# Soft Classes and Soft Rough Classes with Applications in Decision Making

Authors: Faruk Karaaslan
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

Rough set was defined by Pawlak in 1982. Concept of soft set was proposed as a mathematical tool to cope with uncertainty and vagueness by Molodtsov in 1999. Soft sets were combined with rough sets by Feng et al. in 2011. Feng et al. investigated relationships between a subset of initial universe of soft set and a soft set. Feng et al. defined the upper and lower approximations of a subset of initial universe over a soft set. In this study, we firstly define concept of soft class and soft class operations such as union, intersection, and complement. Then we give some properties of soft class operations. Based on definition and operations of soft classes, we define lower and upper approximations of a soft set. Subsequently, we introduce concept of soft rough class and investigate some properties of soft rough classes. Moreover, we give a novel decision making method based on soft class and present an example related to novel method.


---

## Body

## 1. Introduction

The concept of soft set was introduced by Molodtsov [1] in 1999 as a general mathematical tool for dealing with problems involving uncertain data. Maji et al. [2] defined some concepts and operations on soft sets such as soft subset, soft equality, soft union, soft intersection, and soft complement. Çağman and Enginoğlu [3] redefined soft set operations suggested by Maji et al. [2] and developed a decision making method called uni-int decision making method. Çağman [4] made some contributions to the theory of soft sets to fill gaps of former definition and operations.Rough set theory was proposed by Pawlak [5] as an alternative approach to fuzzy sets theory and tolerance theory and has been applied successfully to a lot of fields such as machine learning, pattern recognition, and data mining. Dubois and Prade [6] defined lower and upper approximations of a fuzzy set to extend concept of rough set and they proposed the rough fuzzy sets. Soft sets were combined with fuzzy sets and rough sets by Feng et al. [7]. In 2011, Feng et al. [8] introduced soft rough approximation space and soft rough set based on the novel granulation structures called soft approximation spaces and presented basic properties of soft rough approximations supported by some illustrative example. They also defined some new types of soft sets such as full soft sets, intersection complement softs set, and partition soft sets. Meng et al. [9] proposed a new soft rough set model and derived its properties. They also established a more general model called soft rough fuzzy set. Irfan Ali [10] discussed concept of approximation space associated with each parameter in a soft set and defined an approximation space associated with the soft sets and established connection between soft set, fuzzy soft set, and rough sets. Feng [11] gave an application of soft rough approximations in multicriteria group decision making problems. Zhang [12] defined a new rough set model and investigated its some fundamental properties. He also presented a decision making method for intuitionistic fuzzy soft sets based on this new rough set approach. Zhang [13] studied parameter reduction of fuzzy soft sets based on soft fuzzy rough set and defined some new concepts such as lower soft fuzzy rough approximation operator and upper soft fuzzy rough approximation operator. To find approximation of a set, Shabir et al. [14] proposed modified soft rough sets. Sun and Ma [15] proposed a new concept of soft fuzzy rough set by combining the fuzzy soft set with the traditional fuzzy rough set. They also defined concept of the pseudofuzzy binary relation and based on this concept they defined the soft fuzzy rough lower and upper approximation operators of any fuzzy subset in the parameter set. In this paper, we define concept of soft class and soft class operations based on decision makers set and investigate some fundamental properties of soft class operations. Then, we define soft rough class approximations and soft rough class and investigate some properties of them. Furthermore, we present a method to evaluate the decision makers and give an example to illustrate the process of this method. Proposed method can be used in many areas such as industrial engineering, economy, and social sciences. In particular, in industrial engineering, it can be used effectively for Quality Lifecycle Management and Choosing Product.


## 2. Preliminary

LetU be an initial universe, let E be the universe of all possible parameters related to the objects in U, and let P
(
U
) be power set of U.Definition 1 (see [1]).
Consider a nonempty setA such that A
⊆
E. A pair (
f
,
A
) is called a soft set over U, where f is a mapping given by f
:
A
→
P
(
U
).In this paper, we will use the following definition given by Çağman [4] for basic set operations on soft sets.Definition 2 (see [4]).
A soft setf over U is a set valued function from E to P
(
U
). It can be written as a set of ordered pairs:(1)
f
=
e
,
f
e
:
e
∈
E
.Note that if f
(
e
)
=
∅, then the element (
e
,
f
(
e
)
) will not appear in soft set f. Set of all soft sets over U will be denoted by S
(
U
).Example 3.
LetU
=
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
u
8
} be the universe containing eight houses and let E
=
{
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
} be the set of parameters. Here, e
i
(
i
=
1,2
,
3,4
,
5,6
) stand for the parameters “modern,” “with parking,” “expensive,” “cheap,” “large,” and “near to city,” respectively. Then, the following soft sets are described by Mr. A and Mr. B who want to buy a house, respectively:(2)
f
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
1
,
u
4
,
u
7
,
u
8
,
e
3
,
u
1
,
u
2
,
u
3
,
u
8
,
g
=
e
2
u
1
,
u
3
,
u
6
,
e
3
,
U
,
e
5
,
u
2
,
u
4
,
u
5
,
u
6
.Definition 4 (see [4]).
Letf
,
g
∈
S
(
U
). Then,(1)
iff
(
e
)
=
∅, for all e
∈
E, f is said to be a null soft set, denoted by Φ;
(2)
iff
(
e
)
=
U, for all e
∈
E, f is said to be absolute soft set, denoted by U
^;
(3)
f is soft subset of g, denoted by f
⊆
~
g, if f
(
e
)
⊆
g
(
e
) for all e
∈
E;
(4)
f
=
g, if f
⊆
~
g and g
⊆
~
f;
(5)
soft union off and g, denoted by f
∪
~
g, is a soft set over U and is defined by f
∪
~
g
:
E
→
P
(
U
) such that (
f
∪
~
g
)
(
e
)
=
f
(
e
)
∪
g
(
e
) for all e
∈
E;
(6)
soft intersection off and g, denoted by f
∩
~
g, is a soft set over U and is defined by f
∩
~
g
:
E
→
P
(
U
) such that (
f
∩
~
g
)
(
e
)
=
f
(
e
)
∩
g
(
e
) for all e
∈
E;
(7)
soft complement off is denoted by f
c
~ and is defined by f
c
~
:
E
→
P
(
U
) such that f
c
~
(
e
)
=
U
∖
f
(
e
) for all e
∈
E.Example 5.
Let us consider soft setsf and g given in Example 3. Then, (3)
f
∪
~
g
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
1
,
u
3
,
u
4
,
u
6
,
u
7
,
u
8
,
e
3
,
U
,
e
5
,
u
2
,
u
4
,
u
5
,
u
6
,
f
∩
~
g
=
e
2
u
1
,
e
3
,
u
1
,
u
2
,
u
3
,
u
8
,
f
c
~
=
e
1
,
u
2
,
u
5
,
u
6
,
u
7
,
u
8
,
e
2
,
u
2
,
u
3
,
u
5
,
u
6
,
e
3
,
u
4
,
u
5
,
u
6
,
u
7
,
e
4
,
U
,
e
5
,
U
,
e
6
,
U
.Definition 6 (see [8]).
LetS
=
(
f
,
A
) be a soft set over U. Then, the pair P
=
(
U
,
S
) is called soft approximation space. Based on the soft approximation space P, we define the two operations,(4)
apr
_
P
X
=
u
∈
U
:
∃
a
∈
A
,
u
∈
f
a
⊆
X
,
apr
¯
P
X
=
u
∈
U
:
∃
a
∈
A
,
u
∈
f
a
,
f
a
∩
X
≠
∅
,assigning to every subset X
⊆
U two sets a
p
r
_
P
(
X
) and a
p
r
¯
P
(
X
), which are called the soft P-lower approximation and the soft P-upper approximation of X, respectively. In general, we refer to a
p
r
_
P
(
X
) and a
p
r
¯
P
(
X
) as soft rough approximations of X with respect to P. Moreover, the sets(5)
POS
P
X
=
apr
_
P
X
,
NEG
P
X
=
-
apr
¯
P
X
,
BND
P
X
=
apr
¯
P
X
-
apr
_
P
Xare called the soft P-positive region, the soft P-negative region, and the soft P-boundary region of X, respectively. If a
p
r
_
P
(
X
)
=
a
p
r
¯
P
(
X
), X is said to be soft P-definable; otherwise X is called a soft P-rough set.Example 7 (see [8]).
LetU
=
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
}, let E
=
{
e
1
,
e
2
,
e
3
,
e
4
,
e
5
,
e
6
}, and let A
=
{
e
1
,
e
2
,
e
3
,
e
4
}
⊆
E. Let S
=
(
f
,
A
) be a soft set over U given by Table 1 and the approximation space P
=
(
U
,
S
).
ForX
=
{
u
3
,
u
4
,
u
5
}
⊆
U, we have a
p
r
_
P
(
X
)
=
{
u
3
} and a
p
r
¯
P
(
X
)
=
{
u
1
,
u
2
,
u
3
,
u
4
}. Thus a
p
r
_
P
(
X
)
≠
a
p
r
¯
P
(
X
)
=
{
u
3
} and X is a soft P-rough set. Note that X
=
{
u
3
,
u
4
,
u
5
}
⊈
a
p
r
¯
P
(
X
)
=
{
u
1
,
u
2
,
u
3
,
u
5
} in this case. Moreover, it is easy to see that P
O
S
P
(
X
)
=
{
u
3
}, N
E
G
P
(
X
1
)
=
{
u
3
,
u
4
}, and B
N
D
P
(
X
)
=
{
u
1
,
u
2
,
u
3
}. On the other hand, one can consider X
1
=
{
u
3
,
u
4
}
⊆
U. Since a
p
r
_
P
(
X
1
)
=
{
u
3
}
=
a
p
r
¯
P
(
X
1
), by definition, X
1 is a soft P-definable set.Table 1
The tabular representation of the soft setS.
u
1
u
2
u
3
u
4
u
5
u
6
e
1
1
0
0
0
0
1
e
2
0
0
1
0
0
0
e
3
0
0
0
0
0
0
e
4
1
1
0
0
1
0


