entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets
TRANSCRIPT
Accepted Manuscript
Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets
Yuncheng Jiang, Yong Tang, Hai Liu, Zhenzhou Chen
PII: S0020-0255(13)00268-5
DOI: http://dx.doi.org/10.1016/j.ins.2013.03.052
Reference: INS 10036
To appear in: Information Sciences
Received Date: 20 March 2012
Revised Date: 19 February 2013
Accepted Date: 18 March 2013
Please cite this article as: Y. Jiang, Y. Tang, H. Liu, Z. Chen, Entropy on intuitionistic fuzzy soft sets and on interval-
valued fuzzy soft sets, Information Sciences (2013), doi: http://dx.doi.org/10.1016/j.ins.2013.03.052
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Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets
Yuncheng Jiang , Yong Tang, Hai Liu, Zhenzhou Chen
School of Computer Science, South China Normal University, Guangzhou 510631, PR China
Abstract
Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for
dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with
imprecise and fuzzy parameters. In order to handle these types of problem parameters, some fuzzy (or intuitionistic
fuzzy, interval-valued fuzzy) extensions of soft set theory are presented, yielding fuzzy (or intuitionistic fuzzy,
interval-valued fuzzy) soft set theory. In this paper, we define the distance measures between intuitionistic fuzzy
soft sets and give an axiom definition of intuitionistic entropy for an intuitionistic fuzzy soft set and a theorem
which characterizes it. Furthermore, we discuss the relationship between intuitionistic fuzzy soft sets and
interval-valued fuzzy soft sets and transform the structure of entropy for intuitionistic fuzzy soft sets to the
interval-valued fuzzy soft sets.
Keywords: Soft sets; Fuzzy soft sets; Intuitionistic fuzzy soft sets; Interval-valued fuzzy soft sets; Entropy
1. Introduction
Soft set theory, proposed by Molodtsov [51], is a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. Concretely, a soft set is a collection of approximate descriptions of an object. This initial description of the object has an approximate nature, and we do not need to introduce the notion of exact solution. The absence of any restrictions on the approximate description in soft sets make this theory very convenient and easily applicable in practice [35]. This theory has proven useful in many different fields such as decision making [10][11][16][22][36][46][55], data analysis [72], forecasting [62], simulation [34], optimization [37], and texture classification [52].
Up to the present, research on soft sets has been very active and many results have been achieved in the theoretical and application aspects (see Section 5 for more details) [8][25][53][64]. Especially, Maji et al. [43] initiated the study on hybrid structures involving both fuzzy sets and soft sets. In [43] the notion of fuzzy soft sets was introduced as a fuzzy generalization of classical soft sets and some basic properties were discussed in detail. Afterwards, many researchers have worked on this concept. For example, various kinds of extended fuzzy soft sets such as generalised fuzzy soft sets [48], intuitionistic fuzzy soft sets [42][44][47], vague soft sets [63], interval-valued fuzzy soft sets [65], and interval-valued intuitionistic fuzzy soft sets [24] were presented by extending the classical fuzzy soft sets [43]. In particular, intuitionistic fuzzy soft sets [42][44][47] and interval-valued fuzzy soft sets [65] are two intuitively straightforward extensions of fuzzy soft sets proposed by Maji et al. [43]. These two extended fuzzy soft sets were independently introduced to alleviate some drawbacks of fuzzy soft sets. Concretely, an Corresponding author. E-mail addresses: [email protected], [email protected] (Yuncheng Jiang).
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intuitionistic fuzzy soft set allocates to each element of the universe both a degree of membership and a degree of non-membership such that + 1, thus relaxing the enforced condition = 1 from fuzzy soft set theory. Interval-valued fuzzy soft set theory emerges from the observation that in a lot of cases, no objective procedure is available to select the crisp membership degrees of elements in a fuzzy soft set. In other words, in many real applications, the membership degree in a fuzzy soft set cannot be lightly confirmed. It is more reasonable to give an interval-valued data to describe membership degree. Thus, it was suggested to specify an interval-valued degree of membership [ , +] to each element of the universe. Intuitionistic fuzzy soft set theory and interval-valued fuzzy soft set theory have the virtue of complementing fuzzy soft sets, which are able to model vagueness and uncertainty precisely. In fact, this is the same as that of intuitionistic fuzzy sets [4][5] and interval-valued fuzzy sets [19][58][71]. It should be noted that although intuitionistic fuzzy sets have nothing to do with intuitionism [13], in this paper we still use the terminology of “intuitionistic fuzzy sets”.
