‘Ž) = {π‘Ž}∗
(133)
for each π‘Ž ∈ 𝐴. Then, (𝑓FC )𝐴 = (𝑓FC , 𝐴) is called the soft set
over π‘ˆ induced by FC. We denote (𝑓FC )𝐴 by 𝑆FC .
14
Journal of Applied Mathematics
Definition 62. Let 𝑆 = 𝑓𝐴 be a soft set over π‘ˆ. Define a binary
relation 𝐼𝑆 ∈ 2π‘ˆ × π΄ by
(𝑒, π‘Ž) ∈ 𝐼𝑆 ⇐⇒ 𝑒 ∈ 𝑓 (π‘Ž)
(134)
for each 𝑒 ∈ π‘ˆ and each π‘Ž ∈ 𝐴. Then, (π‘ˆ, 𝐴, 𝐼𝑆 ) is called the
formal context induced by 𝑆. We denote (π‘ˆ, 𝐴, 𝐼𝑆 ) by FC𝑆 .
Lemma 63. Let 𝐹𝐢 = (π‘ˆ, 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,
FC = FC𝑆FC .
(135)
Proof. Obviously, FC𝑆FC = (π‘ˆ, 𝐴, 𝐼𝑆FC ).
For each (𝑒, π‘Ž) ∈ π‘ˆ × π΄,
(𝑒, π‘Ž) ∈ 𝐼𝑆FC ⇐⇒ 𝑒 ∈ 𝑓FC (π‘Ž) .
(136)
𝑓FC (π‘Ž) = {π‘Ž}∗ = {𝑒 ∈ π‘ˆ : π‘’πΌπ‘Ž} = {𝑒 ∈ π‘ˆ : (𝑒, π‘Ž) ∈ 𝐼} .
(137)
Then,
(138)
Thus, for each (𝑒, π‘Ž) ∈ π‘ˆ × π΄, (𝑒, π‘Ž) ∈ 𝐼𝑆FC ⇔ (𝑒, π‘Ž) ∈ 𝐼.
This implies 𝐼𝑆FC = 𝐼.
Hence,
FC = FC𝑆FC .
(139)
Lemma 64. Let 𝑆 = 𝑓𝐴 be a soft set over π‘ˆ, let 𝐹𝐢𝑆 be the
formal context induced by 𝑆, and let
𝑆FC𝑆
(140)
be the soft set induced by FC𝑆 . Then,
𝑆 = 𝑆FC𝑆 .
(141)
Proof. Obviously, 𝑆FC𝑆 = (𝑓FC𝑆 , 𝐴).
Since FC𝑆 = (π‘ˆ, 𝐴, 𝐼𝑆 ), then
𝑓FC𝑆 (π‘Ž) = {𝑒 ∈ π‘ˆ : (𝑒, π‘Ž) ∈ 𝐼𝑆 }
for each π‘Ž ∈ 𝐴.
(142)
So, (𝑒, π‘Ž) ∈ 𝐼𝑆 ⇔ 𝑒 ∈ 𝑓(π‘Ž).
Obviously, 𝑓(π‘Ž) = {𝑒 ∈ π‘ˆ : 𝑒 ∈ 𝑓(π‘Ž)} for each π‘Ž ∈ 𝐴.
Thus, for each π‘Ž ∈ 𝐴, 𝑓FC (π‘Ž) = 𝑓(π‘Ž).
Hence,
𝑆 = 𝑆FC𝑆 .
(143)
Theorem 65. Let
Γ = {𝑆 = 𝑓𝐴 : 𝑆 is a soft set over π‘ˆ} ,
Σ = {FC = (π‘ˆ, 𝐴, 𝐼) : FC is a formal context} .
π‘˜ (FC) = 𝑆FC ,
𝑙 (𝑆) = FC𝑆 ,
for any FC = (π‘ˆ, 𝐴, 𝐼) ∈ Σ,
for any 𝑆 = 𝑓𝐴 ∈ Γ.
(145)
By Lemma 63,
𝑙 ∘ π‘˜ = π‘–Σ ,
(146)
where 𝑙 ∘ π‘˜ is the composition of π‘˜ and 𝐺, and π‘–Σ is the identity
mapping on Σ.
By Lemma 64,
π‘˜ ∘ 𝑙 = 𝑖à ,
For each π‘Ž ∈ 𝐴,
𝑒 ∈ 𝑓FC (π‘Ž) ⇐⇒ (𝑒, π‘Ž) ∈ 𝐼.
Proof. Two mappings π‘˜ : Σ → Γ and 𝑙 : Γ → Σ are defined
as follows:
(147)
where 𝑙 ∘ 𝐹 is the composition of 𝑙 and π‘˜, and 𝑖à 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. Theorem 65 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.
Acknowledgment
This work is supported by the National Natural Science
Foundation of China (no. 11061004, 11226085), the Science
Research Project of Guangxi University for Nationalities (no.
2011QD015) and the Science Foundation of Guangxi College
of Finance and Economics (no. 2012ZD001).
References
(144)
Then, there exists a one-to-one correspondence between Γ
and Σ.
[1] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about
Data, Kluwer Academic, Boston, Mass, USA, 1991.
[2] D. Molodtsov, “Soft set theory—first results,” Computers &
Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.
Journal of Applied Mathematics
[3] P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers
& Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562,
2003.
[4] P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets
in a decision making problem,” Computers & Mathematics with
Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002.
[5] P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” Journal of
Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001.
[6] A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach
to decision making problems,” Journal of Computational and
Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007.
[7] Y. Jiang, Y. Tang, Q. Chen, J. Wang, and S. Tang, “Extending soft
sets with description logics,” Computers & Mathematics with
Applications, vol. 59, no. 6, pp. 2087–2096, 2010.
[8] H. AktasΜ§ and N. CΜ§agΜ†man, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007.
[9] F. Feng, C. Li, B. Davvaz, and M. I. Ali, “Soft sets combined with
fuzzy sets and rough sets: a tentative approach,” Soft Computing,
vol. 14, no. 9, pp. 899–911, 2010.
[10] F. Feng, X. Liu, V. Leoreanu-Fotea, and Y. B. Jun, “Soft sets and
soft rough sets,” Information Sciences, vol. 181, no. 6, pp. 1125–
1137, 2011.
[11] M. Shabir and M. Naz, “On soft topological spaces,” Computers
& Mathematics with Applications, vol. 61, no. 7, pp. 1786–1799,
2011.
[12] X. Ge, Z. Li, and Y. Ge, “Topological spaces and soft sets,”
Journal of Computational Analysis and Applications, vol. 13, no.
5, pp. 881–885, 2011.
[13] Z. Pawlak and A. Skowron, “Rough sets: some extensions,”
Information Sciences, vol. 177, no. 1, pp. 28–40, 2007.
[14] Y. Y. Yao and T. Y. Lin, “Generalization of rough sets using
modal logic,” Intelligent Automation and Soft Computing, vol. 2,
no. 2, pp. 103–120, 1996.
[15] W. Zhang, W. Wu, J. Liang, and D. Li, Rough Sets Theorey and
Methods, Chinese Scientific, Beijing, China, 2001.
[16] M. Kondo, “On the structure of generalized rough sets,” Information Sciences, vol. 176, no. 5, pp. 589–600, 2006.
[17] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical
Foundations, Springer, New York, NY, USA, 1999.
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