## 3. Soft Classes

In this section, we define concept of soft class and soft class operations. Also we obtain some basic properties of soft class operations.Definition 8.
LetE be a parameter set, let U be an initial universe, and let D
=
{
d
i
:
i
=
1,2
,
…
,
n
} be a set of decision makers. Indexed class of soft sets {
f
d
i
:
f
d
i
:
E
→
P
(
U
)
,
d
i
∈
D
} is called a soft class and is denoted by f
D.
If, for anyd
i
∈
D, f
d
i
=
Φ, the soft set f
d
i does not appear in soft class f
D.Throughout this studyE, U, and D denote parameter set, initial universe, and decision makers set, respectively.From now on, all soft classes over parameter setE, initial universe U, and decision makers set D will be denoted by S
C
D
E
(
U
).Example 9.
LetE
=
{
e
1
,
e
2
,
e
3
,
e
4
} be a parameter set, let U
=
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
u
8
} be an initial universe, and let D
=
{
d
1
,
d
2
,
d
3
} be a set of decision makers. If we consider soft sets f
d
1
,
f
d
2
,
f
d
3 given as (6)
f
d
1
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
1
,
u
4
,
u
5
,
u
6
,
e
3
,
u
1
,
u
2
,
u
3
,
u
8
,
e
4
,
,
f
d
2
=
e
1
,
u
1
,
u
2
,
e
2
,
u
3
,
u
6
,
e
3
,
U
,
e
4
,
u
1
,
u
3
,
u
7
,
u
8
,
f
d
3
=
e
1
,
u
2
,
u
3
,
u
5
,
e
2
,
u
1
,
u
4
,
u
7
,
e
3
,
,
e
4
,
u
5
,
u
8
,then f
D
=
{
f
d
1
,
f
d
2
,
f
d
3
} is a soft class. We can represent a soft class in tabular form as shown in Table 2.Table 2
The tabular representation of the soft classf
D.
f
D
f
d
1
f
d
2
f
d
3
e
1
{
u
1
,
u
3
,
u
4
}
{
u
1
,
u
2
}
{
u
2
,
u
3
,
u
5
}
e
2
{
u
1
,
u
4
,
u
5
,
u
6
}
{
u
3
,
u
6
}
{
u
1
,
u
4
,
u
7
}
e
3
{
u
1
,
u
2
,
u
3
,
u
8
}
U
{
}
e
4
{
}
{
u
1
,
u
3
,
u
7
,
u
8
}
{
u
5
,
u
8
}Definition 10.
Letf
D
∈
S
C
D
E
(
U
). If, for all d
i
∈
D, f
d
i
=
Φ, then f
D is called an empty soft class and is denoted by Ø.Definition 11.
Letf
D
∈
S
C
D
E
(
U
). If, for all d
i
∈
D, f
d
i
=
U
^, then f
D is called a universal soft class and is denoted by U.Definition 12.
Letf
D
,
g
D
∈
S
C
D
E
(
U
). Then, f
D is a soft subclass of g
D, denoted by f
D
⊑
g
D, if, for all d
i
∈
D, f
d
i
⊆
~
g
d
i.Example 13.
LetE
=
{
e
1
,
e
2
,
e
3
,
e
4
} be a parameter set, let U
=
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
u
8
} be an initial universe, and let D
=
{
d
1
,
d
2
,
d
3
} be a decision makers set. If (7)
f
d
1
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
1
,
u
4
,
u
5
,
u
6
,
e
3
,
u
1
,
u
2
,
u
3
,
u
8
,
e
4
,
,
f
d
2
=
e
1
,
u
1
,
u
2
,
e
2
,
u
3
,
u
6
,
e
3
,
U
,
e
4
,
u
1
,
u
3
,
u
7
,
u
8
,
f
d
3
=
e
1
,
u
2
,
u
3
,
u
5
,
e
2
,
u
1
,
u
4
,
u
7
,
e
3
,
,
e
4
,
u
5
,
u
8
,
g
d
1
=
e
1
,
u
1
,
u
3
,
e
2
,
u
1
,
u
4
,
u
6
,
e
3
,
u
1
,
u
2
,
u
8
,
e
4
,
,
g
d
2
=
e
1
,
u
1
,
e
2
,
u
3
,
u
6
,
e
3
,
u
1
,
u
2
,
u
3
,
u
6
,
e
4
,
u
1
,
u
3
,
u
7
,
g
d
3
=
e
1
,
u
2
,
u
5
,
e
2
,
u
1
,
u
4
,
e
3
,
,
e
4
,
u
5
,
u
8
,then soft classes can be written as f
D
=
{
f
d
1
,
f
d
2
,
f
d
3
} and g
D
=
{
g
d
1
,
g
d
2
,
g
d
3
}.
Note that, for alld
i
∈
D, since g
d
i
⊆
~
f
d
i, g
D
⊑
f
D.Proposition 14.
Iff
D
,
g
D, and h
D
∈
S
C
D
E
(
U
), then(1)
f
D
⊑
U;
(2)
Ø
⊑
f
D;
(3)
f
D
⊑
f
D;
(4)
f
D
⊑
g
D and g
D
⊑
h
D
⇒
f
D
⊑
h
D.Proof.
Iff
D, g
D, and h
D
∈
S
C
D
E
(
U
), then, for all d
i
∈
D,(1)
f
d
i
⊆
~
U
^
⇒
f
D
⊑
U;
(2)
Φ
⊆
~
f
d
i
⇒
Ø
⊑
f
D;
(3)
f
d
i
⊆
~
f
d
i
⇒
f
D
⊑
f
D;
(4)
f
d
i
⊆
~
g
d
i and g
d
i
⊆
~
h
d
i
⇒
f
d
i
⊆
~
h
d
i; then, f
D
⊑
g
D and g
D
⊑
h
D
⇒
f
D
⊑
h
D.Definition 15.
Letf
D
,
g
D
∈
S
C
D
E
(
U
). Then, f
D and g
D are equal soft classes if and only if f
D
⊑
g
D and f
D
⊒
g
D. This relation is denoted by f
D
=
g
D.Definition 16.
Letf and g
∈
S
(
U
) and let f
⊆
~
g. Then, according to the soft set g, degree of subsethood of soft set f, denoted by f
g
∘, is defined as follows: (8)
f
g
∘
=
1
E
g
∑
e
∈
E
f
e
g
e
,
g
e
≠
∅
.Here,E
g is set of parameters such that g
(
e
)
≠
∅.Definition 17.
Letf
D
,
g
D
∈
S
C
D
E
(
U
), let f
D
≠
Ø, and let D
1 and D
2 be two subsets of D such that D
1
∪
D
2
=
D and D
1
∩
D
2
=
∅. If ∀
d
i
∈
D
1, f
d
i
⊆
~
g
d
i, and ∀
d
i
∈
D
2, f
d
i
⊈
~
g
d
i, then f
D is called almost-subclass of soft class g
D and is denoted by f
D
⊑
a
g
D.From now on, decision makers setD