Entropy is an important topic in fuzzy set theory. The entropy of fuzzy sets describes the fuzziness degree of fuzzy sets. In 1965, Zadeh introduced the fuzzy entropy for the first time [67]. Luca and Termini [41] introduced the axiom construction of entropy of fuzzy sets and referred to Shannon’s probability entropy, interpreting it as a measure of the amount of information. Kaufmann [33] pointed out that an entropy of a fuzzy set can be gotten through the distance between the fuzzy set and it’s nearest non-fuzzy set, whereas Higashi and Klir [20] did it using the distance from a fuzzy set to its complement. Trillas and Riera [57] proposed general expressions for this entropy, and Loo [40] proposed a definition of entropy which includes those given by Luca and Termini [41] and Kaufmann [33]. Liu [39] gave the well-known axiomatic definitions of fuzzy similarity, distance and entropy. Fan and Ma [15] presented a general conclusion about fuzzy entropy induced by distance measure based on the axiom definitions of fuzzy entropy and distance measure. Mi et al. [50] introduced a generalized axiomatic definition of entropy of a fuzzy set with a distance based on the axiomatic definition of Liu [39].
Some authors have investigated entropy of intuitionistic fuzzy sets and interval-valued fuzzy sets. Burillo and Bustince [9] introduced the concept of entropy of intuitionistic fuzzy sets, which allows us to measure the degree of fuzziness of an intuitionistic fuzzy set. Szmidt and Kacprzyk [56] proposed a nonprobilistic-type entropy measure with a geometric interpretation of intuitionistic fuzzy sets. Zeng and Li [69] expressed the axioms of Szmidt and Kacprzyk [56] using the notion of interval-valued fuzzy sets and discussed the relationship between similarity measure and entropy. Hung and Yang [21] exploited the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets, proposed two families of entropy measures for intuitionistic fuzzy sets and also constructed the axiom definition and properties. Vlachos et al. [59] presented a unified framework for subsethood, entropy, and cardinality for interval-valued fuzzy sets. An axiomatic skeleton for subsethood was introduced and some subsethood and entropy measures in the interval-valued fuzzy setting were proposed. Zhang et al. [71] proposed a new axiomatic definition of entropy for interval-valued fuzzy sets based on distance which is consistent with the axiomatic definition of entropy of a fuzzy set introduced by Luca and Termini [41] and Liu [39].
As is shown in [2], every fuzzy set may be considered as a (classical) soft set. Since fuzzy soft sets are fuzzy generalizations of classical soft sets, thus, every fuzzy set may be considered as a fuzzy soft set. That is to say, we can regard fuzzy soft sets as extensions of fuzzy sets from the
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soft set theoretical viewpoint [16]. Correspondingly, intuitionistic fuzzy soft sets [42][44][47] and interval-valued fuzzy soft sets [65] should be regarded as extensions of intuitionistic fuzzy sets and interval-valued fuzzy sets from the soft set theoretical viewpoint, respectively. Thus, the following natural question arises: How can we appraise the fuzziness associated with an intuitionistic fuzzy soft set or an interval-valued fuzzy soft set? It is well-known that a measure of fuzziness often used in the literature is an entropy. Naturally, we may use entropy to measure the fuzziness of an intuitionistic fuzzy soft set or an interval-valued fuzzy soft set. The purpose of this paper is to investigate entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets.
The rest of this paper is organized as follows. The following section briefly reviews some basic notions and background on soft sets, fuzzy soft sets, intuitionistic fuzzy soft sets, and interval-valued fuzzy soft sets. In Section 3, we propose several distance measures for intuitionistic fuzzy soft sets and present the structure of entropy for intuitionistic fuzzy soft sets. In Section 4, we discuss the relationship between intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets and transform the structure of entropy for intuitionistic fuzzy soft sets to the interval-valued fuzzy soft sets. In Section 5, we review some previous work on soft sets and (extended) fuzzy soft sets. Finally, in Section 6, we draw the conclusion and present some topics for future research. 2. Preliminaries
In the current section we will briefly recall the notions of soft sets, fuzzy soft sets, intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets. See especially [42][43][44][45][47] [51][65] for further details and background.