1 will denote set of d
i
∈
D such that f
d
i
⊆
~
g
d
i.Definition 18.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) and let f
D
⊑
a
g
D. Then, according to the soft class g
D, degree of subclasshood of soft class f
D, denoted by a
(
f
D
,
g
D
), is defined as follows: (9)
a
f
D
,
g
D
=
D
1
D
∑
d
i
∈
D
1
f
g
∘
d
i
.Here, for all d
i
∈
D
1, f
d
i
⊆
~
g
d
i and f
g
∘
(
d
i
)
=
1
/
E
g
∑
e
∈
E
(
f
d
i
e
/
g
d
i
e
), such that g
d
i
(
e
)
≠
∅.Example 19.
Let us consider soft classf
D given in Example 13 and soft class g
D given as follows: (10)
g
d
1
=
e
1
,
u
1
,
u
3
,
u
4
,
u
7
,
e
2
,
u
1
,
u
2
,
u
4
,
u
5
,
u
6
,
u
7
,
e
3
,
u
1
,
u
2
,
u
3
,
u
4
,
u
8
,
e
4
,
u
1
,
g
d
2
=
e
1
,
u
1
,
u
2
,
u
5
,
u
6
,
e
2
,
u
3
,
u
6
,
u
7
,
e
3
,
U
,
e
4
,
u
1
,
u
2
,
u
3
,
u
6
,
u
7
,
u
8
,
g
d
3
=
e
1
,
u
2
,
u
3
,
e
2
,
u
1
,
u
4
,
u
7
,
e
3
,
,
e
4
,
u
5
.Here, since f
d
1
⊆
~
g
d
1, f
d
2
⊆
~
g
d
2, and f
d
3
⊈
~
g
d
3, |
D
1
|
=
2. Then, (11)
f
g
∘
d
1
=
1
E
g
∑
e
∈
E
f
d
1
e
g
d
1
e
=
1
4
0.75
+
0.66
+
0.80
+
0
=
0.55
,
f
g
∘
d
2
=
1
E
g
∑
e
∈
E
f
d
2
e
g
d
2
e
=
1
4
0.50
+
0.66
+
1
+
0.66
=
0.71
.Thus, (12)
a
f
D
,
g
D
=
2
3
0.55
+
0.71
=
0.84 and f
D
⊑
a
g
D.Corollary 20.
Letf
D
,
g
D
∈
S
C
D
E
(
U
). Then(1)
if∀
d
i
∈
D, f
d
i
=
g
d
i, then a
(
f
D
,
g
D
)
=
1;
(2)
iff
D
⊑
g
D, f
D may be a
l
m
o
s
t-subclass of soft class g
D;
(3)
iff
D
⊑
a
g
D, f
D may not be a subclass of soft class g
D.Definition 21.
Letf
D
,
g
D
∈
S
C
D
E
(
U
). Then, union of soft classes f
D and g
D, denoted by f
D
⊔
g
D, is defined by class of soft sets as follows: (13)
f
D
⊔
g
D
=
f
d
i
∪
~
g
d
i
:
d
i
∈
D
.Example 22.
LetE
=
{
e
1
,
e
2
,
e
3
,
e
4
} be a parameter set, let U
=
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
u
8
} be an initial universe, and let D
=
{
d
1
,
d
2
,
d
3
} be a decision makers set. If (14)
f
d
1
=
e
1
,
u
1
,
u
2
,
u
5
,
e
2
,
u
3
,
u
4
,
u
7
,
u
8
,
e
3
,
u
1
,
u
3
,
u
5
,
u
6
,
e
4
,
,
f
d
2
=
e
1
,
u
1
,
u
3
,
e
2
,
u
3
,
u
5
,
e
3
,
u
2
,
u
4
,
e
4
,
u
5
,
u
7
,
u
8
,
f
d
3
=
e
1
,
u
4
,
u
5
,
u
6
,
u
7
,
e
2
,
u
1
,
u
6
,
u
8
,
e
3
,
,
e
4
,
u
1
,
u
4
,
u
5
,
g
d
1
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
4
,
u
6
,
e
3
,
u
3
,
u
5
,
u
8
,
e
4
,
u
3
,
u
7
,
g
d
2
=
e
1
,
u
2
,
u
3
,
e
2
,
u
3
,
u
6
,
u
8
,
e
3
,
u
1
,
u
2
,
u
3
,
u
6
,
e
4
,
u
1
,
u
3
,
u
7
,
u
8
,
g
d
3
=
e
1
,
u
2
,
u
5
,
u
6
,
e
2
,
u
1
,
u
4
,
e
3
,
,
e
4
,
u
5
,
u
7
,
u
8
,then soft classes can be written as f
D
=
{
f
d
1
,
f
d
2
,
f
d
3
} and g
D
=
{
g
d
1
,
g
d
2
,
g
d
3
}.
Here,(15)
f
D
⊔
g
D
=
f
d
1
∪
~
g
d
1
=
e
1
,
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
e
2
,
u
3
,
u
4
,
u
6
,
u
7
,
u
8
,
e
3
,
u
1
,
u
3
,
u
5
,
u
6
,
u
8
,
e
4
,
u
3
,
u
7
f
d
2
∪
~
g
d
2
=
e
1
,
u
1
,
u
2
,
u
3
,
e
2
,
u
3
,
u
5
,
u
6
,
u
8
,
e
3
,
u
1
,
u
2
,
u
3
,
u
5
,
u
6
,
e
4
,
u
1
,
u
3
,
u
5
,
u
7
,
u
8
f
d
3
∪
~
g
d
3
=
e
1
,
u
2
,
u
4
,
u
5
,
u
6
,
u
7
,
e
2
,
u
1
,
u
4
,
u
6
,
u
8
,
e
3
,
,
e
4
,
u
1
,
u
4
,
u
5
,
u
7
,
u
8
.Proposition 23.
Iff
D
,
g
D, and h
D
∈
S
C
D
E
(
U
), then(1)
f
D
⊔
f
D
=
f
D;
(2)
f
D
⊔
Ø
=
f
D;
(3)
f
D
⊔
U
=
U;
(4)
f
D
⊔
f
D
c
~
=
U;
(5)
f
D
⊔
g
D
=
g
D
⊔
f
D;
(6)
(
f
D
⊔
g
D
)
⊔
h
D
=
f
D
⊔
(
g
D
⊔
h
D
).Proof.
Letf
D
,
g
D, and h
D
∈
S
C
D
E
(
U
). Then, for all d
i
∈
D,(1)
f
D
⊔
f
D
=
f
D, since f
d
i
∪
~
f
d
i
=
f
d
i;
(2)
f
D
⊔
Ø
=
f
D, since f
d
i
∪
~
Φ
=
f
d
i;
(3)
f
D
⊔
U
=
U, since f
d
i
∪
~
U
^
=
U
^;
(4)
f
D
⊔
f
D
c
~
=
U, since f
d
i
∪
~
f
d
i
c
~
=
U
^;
(5)
f
D
⊔
g
D
=
g
D
⊔
f
D, since f
d
i
∪
~
g
d
i
=
g
d
i
∪
~
f
d
i;
(6)
(
f
D
⊔
g
D
)
⊔
h
D
=
f
D
⊔
(
f
D
⊔
h
D
), since (
f
d
i
∪
~
g
d
i
)
∪
~
h
d
i
=
f
d
i
∪
~
(
g
d
i
∪
~
h
d
i
).Definition 24.
Letf
D
,
g
D
∈
S
C
D
E
(
U
). Then, intersection of soft classes f
D and g
D, denoted by f
D
⊓
g
D, is defined by class of soft sets as follows: (16)
f
D
⊓
g
D
=
f
d
i
∩
~
g
d
i
:
d
i
∈
D
.Example 25.
Let us consider soft classesf
D and g
D given in Example 22. Then, (17)
f
D
⊓
g
D
=
f
d
1
∩
~
g
d
1
=
e
1
,
u
1
,
e
2
,
u
4
,
e
3
,
u
3
,
u
5
,
e
4
,
f
d
2
∩
~
g
d
2
=
e
1
,
u
3
,
e
2
,
u
3
,
e
3
,
u
2
,
e
4
,
u
7
,
u
8
f
d
3
∩
~
g
d
3
=
e
1
,
u
5
,
u
6
,
e
2
,
u
1
,
e
3
,
,
e