First we recall the basic definitions of fuzzy sets. The theory of fuzzy sets, first developed by Zadeh in 1965 [68], provides an appropriate framework for representing and processing vague concepts by allowing partial memberships. Let U be a nonempty set, called universe. The family of all fuzzy sets of U is denoted by F(U).
A fuzzy set A on the universe U is described by its membership function A, which is a mapping A: U [0, 1]. A(x) essentially specifies the grade of element x, or the degree to which x U belongs to the fuzzy set A. A fuzzy set A can be completely characterized by the set of ordered pairs A={ x, A(x) | x U}. There are many different definitions for fuzzy set operations. With the min-max system proposed by Zadeh [14], fuzzy intersection, union, and complement are defined as follows:
A B(x) = min{ A(x), B(x)}; A B(x) = max{ A(x), B(x)}; A(x) = 1 A(x);
where A, B F(U) and x U. By A B, we mean that A(x) B(x) for all x U. Clearly A=B if both A B and B A, i.e., A(x)= B(x) for all x U.
Now we recall the notion of soft sets. Molodtsov [51] defined the soft set in the following way. Let U be an initial universe of
objects and E the set of parameters in relation to objects in U. Parameters are often attributes, characteristics, or properties of objects. Let P(U) denote the power set of U and A E.
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Definition 1. A pair F, A is called a soft set over U, where F is a mapping given by F: A P(U).
In other words, the soft set is not a kind of set in ordinary sense, but a parameterized family of subsets of the set U [45][51][65][66]. For any parameter A, F( ) U may be considered as the set of -approximate elements of the soft set F, A .
Maji et al. [43] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of (classical) soft sets. Maji et al. [43] proposed the concept of the fuzzy soft sets as follows.
Definition 2. Let F(U) be the set of all fuzzy subsets of U. Let E be a set of parameters and A E. A pair F, A is called a fuzzy soft set over U, where F is a mapping given by F: A F(U).
It is easy to see that every (classical) soft set may be considered as a fuzzy soft set [16]. Generally speaking, for any parameter A, F( ) is a fuzzy subset of U and it is called fuzzy value set of parameter . If for any parameter A, F( ) is a crisp subset of U, then F, A is degenerated to be the standard soft set. Let us denote F( )(x) by the membership degree that object x holds parameter where x U and A, then F( ) can be written as a fuzzy set such that F( )={ x, F( )(x) | x U}. Let FSS(U) denote the set of all fuzzy soft sets over U.
Let U be an initial universe and E be a set of parameters. Suppose that A, B E, F, A and G, B are two fuzzy soft sets, we say that F, A is a fuzzy soft subset of G, B if and only if (1) A B, and (2) A, F( ) is a fuzzy subset of G( ), that is, for all x U and A, F( )(x) G( )(x).
This relationship is denoted by F, A ⋐ G, B . Similarly, F, A is said to be a fuzzy soft superset of G, B , if G, B is a fuzzy soft subset of F, A . We denote it by F, A ⋑ G, B .
Let F, A and G, B be two fuzzy soft sets over a universe U, F, A and G, B are said to be fuzzy soft equal if and only if (1) F, A is a fuzzy soft subset of G, B , and (2) G, B is a fuzzy soft subset of F, A . We write F, A = G, B .
Before introduce the notion of the intuitionistic fuzzy soft sets, let us give the concept of intuitionistic fuzzy sets [4][5].
Let a set E be fixed. An intuitionistic fuzzy set or IFS in E is an object having the form A={ x, A(x), A(x) | x E}, where the functions A: E [0, 1] and A: E [0, 1] define the degree of membership and the degree of non-membership respectively of the element x ( E) to the set A. For any x E, 0 A(x)+ A(x) 1. For any intuitionistic fuzzy set A, we will call the intuitionistic index of the element x in the set A the following expression: A(x)=1 A(x) A(x). Clearly, 0 A(x) 1 for all x.