4
,
u
5
.Proposition 26.
Iff
D
,
g
D, and h
D
∈
S
C
D
E
(
U
), then(1)
f
D
⊓
f
D
=
f
D;
(2)
f
D
⊓
Ø
=
Ø;
(3)
f
D
⊓
U
=
f
D;
(4)
f
D
⊓
f
D
c
~
=
Ø;
(5)
f
D
⊓
g
D
=
g
D
⊓
f
D;
(6)
(
f
D
⊓
g
D
)
⊓
h
D
=
f
D
⊓
(
g
D
⊓
h
D
).Proof.
Letf
D
,
g
D, and h
D
∈
S
C
D
E
(
U
). Then, for all d
i
∈
D,(1)
f
D
⊓
f
D
=
f
D, since f
d
i
∩
~
f
d
i
=
f
d
i;
(2)
f
D
⊓
Ø
=
Ø, since f
d
i
∩
~
Φ
=
Φ;
(3)
f
D
⊓
U
=
f
D, since f
d
i
∩
~
U
^
=
f
d
i;
(4)
f
D
⊓
f
D
c
~
=
Ø, since f
d
i
∩
~
f
d
i
c
~
=
Φ;
(5)
f
D
⊓
g
D
=
g
D
⊓
f
D, since f
d
i
∩
~
g
d
i
=
g
d
i
∩
~
f
d
i;
(6)
(
f
D
⊓
g
D
)
⊓
h
D
=
f
D
⊓
(
f
D
⊓
h
D
), since (
f
d
i
∩
~
g
d
i
)
∩
~
h
d
i
=
f
d
i
∩
~
(
g
d
i
∩
~
h
d
i
).Proposition 27.
Iff
D
,
g
D, and h
D
∈
S
C
D
E
(
U
), then(1)
f
D
⊔
(
g
D
⊓
h
D
)
=
(
f
D
⊔
g
D
)
⊓
(
f
D
⊔
h
D
);
(2)
f
D
⊓
(
g
D
⊔
h
D
)
=
(
f
D
⊓
g
D
)
⊔
(
f
D
⊓
h
D
).Proof.
The proof can be easily obtained from Definitions21 and 24.Definition 28.
Letf
D
∈
S
C
D
E
(
U
). Then, soft complement of soft class f
D, denoted by f
D
c
~, is defined by class of soft sets as follows: (18)
f
D
c
~
=
f
d
i
c
~
:
d
i
∈
D
.Here, f
d
i
c
~
=
U
^
∖
~
f
d
i for all d
i
∈
D.Proposition 29.
Iff
D
∈
S
C
D
E
(
U
), then(1)
(
f
D
c
~
)
c
~
=
f
D;
(2)
Ø
c
~
=
U.Proof.
The proof can be easily obtained from Definition28.Proposition 30.
Iff
D
,
g
D
∈
S
C
D
E
(
U
), then(1)
(
f
D
⊔
g
D
)
c
~
=
f
D
c
~
⊓
g
D
c
~;
(2)
(
f
D
⊓
g
D
)
c
~
=
f
D
c
~
⊔
g
D
c
~.Proof.
Letf
D
,
g
D
∈
S
C
D
E
(
U
). Then, for all d
i
∈
D,(1)
(
f
D
⊔
g
D
)
c
~
=
f
D
c
~
⊓
g
D
c
~, since (
f
d
i
∪
~
g
d
i
)
c
~
=
f
d
i
c
~
∩
~
g
d
i
c
~;
(2)
(
f
D
⊓
g
D
)
c
~
=
f
D
c
~
⊓
g
D
c
~, since (
f
d
i
∩
~
g
d
i
)
c
~
=
f
d
i
c
~
∪
~
g
d
i
c
~.Proposition 31.
Iff
j
D
∈
S
C
D
E
(
U
)  (
j
=
1,2
,
…
,
n
), then(1)
(
⋃
~
j
=
1
n
f
j
d
i
)
c
~
=
⋂
~
j
=
1
n
(
f
j
d
i
)
c
~, for all d
i
∈
D;
(2)
(
⋂
~
j
=
1
n
f
j
d
i
)
c
~
=
⋃
~
j
=
1
n
(
f
j
d
i
)
c
~, for all d
i
∈
D.Proof.
Sincef
j
d
i are soft sets for all d
i
∈
D  (
j
=
1,2
,
…
,
n
), the proof is clear.Proposition 32.
Letf
j
D
∈
S
C
D
E
(
U
). Then,(1)
(
⨆
j
=
1
n
f
j
D
)
c
~
=
⨅
j
=
1
n
(
f
j
D
)
c
~;
(2)
(
⨅
j
=
1
n
f
j
D
)
c
~
=
⨆
j
=
1
n
(
f
j
D
)
c
~.Proof.
The proof is obvious from Propositions30 and 31.Definition 33.
Letf
D
∈
S
C
D
E
(
U
) and let g
∈
S
(
U
). Then, f
D is called soft partition of soft set g if and only if all of the following conditions hold:(1)
Φ
∉
f
D.
(2)
⋃
d
i
∈
D
f
d
i
(
e
)
=
g
(
e
), for all e
∈
E.
(3)
Iff
d
i
,
f
d
j
∈
f
D and i
≠
j, then f
d
i
(
e
)
∩
f
d
j
(
e
)
=
∅, for all e
∈
E.Definition 34.
Letf
D
∈
S
C
D
E
(
U
) and let g
∈
S
(
U
). If, for all e
∈
E, (19)
g
e
⊆
⋃
d
i
∈
D
f
d
i
e
,then soft class f
D is called soft cover of soft set g.Example 35.
Let us consider soft classf
D given in Example 22. Then, soft class f
D is soft cover of soft set g given as follows: (20)
g
=
e
1
,
u
1
,
u
6
,
e
2
,
u
4
,
u
5
,
u
8
,
e
3
,
u
2
,
u
5
,
e
4
,
u
4
,
u
5
.Definition 36.
Letf
D
∈
S
C
D
E
(
U
). If, for all e
∈
E and d
i
∈
D, ⋃
d
i
∈
D
f
d
i
(
e
)
=
U
^, then soft class f
D is called full soft class and is denoted by f
^
D.Proposition 37.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) be two soft covers of soft set h
∈
S
(
U
). Then, f
D
⊓
g
D is a soft cover of soft set h.Proof.
Assume thatf
D and g
D be two soft covers of soft set h; then, for all e
∈
E, h
(
e
)
⊆
⋃
d
i
∈
D
f
d
i
(
e
) and h
(
e
)
⊆
⋃
d
i
∈
D
g
d
i
(
e
). Hence, h
(
e
)
⊆
(
⋃
d
i
∈
D
f
d
i
(
e
)
)
∩
(
⋃
d
i
∈
D
g
d
i
(
e
)
)
=
⋃
d
i
∈
D
(
f
d
i
(
e
)
∩
g
d
i
(
e
)
) for all e
∈
E. So, soft class f
D
⊓
g
D is a soft cover of soft set h.Proposition 38.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) be two soft covers of soft set h
∈
S
(
U
). Then, f
D
⊔
g
D is a soft cover of soft set h.Proof.
Assume thatf
D and g
D be two soft covers of soft set h; then, for all e
∈
E, h
(
e
)
⊆
⋃
d
i
∈
D
f
d
i
(
e
) and h
(
e
)
⊆
⋃
d
i
∈
D
g
d
i
(
e
). Hence, h
(
e
)
⊆
(
⋃
d
i
∈
D
f
d
i
(
e
)
)
∪
(
⋃
d
i
∈
D
g
d
i
(
e
)
)
=
⋃
d
i
∈
D
(
f
d
i
(
e
)
∪
g
d
i
(
e
)
) for all e
∈
E. So soft class f
D
⊔
g
D is a soft cover of soft set h.Corollary 39.
Letf
^
D
∈
S
C
D
E
(
U
). Then, for all g
∈
S
(
U
), f
^
D is a soft cover of soft set g.


## 4. Soft Rough Classes

In this section, we define soft rough class and investigate its some properties.Definition 40.
Letf
D
∈
S
C
D
E
(
U
). For e
j
∈
E, parameterized class (e
j-class) of soft class f
D, denoted by C
f
D
(
e
j
), is defined as follows: (21)
C
f
D
e
j
=
f
d
i
e
j
:
d
i
∈
D
.Example 41.
LetE
=
{
e
1
,
e
2
,
e
3
,
e
4
} be a parameter set, let U
=
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
u
8
} be an initial universe, and let {
d
1
,
d
2
,
d
3
} be a set of decision makers. Let us consider soft sets f
d
1
,
f
d
2, and f
d
3 given as follows: (22)
f
d
1
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
1
,
u
4
,
u
5
,
u
6
,
e
3
,
u
1
,
u
2
,
u
3
,
u
8
,
e
4
,
,
f
d
2
=
e
1
,
u
1
,
u
2
,
e
2
,
u
3
,
u
6
,
e
3
,
U
,
e
4
,
u
1
,
u
3
,
u
7
,
u
8
,
f
d
3
=
e
1
,
u
2
,
u
3
,
u
5
,
e
2
,
u
1
,
u
4
,
u
7
,
e
3
,
,
e
4
,
u
5
,
u
8
;then all of parameterized classes of f
D are as follows: (23)
C
f
D
e
1
=
u
1
,
u
3
,
u
4
,
u
1
,
u
2
,
u
2
,
u
3
,
u
5
,
C
f
D
e
2
=
u
1
,
u
4
,
u
5
,
u
6
,
u
3
,
u
6
,
u
1
,
u
4
,
u
7
,
C
f
D
e
3
=
u
1
,
u
2
,
u
3
,
u
8
,
U
,
,
C
f
D
e
4
=
,
u
1
,
u
3
,
u
7
,
u
8
,
u
5
,
u
8
.Definition 42.
Letf
D
∈
S
C
D
E
(
U
). Then, for g
∈
S
(
U
) and e
∈
E, e-lower approximation, denoted by g
_
f
D
(
e
), is defined as follows: (24)
g
_
f
D
e
=
u
∈
U
:
∃
f
d
i
e
∈
C
f
D
e
,
u
∈
f
d
i
e
⊆
g
e
.
e-upper approximation, denoted by g
¯
f
D
(
e
), is defined as follows: (25)
g
¯
f
D
e
=
u
∈
U
:
∃
f
d
i
e
∈
C
f
D
e
,
u
∈
f
d
i
e
,
f
d
i
e
∩
g
e
≠
∅
.Moreover, the sets(26)
POS
f
D
g
e
=
g
_
f
D
e
,
NEG
f
D
g
e
=
-
g
¯
f
D
e
,
BND
f
D
g
e
=
g
¯
f
D
e
-
g
_
f
D
eare called the e-positive region, the e-negative region, and e-boundary region of g
∈
S
C
D
E
(
U
). Here -
g
f
D
(
e
) is complement of set g
f
D
(
e
). If g
_
f
D
(
e
)
=
g
¯
f
D
(
e
), g is said to be e-definable; otherwise g is called e-rough set.Proposition 43.
Letf
D
∈
S
C
D
E
(
U
). Then we have (27)
g
_
f
D
e
=
⋃
d
i
∈
D
f
d
i
e
:
f
d
i
e
⊆
g
e
,
g
¯
f
D
e
=
⋃
d
i
∈
D
f
d
i
e
:
f
d
i
e
∩
g
e
≠
∅for all g
∈
S
(
U
).Definition 44.
Letf
D
∈
S
C
D
E
(
U
) and let g
∈
S
(
U
). Then, soft f
D-lower approximation, denoted by a
p
r
_
f
D
g, is defined as follows: (28)
apr
_
f
D
g
=
e
,
g
_
f
D
e
:
e
∈
E
.Also, soft f
D-upper approximation, denoted by a
p
r
¯
f
D
g, is defined as follows: (29)
apr
¯
f
D
g
=
e
,
g
¯
f
D
e
:
e
∈
E
.Moreover, the sets(30)
POS
f
D
g
=
apr
_
f
D
g
=
e
,
g
_
f
D
e
:
e
∈
E
,
NEG
f
D
g
=
-
apr
¯
f
D
g
=
e
,
-
g
¯
f
D
e
:
e
∈
E
,
BND
f
D
g
=
apr
¯
f
D
g
-
apr
_
f
D
g
=
e
,
g
_
f
D
e
-
g
¯
f
D
e
:
e
∈
Eare called the soft f
D-positive region, the soft f
D-negative region, and soft f
D-boundary region of g
∈
S
(
U
), respectively. If a
p
r
_
f
D
g
=
a
p
r
¯
f
D
g, f
D is said to be soft f
D-definable; otherwise g is called a soft f
D-rough class.Example 45.
Let us consider soft classf
D given in Example 41. Let g
=
{
(
e
1
,
{
u
1
,
u
3
,
u
4
,
u
5
}
), (
e
2
,
{
u
3
,
u
6
,
u
7
}
)
,
(
e
3
,
U
)
,
(
e
4
,
{
u
5
,
u
7
,
u
8
}
)
}
∈
S
(
U
). Then, (31)
apr
_
f
D
g
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
3
,
u
6
,
e
3
,
U
,
e
4
,
u
5
,
u
8
,
apr
¯
f
D
g
=
e
1
,
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
e
2
,
u
1
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
e
3
,
U
,
e
4
,
u
1
,
u
3
,
u
5
,
u
7
,
u
8
,
POS
f
D
g
=
e
1
,
u
1
,
u
3
,
u
4
,
e
2
,
u
3
,
u
6
,
e
3
,
U
,
e
4
,
u
5
,
u
8
,
NEG
f
D
g
=
e
1
,
u
6
,
u
7
,
u
8
,
e
2
,
u
2
,
u
8
,
e
3
,
,
e
4
,
u
2
,
u
4
,
u
6
,
BND
f
D
g
=
e
1
,
u
2
,
u
5
,
e
2
,
u
1
,
u
4
,
u
5
,
u
7
,
e
3
,
,
e
4
,
u
1
,
u
3
,
u
7
.Lemma 46.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) and let h
,
k
∈
S
(
U
)
. Then, for all e
∈
E and for all d
i
∈
D,(1)
ifh
≠
U
^, h
≠
Φ,h
_
U
(
e
)
=
∅ and h
¯
U
(
e
)
=
U;
(2)
ifh
=
U
^, h
_
U
(
e
)
=
h
¯
U
(
e
)
=
U;
(3)
h
_
Ø
(
e
)
=
h
¯
Ø
(
e
)
=
∅;
(4)
h
_
f
D
⊔
g
D
(
e
)
=
h
_
f
D
(
e
)
∪
h
_
g
D
(
e
);
(5)
h
¯
f
D
⊔
g
D
(
e
)
⊇
h
¯
f
D
(
e
)
∪
h
¯
g
D
(
e
);
(6)
h
_
f
D
⊓
g
D
(
e
)
⊇
h
_
f
D
(
e
)
∩
h
_
g
D
(
e
);
(7)
h
¯
f
D
⊓
g
D
(
e
)
=
h
¯
f
D
(
e
)
∩
h
¯
g
D
(
e
).Proof.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) and let h
,
k
∈
S
(
U
)
. The proofs of (1), (2), and (3) are clear from definitions of e-upper and e-lower approximations: (4)
Letx
∈
h
_
f
D
⊔
g
D
(
e
). Then, x
∈
f
d
i
(
e
)
∪
g
d
i
(
e
)
⊆
h
(
e
) for some d
i
∈
D. Thus x
∈
f
d
i
(
e
)
⊆
h
(
e
) or x
∈
g
d
i
(
e
)
⊆
h
(
e
). So x
∈
h
_
f
D
(
e
)
∪
h
_
g
D
(
e
) and h
_
f
D
⊔
g
D
(
e
)
⊆
h
_
f
D
(
e
)
∪
h
_
g
D
(
e
). To prove the reverse inclusion, assume that x
∈
h
_
f
D
(
e
)
∪
h
_
g
D
(
e
). Then, x
∈
h
_
f
D
(
e
) or x
∈