By introducing the concept of intuitionistic fuzzy sets into the theory of soft sets, Maji et al. [42][44][47] proposed the concept of the intuitionistic fuzzy soft sets as follows.
Definition 3. Consider U and E as a universe set and a set of parameters respectively. IF(U) denotes the set of all intuitionistic fuzzy sets of U. Let A E. A pair F, A is an intuitionistic fuzzy soft set over U, where F is a mapping given by F: A IF(U).
Generally speaking, for any parameter A, F( ) is an intuitionistic fuzzy subset of U and it is called intuitionistic fuzzy value set of parameter . Clearly, F( ) can be written as an intuitionistic fuzzy set such that F( )={ x, F( )(x), F( )(x) | x U}, where F( )(x) and F( )(x) be the membership and non-membership functions respectively. If x U, F( )(x)=1 F( )(x), then F( ) will degenerated to be a
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standard fuzzy set and then F, A will be degenerated to be a traditional fuzzy soft set. Let IFSS(U) denote the set of all intuitionistic fuzzy soft sets over U.
Suppose that A, B E, F, A and G, B are two intuitionistic fuzzy soft sets, we say that F, A is an intuitionistic fuzzy soft subset of G, B if and only if (1) A B, and (2) A, F( ) is an intuitionistic fuzzy subset of G( ), i.e., for all x U and A, F( )(x) G( )(x) and F( )(x) G( )(x).
This relationship is denoted by F, A ⋐ G, B . Similarly, F, A is said to be an intuitionistic fuzzy soft superset of G, B , if G, B is an intuitionistic fuzzy soft subset of F, A . We denote it by F, A ⋑ G, B .
Let F, A and G, B be two intuitionistic fuzzy soft sets over a universe U, F, A and G, B are said to be intuitionistic fuzzy soft equal if and only if (1) F, A is an intuitionistic fuzzy soft subset of G, B , and (2) G, B is an intuitionistic fuzzy soft subset of F, A . We write F, A = G, B .
In the following, we will introduce the notion of interval-valued fuzzy soft sets [65]. First, let us briefly introduce the concept of the interval-valued fuzzy sets [19].
An interval-valued fuzzy set X on a universe U is a mapping such that X: U Int([0, 1]), where Int([0, 1]) stands for the set of all closed subintervals of [0, 1], the set of all interval-valued fuzzy sets on U is denoted by IVF(U).
Suppose that X IVF(U), x U, X(x)=[ X (x), X+(x)] is called the degree of membership of an element x to X. X (x) and X+(x) are referred to as the lower and upper degrees of membership of x to X where 0 X (x) X+(x) 1. WX(x)= X+(x) X (x) will be the width.
Definition 4. Let U be an initial universe and E be a set of parameters. IVF(U) denotes the set of all interval-valued fuzzy sets of U. Let A E. A pair F, A is an interval-valued fuzzy soft set over U, where F is a mapping given by F: A IVF(U).
An interval-valued fuzzy soft set is a parameterized family of interval-valued fuzzy subsets of U, thus, its universe is the set of all interval-valued fuzzy sets of U, i.e., IVF(U). An interval-valued fuzzy soft set is also a special case of a soft set because it is still a mapping from parameters to IVF(U). For any parameter A, F( ) is referred as the interval fuzzy value set of parameter , it is actually an interval-valued fuzzy set of U where x U and A, it can be written as: F( )={ x, F( )(x) }, here, F( )(x) is the interval-valued fuzzy membership degree that object x holds on parameter . If x U, F( )
(x)= F( )+(x), then F( ) will be degenerated to be a standard fuzzy set and
then F, A will be degenerated to be a traditional fuzzy soft set. Let IVFSS(U) denote the set of all interval-valued fuzzy soft sets over U.
Suppose that A, B E, F, A and G, B are two interval-valued fuzzy soft sets, we say that F, A is an interval-valued fuzzy soft subset of G, B if and only if (1) A B, and (2) A, F( ) is an interval-valued fuzzy subset of G( ), i.e., for all x U and A, F( )
(x) G( ) (x) and F( )
+(x) G( )+(x).
This relationship is denoted by F, A ⋐ G, B . Similarly, F, A is said to be an interval-valued fuzzy soft superset of G, B , if G, B is an interval-valued fuzzy soft subset of F, A . We denote it by F, A ⋑ G, B .
Let F, A and G, B be two interval-valued fuzzy soft sets over a universe U, F, A and G, B are said to be interval-valued fuzzy soft equal if and only if (1) F, A is an interval-valued fuzzy soft subset of G, B , and (2) G, B is an interval-valued fuzzy soft subset of F, A . We write F, A = G, B .
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3. Entropy on intuitionistic fuzzy soft sets
In the current section we give the structure of entropy on intuitionistic fuzzy soft sets by extending the structure to entropy on intuitionistic fuzzy sets [9]. Concretely, we define the distance measures between intuitionistic fuzzy soft sets and we give an axiom definition of intuitionistic fuzzy soft entropy and a theorem which characterizes it.
3.1. Distance measures between intuitionistic fuzzy soft sets
At first, we introduce the axiom definition of distance measure between intuitionistic fuzzy soft sets.
Definition 5. Let = F, A and = G, A be two intuitionistic fuzzy soft sets over U, i.e., , IFSS(U). Let d be a mapping d: IFSS(U) IFSS(U) R+ {0} (where R+ {0} denotes the set of the non-negative real numbers). If d( , ) satisfies the following properties ((P1) (P4)):
(P1) d( , ) 0; (P2) d( , )=d( , ); (P3) d( , )=0 if and only if = ; (P4) for any = H, A IFSS(U), d( , )+d( , ) d( , ).
Then d( , ) is a distance measure between intuitionistic fuzzy soft sets and .
To propose the distance measures for intuitionistic fuzzy soft sets, we define intuitionistic fuzzy soft matrices which are representative of intuitionistic fuzzy soft sets. To express the concept of intuitionistic fuzzy soft matrix, we redefine the notion of intuitionistic fuzzy soft sets with a new style firstly.
Definition 6. Let IF(U) be the set of all intuitionistic fuzzy sets over U. Let E be a set of parameters and A E. An intuitionistic fuzzy soft set FA, E on U is defined by the set of ordered pair , FA( ) , where E, FA( ) IF(U). Formally, FA, E ={ , FA( ) | E, FA( ) IF(U)}, where FA: E
IF(U) such that FA( )= (i.e., )(εµAF (u)=0 and )(εγ
AF (u)=1 for any u U) if A.
Definition 7. Let FA, E be an intuitionistic fuzzy soft set over U. Then an intuitionistic fuzzy relation form of FA, E is defined by
RA={ u, , FA( )(u) | E, u U}={ u, , ( )(εµAF (u), )(εγ
AF (u)) | E, u U}.
If U={u1, u2, …, um}, E={ 1, 2, …, n} and A E, then the RA can be presented by a table as in the following form:
RA 1 2 … n
u1
u2 . . . um
( )( 1εµAF (u1), )( 1εγ
AF (u1)) ( )( 2εµAF (u1), )( 2εγ
AF (u1)) … ( )( nAF εµ (u1), )( nAF εγ (u1)) ( )( 1εµ
AF (u2), )( 1εγAF (u2)) ( )( 2εµ
AF (u2), )( 2εγAF (u2)) … ( )( nAF εµ (u2), )( nAF εγ (u2))
. . . . . . . . . ( )( 1εµ
AF (um), )( 1εγAF (um)) ( )( 2εµ
AF (um), )( 2εγAF (um)) … ( )( nAF εµ (um), )( nAF εγ (um))
If aij=( )( jAF εµ (ui), )( jAF εγ (ui)), we can define a matrix as follows:
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⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=×
mnmm
n
n
nmij
aaa
aaaaaa
a
...
......
][
21
22221
11211
MMM,
which is called an m n intuitionistic fuzzy soft matrix of the intuitionistic fuzzy soft set FA, E over U.