h
_
g
D
(
e
). So, for some d
i
∈
D, x
∈
f
d
i
(
e
)
⊆
h
(
e
) or x
∈
g
d
i
(
e
)
⊆
h
(
e
) and x
∈
f
d
i
(
e
)
∪
g
d
i
(
e
)
⊆
h
(
e
). Thus, x
∈
h
_
f
D
⊔
g
D
(
e
). Then we have h
_
f
D
⊔
g
D
(
e
)
⊆
h
_
f
D
(
e
)
∪
h
_
g
D
(
e
)
h
¯
f
D
⊔
g
D
(
e
)
=
h
¯
f
D
(
e
)
∪
h
¯
g
D
(
e
).
(5)
Letx
∈
h
¯
f
D
(
e
)
∪
h
¯
g
D
(
e
). Then we have that x
∈
h
¯
f
D
(
e
) or x
∈
h
¯
g
D
(
e
). By definition, there exists some d
i
∈
D such that x
∈
f
d
i
(
e
) and f
d
i
(
e
)
∩
h
(
e
)
≠
∅ or x
∈
g
d
i and g
d
i
(
e
)
∩
h
(
e
)
≠
∅. So x
∈
f
d
i
(
e
)
∪
g
d
i
(
e
) and (
f
d
i
(
e
)
∪
g
d
i
(
e
)
)
∩
h
(
e
)
≠
∅. Thus, x
∈
h
¯
f
D
⊔
g
D
(
e
). We concluded that h
¯
f
D
⊔
g
D
(
e
)
⊇
h
¯
f
D
(
e
)
∪
h
¯
g
D
(
e
).
(6)
Letx
∈
h
_
f
D
(
e
)
∩
h
_
g
D
(
e
). By definition, there exists some d
i
∈
D such that x
∈
f
d
i
(
e
)
⊆
h
(
e
) and x
∈
g
d
i
(
e
)
⊆
h
(
e
). So x
∈
(
f
d
i
(
e
)
∩
g
d
i
)
(
e
)
⊆
h
(
e
). Hence, x
∈
h
_
f
D
⊓
g
D
(
e
). Thus, we conclude that h
_
f
D
⊓
g
D
(
e
)
⊇
h
_
f
D
(
e
)
∩
h
_
g
D
(
e
).
(7)
Letx
∈
h
¯
f
D
(
e
)
∩
h
¯
g
D
(
e
). Then we have that x
∈
h
¯
f
D
(
e
) and x
∈
h
¯
g
D
(
e
). By definition, there exists some d
i
∈
D such that x
∈
f
d
i
(
e
) and f
d
i
(
e
)
∩
h
(
e
)
≠
∅ and x
∈
g
d
i and g
d
i
(
e
)
∩
h
(
e
)
≠
∅. So x
∈
f
d
i
(
e
)
∩
g
d
i
(
e
) and (
f
d
i
(
e
)
∩
d
d
i
(
e
)
)
∩
h
(
e
)
≠
∅. Thus, x
∈
h
¯
f
D
⊓
g
D
(
e
). We concluded that h
¯
f
D
⊓
g
D
(
e
)
⊇
h
¯
f
D
(
e
)
∩
h
¯
g
D
(
e
). To prove the reverse inclusion, assume that x
∈
h
¯
f
D
⊓
g
D
(
e
); then x
∈
f
d
i
(
e
)
∩
g
d
i
(
e
) such that (
f
d
i
(
e
)
∩
g
d
i
(
e
)
)
∩
h
(
e
)
≠
∅ and (
f
d
i
(
e
)
∩
h
d
i
(
e
)
)
∩
(
g
d
i
(
e
)
∩
h
(
e
)
)
≠
∅. Hence, (
f
d
i
(
e
)
∩
h
d
i
(
e
)
)
≠
∅ and (
g
d
i
(
e
)
∩
h
d
i
(
e
)
)
≠
∅. Since x
∈
f
d
i
(
e
) and (
f
d
i
(
e
)
∩
h
d
i
(
e
)
)
≠
∅, x
∈
h
¯
f
D
(
e
) and in a similar way x
∈
h
¯
g
D
(
e
). We get that h
¯
f
D
⊓
g
D
(
e
)
⊆
h
¯
f
D
(
e
)
∩
h
¯
g
D
(
e
). Then, h
¯
f
D
⊓
g
D
(
e
)
=
h
¯
f
D
(
e
)
∩
h
¯
g
D
(
e
).Theorem 47.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) and h
∈
S
(
U
). Then,(1)
a
p
r
_
f
D
⊔
g
D
(
h
)
=
a
p
r
_
f
D
(
h
)
∪
~
a
p
r
_
g
D
(
h
);
(2)
a
p
r
¯
f
D
⊔
g
D
(
h
)
⊇
a
p
r
¯
f
D
(
h
)
∪
~
a
p
r
¯
g
D
(
h
);
(3)
a
p
r
_
f
D
⊓
g
D
(
h
)
⊇
a
p
r
_
f
D
(
h
)
∩
~
a
p
r
_
g
D
(
h
);
(4)
a
p
r
¯
f
D
⊓
g
D
(
h
)
=
a
p
r
¯
f
D
(
h
)
∩
~
a
p
r
¯
g
D
(
h
);
(5)
a
p
r
_
Ø
(
h
)
=
Φ;
(6)
a
p
r
¯
Ø
(
h
)
=
Φ;
(7)
if∀
d
i
∈
D, h
≠
U
^, a
p
r
_
U
(
h
)
=
Φ and a
p
r
¯
U
(
h
)
=
U
^;
(8)
if∀
d
i
∈
D, h
=
U
^, a
p
r
_
U
(
h
)
=
U
^ and a
p
r
¯
U
(
h
)
=
U
^.Proof.
By using Lemma46, the proof can be easily made.Theorem 48.
Letf
D
∈
S
C
D
E
(
U
) and let h
,
k
∈
S
(
U
). Then,(1)
a
p
r
_
f
D
(
Φ
)
=
a
p
r
¯
f
D
(
Φ
)
=
Φ;
(2)
a
p
r
_
f
D
(
U
^
)
=
a
p
r
¯
f
D
(
U
^
)
=
U
^;
(3)
h
⊆
~
k
⇒
a
p
r
_
f
D
(
h
)
⊆
~
a
p
r
_
f
D
(
k
);
(4)
h
⊆
~
k
⇒
a
p
r
¯
f
D
(
h
)
⊆
~
a
p
r
¯
f
D
(
k
);
(5)
a
p
r
_
f
D
(
h
∩
~
k
)
⊆
~
a
p
r
_
f
D
(
h
)
∩
~
a
p
r
_
f
D
(
k
);
(6)
a
p
r
¯
f
D
(
h
∩
~
k
)
⊆
~
a
p
r
¯
f
D
(
h
)
∩
~
a
p
r
¯
f
D
(
k
);
(7)
a
p
r
_
f
D
(
h
∪
~
k
)
⊇
~
a
p
r
_
f
D
(
h
)
∪
~
a
p
r
_
f
D
(
k
);
(8)
a
p
r
¯
f
D
h
∪
~
k
=
a
p
r
¯
f
D
(
h
)
∪
~
a
p
r
¯
f
D
(
k
).Proof.
(1) It is straightforward.
(2) It is straightforward.
(3) Letx
∈
h
_
f
D
(
e
). Then, for some d
i
∈
D, x
∈
f
d
i
(
e
)
⊆
h
(
e
). Since h
⊆
~
k, h
(
e
)
⊆
k
(
e
), and x
∈
f
d
i
(
e
)
⊆
k
(
e
). Therefore x
∈
k
_
f
D
(
e
) and h
_
f
D
(
e
)
⊆
k
_
f
D
(
e
); From definition of soft f
D-lower approximation a
p
r
_
f
D
(
h
)
⊆
~
a
p
r
_
f
D
(
k
)
.
(4) Leth
⊆
~
k. Then, for all d
i
∈
D, h
(
e
)
∩
f
d
i
(
e
)
⊆
k
(
e
)
∩
f
d
i
(
e
) and h
(
e
)
∩
f
d
i
(
e
)
≠
∅, k
(
e
)
∩
f
d
i
(
e
)
≠
∅ for some d
i
∈
D. Therefore h
¯
f
D
(
e
)
⊆
k
¯
f
D
(
e
). From definition of soft f
D-upper approximation a
p
r
¯
f
D
(
h
)
⊆
~
a
p
r
¯
f
D
(
k
).
(5) Sinceh
∩
~
k
⊆
~
h and h
∩
~
k
⊆
~
k, from (
3
), a
p
r
_
f
D
(
h
∩
~
k
)
⊆
~
a
p
r
_
f
D
(
k
) and a
p
r
_
f
D
(
h
∩
~
k
)
⊆
~
a
p
r
_
f
D
(
k
), respectively. Therefore, a
p
r
_
f
D
(
h
∩
~
k
)
⊆
~
a
p
r
_
f
D
(
h
)
∩
~
a
p
r
_
f
D
(
k
)
.
(6) This is similar to proof(
5
).
(7) Sincek
⊆
~
h
∪
~
k, from (
3
),  a
p
r
_
f
D
(
k
)
⊆
~
a
p
r
_
f
D
(
h
∪
~
k
). Similarly, a
p
r
_
f
D
(
h
)
⊆
~
a
p
r
_
f
D
(
h
∪
~
k
). Therefore, a
p
r
_
f
D
(
h
∪
~
k
)
⊇
~
a
p
r
_
f
D
(
k
)
∪
~
a
p
r
_
f
D
(
h
).
(8) Let(
e
,
f
(
e
)
)
∈
a
p
r
¯
f
D
(
h
∪
~
k
). By definition of soft f
D-upper approximation, there exist some e
∈
E such that u
∈
f
(
e
) and f
(
e
)
∩
(
h
∪
~
k
)
(
e
)
≠
∅. Hence, we get that either f
(
e
)
∩
h
(
e
)
≠
∅ or f
(
e
)
∩
k
(
e
)
≠
∅. Then, (
e
,
f
(
e
)
)
∈
a
p
r
¯
f
D
(
h
) or (
e
,
f
(
e
)
)
∈
a
p
r
¯
f
D
(
h
). This shows that(32)
apr
¯
f
D
h
∪
~
k
⊆
~
apr
¯
f
D
h
∪
~
apr
¯
f
D
k
.
To prove the reverse inclusion, note thatk
⊆
~
h
∪
~
k and h
⊆
~
h
∪
~
k; then from (3)   a
p
r
¯
f
D
(
k
)
⊆
~
a
p
r
¯
f
D
(
h
∪
~
k
) and a
p
r
¯
f
D
(
h
)
⊆
~
a
p
r
¯
f
D
(
h
∪
~
k
), respectively. Thus,(33)
apr
¯
f
D
h
∪
~
apr
¯
f
D
k
⊆
~
apr
¯
f
D
h
∪
~
k
. From (32) and (33), (34)
apr
¯
f
D
h
∪
~
k
=
apr
¯
f
D
h
∪
~
apr
¯
f
D
k
.Definition 49.
Letf
D
∈
S
C
D
E
(
U
) and let h
,
k
∈
S
(
U
). We define (35)
h
⌣
f
D
k
⟺
apr
_
f
D
h
=
apr
_
f
D
k
,
h
⌢
f
D
k
⟺
apr
¯
f
D
h
=
apr
¯
f
D
k
,
h
≍
f
D
k
⟺
h
⌣
f
D
k
=
h
⌢
f
D
k
.These binary relations are called the lower soft class rough equal relation and the upper soft class rough equal relation, respectively.Theorem 50.
Letf
D
∈
S
C
D
E
(
U
) and let h
,
k
,
h
′, and k
′
∈
S
(
U
). Then,(1)
h
⌢
f
D
k
⇔
h
⌢
f
D
(
h
∪
~
k
)
⌢
f
D
k;
(2)
h
⌢
f
D
h
′, k
⌢
f
D
k
′
⇒
(
h
∪
~
k
)
⌢
f
D
(
h
′
∪
~
k
′
);
(3)
h
⌢
f
D
k
⇒
h
∪
~
(
U
^
∖
~
k
)
⌢
f
D
U
^;
(4)
h
⊆
~
k
,
k
⌢
f
D
Φ
⇒
h
⌢
f
D
Φ;
(5)
h
⊆
~
k
,
h
⌢
f
D
U
^
⇒
k
⌢
f
D
U
^.Proof.
(1) Assume thath
⌢
f
D
k; then a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
f
D
(
k
). From Theorem 48, we know that a
p
r
¯
f
D
(
h
∪
~
k
)
=
a
p
r
¯
f
D
(
h
)
∪
~
a
p
r
¯
f
D
(
k
). Thus, a
p
r
¯
f
D
(
h
∪
~
k
)
=
a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
f
D
(
k
) and so h
⌢
f
D
(
h
∪
~
k
)
⌢
f
D
k. Conversely, suppose that h
⌢
f
D
(
h
∪
~
k
)
⌢
f
D
k. From transitivity of ⌢
f
D, h
⌢
f
D
k.
(2) Suppose thath
⌢
f
D
h
′ and k
⌢
f
D
k
′; then, from definition, a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
f
D
(
h
′
) and a
p
r
¯
f
D
(
k
)
=
a
p
r
¯
f
D
(
k
′
). From Theorem 48, a
p
r
¯
f
D
(
h
∪
~
k
)
=
a
p
r
¯
f
D
(
h
)
∪
~
a
p
r
¯
f
D
(
k
) and a
p
r
¯
f
D
(
h
′
∪
~
k
′
)
=
a
p
r
¯
f
D
(
h
′
)
∪
~
a
p
r
¯
f
D
(
k
′
). Hence, we get a
p
r
¯
f
D
(
h
∪
~
k
)
=
a
p
r
¯
f
D
(
h
′
∪
~
k
′
) and so (
h
∪
~
k
)
⌢
f
D
(
h
′
∪
~
k
′
)
.
(3) Leth
⌢
f
D
k. Then, from definition, a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
f
D
(
k
). By Theorem 48, a
p
r
¯
f
D
(
h
∪
~
(
U
^
∖
~
k
)
)
=
a
p
r
¯
f
D
(
h
)
∪
~
a
p
r
¯
f
D
(
U
^
∖
~
k
) and a
p
r
¯
f
D
(
U
^
)
=
a
p
r
¯
f
D
(
k
)
∪
~
a
p
r
¯
f
D
(
U
^
∖
~
k
). It follows that a
p
r
¯
f
D
U
^
=
a
p
r
¯
f
D
(
h
∪
~
(
U
^
∖
~
k
)
). Therefore, h
∪
~
(
U
^
∖
~
k
)
⌢
f
D
U
^
.
(4) Leth
⊆
~
k and let k
⌢
f
D
Φ. From Theorem 48, we get a
p
r
¯
f
D
(
h
)
⊆
~
a
p
r
¯
f
D
(
k
)
=
a
p
r
¯
f
D
Φ
=
Φ. Thus, a
p
r
¯
f
D
(
h
)
=
Φ
=
a
p
r
¯
f
D
Φ and so h
⌢
f
D
Φ.
(5) Suppose thath
⊆
~
k and h
⌢
f
D
U
^. By Theorem 48, we have a
p
r
¯
f
D
(
k
)
⊇
~
a
p
r
¯
f
D
h
=
a
p
r
¯
f
D
U
^
. Also, since k
⊆
U
^, a
p
r
¯
f
D
(
k
)
⊆
~
a
p
r
¯
f
D
U
^. Hence, a
p
r
¯
f
D
(
k
)
=
a
p
r
¯
f
D
(
U
^
), and so k
⌢
f
D
U
^.Definition 51.
Letf
D
,
g
D
∈
S
C
D
E
(
U
) and let h
∈
S
(
U
). We define (36)
f
D
⌣
h
g
D
⟺
apr
_
f
D
h
=
apr
_
g
D
h
,
f
D
⌢
h
g
D
⟺
apr
¯
f
D
h
=
apr
¯
g
D
h
,
f
D
≍
h
g
D
⟺
f
D
⌣
h
g
D
=
f
D
⌢
h
g
D
.These binary relations are called the lower soft class rough h-equal relation and the upper soft class rough h-equal relation, respectively.Theorem 52.
Letf
D
,
g
D
,
f
D
′, and g
D
′
∈
S
C
D
E
(
U
) and let h
∈
S
(
U
). Then,(1)
f
D
⌢
h
g
D
⇔
f
D
⌢
h
(
f
D
⊓
g
D
)
⌢
h
g
D;
(2)
f
D
⌢
h
f
D
′, g
D
⌢
h
g
D
′
⇒
(
f
D
⊓
f
D
′
)
⌢
h
(
g
D
⊓
g
D
′
);
(3)
f
D
⌢
h
g
D
⇒
f
D
⊓
(
U
∖
~
g
D
)
⌢
h
U;
(4)
f
D
⊑
~
g
D
,
g
D
⌢
h
Ø
⇒
f
D
⌢
h
Ø;
(5)
f
D
⊑
~
g
D
,
f
D
⌢
h
U
⇒
g
D
⌢
h
U.Proof.
(1) Assume thatf
D
⌢
h
g
D; then a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
g
D
(
h
). From Theorem 47, we know that a
p
r
¯
f
D
⊓
g
D
(
h
)
=
a
p
r
¯
f
D
(
h
)
∩
~
a
p
r
¯
g
D
(
h
). Thus, a
p
r
¯
f
D
⊓
g
D
(
h
)
=
a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
g
D
(
h
) and so f
D
⌢
h
f
D
⊓
g
D
⌢
h
g
D. Conversely, suppose that f
D
⌢
h
(
f
D
⊓
g
D
)
⌢
h
g
D. From transitivity of ⌢
h, f
D
⌢
h
g
D.
(2) Suppose thatf
D
⌢
h
f
D
′ and g
D
⌢
h
g
D
′; then, from definition, a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
f
D
′
(
h
) and a
p
r
¯
g
D
(
h
)
=
a
p
r
¯
g
D
′
(
h
). From Theorem 47, a
p
r
¯
f
D
⊓
g
D
h
=
a
p
r
¯
f
D
h
∩
~
a
p
r
¯
g
D
h
=
a
p
r
¯
f
D
′
h
∩
~
a
p
r
¯
g
D
′
h
. So a
p
r
¯
f
D
⊓
g
D
h
=
a
p
r
¯
f
D
′
⊓
g
D
′
h and (
f
D
⊓
g
D
)
⌢
h
(
f
D
′
⊓
g
D
′
).
(3) The proof can be made by similar way to proof of (1) and (2).
(4) Letf
D
⊑
g
D and g
D
⌢
h
Ø. Then, a
p
r
¯
g
D
h
=
a
p
r
¯
Ø
(
h
) and since f
D
⊑
g
D, a
p
r
¯
f
D
h
⊆
~
a
p
r
¯
Ø
h
=
Φ. Therefore, a
p
r
¯
f
D
(
h
)
=
a
p
r
¯
Ø
h. We have f
D
⌢
h
Ø.
(5) The proof can be made by similar way to (4).


## 5. Decision Making Using Soft Rough Class

In this section, some concepts are defined to construct a decision making method using soft rough class and a decision making algorithm is given. Then, an application of proposed decision making method is made for a real problem.Definition 53.
Letf
D
∈
S
C
D
E
(
U
) and let g be a soft set (reference soft set) over U. Then, consistency degree of soft set g related to parameter e
∈
E and soft class f
D, denoted by γ
f
D
g
(
e
), is formulated as follows:(37)
γ
f
D
g
e
=
g
_
f
D
e
g
¯
f
D
e
.
According to soft classf
D consistency degree of soft set g, denoted by Γ
f
D
g, is formulated as follows:(38)
Γ
f
D
g
=
1
E
∑
e
∈
E
γ
f
D
g
e
.Definition 54.
Letf
D
∈
S
C
D
E
(
U
) and let g be a soft set over U. Then, relative consistence degree (rcd) between soft class f
D
-
{
d
i
} and soft set g related to parameter e
∈
E, denoted by γ
f
D
-
{
f
d
i
}
g
(
e
), is formulated as follows:(39)
γ
f
D
-
f
d
i
g
e
=
g
_
f
D
-
f
d
i
e
g
¯
f
D
-
f
d
i
e
.
Between soft classf
D
-
{
d
i
} and soft set g total relative consistency degree is formulated as follows:(40)
Γ
f
D
-
f
d
i
g
=
1
E
∑
e
∈
E
γ
f
D
-
f
d
i
g
e
.Definition 55.
Letf
D
∈
S
C
D
E
(
U
) and let g be a soft set over U. Then Γ
f
D
g
-
Γ
f
D
-
{
f
d
i
}
g is called effectiveness of decision maker d
i and is denoted by e
(
d
i
).Now we will give relations between two decision makers in decision maker setD.Definition 56.
Letf
D
∈
S
C
D
E
(
U
) and let g be a soft set over U. Effectiveness relations between d
i and d
j are defined as follows:(1)
Ife
(
d
i
)
>
g
e
(
d
j
), d
i is more effective than d
j.
(2)
Ife
(
d
i
)
=
g
e
(
d
j
), d
i has same effect as d
j.
(3)
Ife
(
d
i
)
<
g
e
(
d
j
), d
j is more effective than d
i.Algorithm 57.
Step  1. Construct a soft class f
D and reference soft set g over U.
Step  2. Find the consistency degree of soft set g denoted by γ
f
D
g
(
e
) related to parameter e
∈
E.
Step  3. Find consistency degree of soft set g according to soft class f
D.
Step  4. Find total relative consistency degree between soft class f
D
-
{
d
i
} and soft set g.
Step  5. Find effectiveness of each decision maker d
i
∈
D.
Step  6. Chose effective decision maker.


## 6. Applied Example

Assume that an investment company wants to employ stock market analysts. Five persons apply for this position in the company and the department of human resources wants to make appropriate choice among the applicants. Therefore, department of human resources wants some previous evaluations made by applicantsd
1
,
d
2
,
d
3
,
d
4, and d
5 for firms u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
u
7
,
u
8 in different times t
1
,
t
2
,
t
3
,
t
4
,
t
5 in the last two years.Step 1.
According to the appreciation criteria, evaluations of applicantsd
1, d
2, d
3, d
4, and d
5 performed in different times specified by human resources department are represented by soft sets f
d
1, f
d
2, f
d
3, f
d
4, and f
d
5 given as follows:(41)
f
d
1
=
t
1
,
u
1
,
u
3
,
u
4
,
t
2
,
u
1
,
u
4
,
u
5
,
u
6
,
t
3
,
u
1
,
u
2
,
u
3
,
u
8
,
t
4
,
,
t
5
,
u
2
,
u
4
,
u
7
,
f
d
2
=
t
1
,
u
1
,
u
2
,
t
2
,
u
3
,
u
6
,
t
3
,
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
,
t
4
,
u
1
,
u
3
,
u
7
,
u
8
,
t
5
,
u
1
,
u
2
,
f
d
3
=
t
1
,
u
2
,
u
3
,
u
5
,
t
2
,
u
1
,
u
4
,
u
7
,
t
3
,
,
t
4
,
u
5
,
u
8
,
t
5
,
u
3
,
u
5
,
u
7
,
f
d
4
=
t
1
,
u
1
,
u
3
,
t
2
,
u
1
,
u
2
,
u
8
,
t
3
,
u
1
,
u
3
,
u
4
,
t
4
,
u
7
,
u
8
,
t
5
,
u
2
,
u
5
,
u
8
,
f
d
5
=
t
1
,
u
1
,
u
3
,
u
5
,
t
2
,
,
t
3
,
u
1
,
u
8
,
t
4
,
u
1
,
u
3
,
u
5
,
t
5
,
u
1
,
u
4
,
u
6
.Department of human resources has real results previously obtained in specified times: t
1, t
2, t
3, t
4, and t
5. These real results are represented by soft set g (reference soft set) as follows: (42)
g
=
t
1
,
u
1
,
u
3
,
u
4
,
u
5
,
t
2
,
u
3
,
u
6
,
u
7
,
t
3
,
U
,
t
4
,
u
5
,
u
7
,
u
8
,
t
5
,
u
1
,
u
2
,
u
4
,
u
7
.Tabular representation of soft class f
D
=
f
d
1
,
f
d
2
,
f
d
3
,
f
d
4
,
f
d
5 is shown in Table 3.Table 3
The tabular representation of the soft classf
D.
f
D
f
d
1
f
d
2
f
d
3
f
d
4
f
d
5
t
1
{
u
1
,
u
3
,
u
4
}
{
u
1
,
u
2
}
{
u
2
,
u
3
,
u
5
}
{
u
1
,
u
3
}
{
u
1
,
u
3
,
u
5
}
t
2
{
u
1
,
u
4
,
u
5
,
u
6
}
{
u
3
,
u
6
}
{
u
1
,
u
4
,
u
7
}
{
u
1
,
u
2
,
u
8
}
{
}
t
3
{
u
1
,
u
2
,
u
3
,
u
8
}
{
u
1
,
u
2
,
u
3
,
u
4
,
u
5
,
u
6
}
{
}
{
u
1
,
u
3
,
u
4
}
{
u
1
,
u
8
}
t
4
{
}
{
u
1
,
u
3
,
u
7
,
u
8
}
{
u
5
,
u
8
}
{
u
7
,
u
8
}
{
u
1
,
u
3
,
u
5
}
t
5
{
u
2
,
u
4
,
u
7
}
{
u
1
,
u
2
}
{
u
3
,
u
5
,
u
7
}
{
u
2
,
u
5
,
u
8
}
{
u
1
,
u
4
,
u
6
}Step 2.
Using (37), consistency degree of soft set g for time parameters t
1, t
2, t
3, t
4, and t
5 is obtained as in Table 4.Table 4
The tabular representation of the consistency degree of soft setg for t
i (i = 1, 2, 3, 4, 5).
t
1
t
2
t
3
t
4
t
5
γ
f
D
g
0.800
0.666
1.000
0.600
0.500Step 3.
Using (38) and Table 4, consistency degree of soft set g related to soft class f
D is obtained as Γ
f
D
g
=
0.713.Step 4.
Using (39), relative consistency degrees of soft set g with respect to soft class f
D are as in Table 5. And, from (40), total relative consistency degrees of soft set g with respect to soft class f
D are as in Table 6.Table 5
Relative consistency degrees between soft setg and f
D
-
{
d
i
}.
f
D
t
1
t
2
t
3
t
4
t
5
γ
f
D
-
{
f
d
1
}
g
0.750
0.400
1.000
0.600
0.250
γ
f
D
-
{
f
d
2
}
g
0.750
0.000
1.000
0.600
0.375
γ
f
D
-
{
f
d
3
}
g
0.200
0.400
1.000
0.400
0.571
γ
f
D
-
{
f
d
4
}
g
0.800
0.500
1.000
0.400
0.666
γ
f
D
-
{
f
d
5
}
g
0.600
0.666
1.000
0.600
0.571Table 6
Total relative consistency degrees between soft setg and soft class f
D
-
{
d
i
}.
Total relative consistency formula
Values
Γ
f
D
-
{
f
d
1
}
g
0.600
Γ
f
D
-
{
f
d
2
}
g
0.535
Γ
f
D
-
{
f
d
3
}
g
0.514
Γ
f
D
-
{
f
d
4
}
g
0.673
Γ
f
D
-
{
f
d
5
}
g
0.687Step 5.
Using Definition55, effectiveness of the applicants d
1, d
2, d
3, d
4, and d
5 is obtained as follows: (43)
e
d
1
=
0.113
,
e
d
2
=
0.278
,
e
d
3
=
0.199
,
e
d
4
=
0.040
,
e
d
5
=
0.026
.Step 6.
From Definition56, effectiveness of applicants can be ordered as follows: (44)
e
d
2
>
g
e
d
3
>
g
e
d
1
>
g
e
d
4
>
g
e
d
5
.
Then,d
2 is the most effective decision maker in soft class f
D by soft set g.


## 7. Conclusion

In this paper, we have defined concepts of soft class, soft class operations, and soft rough class. Then we have presented a decision making method based on the soft rough class. Finally, we have provided an example that demonstrated that this decision making method can successfully work. It can be applied to problems of many fields that contain uncertainty. Next, we can define fuzzy soft class and fuzzy soft rough class and their operations as generalization of soft classes and soft rough classes. Also a reduction method can be developed based on soft class and soft rough classes.



---

*Source: 1584528-2016-02-25.xml*