According to this definition, an intuitionistic fuzzy soft set FA, E is uniquely characterized by the matrix [aij]m n. It means that an intuitionistic fuzzy soft set FA, E is formally equal to its intuitionistic fuzzy soft matrix [aij]m n. Therefore, we will identify any intuitionistic fuzzy soft set with its intuitionistic fuzzy soft matrix and use these two concepts as interchangeable. In the following sometimes we shall delete the subscripts m n of [aij]m n, we use [aij] instead of [aij]m n. In fact, this is the same as that of soft set and soft matrix [10].
From Definitions 6 and 7, we know that for any intuitionistic fuzzy soft set F, A , we can obtain an intuitionistic fuzzy soft set FA, E (where E is a set of parameters and A E) and vice versa. To the ease of presentation, in the rest of this paper we only consider the intuitionistic fuzzy soft sets for all parameters E such as F, E and G, E .
To illustrate the idea of Definition 7, let us continue to consider the shirt example [48][49].
Example 1. Let U be a set of 3 shirts u1, u2, u3, i.e., U={u1, u2, u3}. Let E={ 1, 2, 3}, where 1=bright, 2=colorful, 3=fade.
Let = F, E be an intuitionistic fuzzy soft set over U defined as follows:
F( 1)(u1)=(0.6, 0.3), F( 2)(u1)=(0.8, 0.1), F( 3)(u1)=(0.5, 0.4), F( 1)(u2)=(0.5, 0.5), F( 2)(u2)=(0.7, 0.2), F( 3)(u2)=(0.6, 0.2), F( 1)(u3)=(0.8, 0.2), F( 2)(u3)=(0.6, 0.2), F( 3)(u3)=(0.7, 0.1).
Thus, according to Definition 7 we get an intuitionistic fuzzy soft matrix as follows:
[aij]=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)1.0,7.0()2.0,6.0()2.0,8.0()2.0,6.0()2.0,7.0()5.0,5.0()4.0,5.0()1.0,8.0()3.0,6.0(
.
Then with the above interpretation the intuitionistic fuzzy soft set F, E is represented by the matrix [aij]m n and we write F, E =[aij]m n. Let F, E =[aij]m n and G, E =[bij]m n be two intuitionistic fuzzy soft sets, clearly, F, E = G, E iff [aij]m n=[bij]m n.
Now we define some distance measures between two intuitionistic fuzzy soft sets = F, E and = G, E over U. Our methods are based on the distance measures of intuitionistic fuzzy sets proposed by Burillo and Bustince [9].
Definition 8. Let = F, E =[aij]m n and = G, E =[bij]m n be two intuitionistic fuzzy soft sets over U. We define the Hamming distance between and as follows:
∑∑= =
−+−=
n
j
m
i
iGiFiGiFHIFSS
uuuud jjjj
1 1
)()()()(
2
|)()(||)()(|),( εεεε γγµµ
ϖω .
We define the normalized Hamming distance between and by the following formula:
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mn
ddHIFSSnH
IFSS
),(),( ϖωϖω = .
The so-called Euclidean distance between and is defined as follows:
∑∑= =
−+−=
n
j
m
i
iGiFiGiFEIFSS
uuuud jjjj
1 1
2)()(
2)()(
2
))()(())()((),( εεεε γγµµ
ϖω .
The normalized Euclidean distance between and is as follows:
mn
ddEIFSSnE
IFSS
),(),( ϖωϖω = .
It is easy to verify that the aforementioned distances (Definition 8) satisfy the properties of distance (i.e., (P1) (P4) of Definition 5). It is obvious that we have the following properties:
0 ),( ϖωHIFSSd mn, 0 ),( ϖωnH
IFSSd 1, 0 ),( ϖωEIFSSd mn , and 0 ),( ϖωnE
IFSSd 1.
Example 2. Assume that an intuitionistic fuzzy soft set = F, E =[aij]3 3 is given as in Example 1. An intuitionistic fuzzy soft set = G, E =[bij]3 3 is given as follows:
[bij]=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)2.0,7.0()2.0,7.0()1.0,8.0()2.0,6.0()3.0,7.0()4.0,5.0()3.0,6.0()2.0,8.0()2.0,7.0(
.
By Definition 8, the Hamming distance between and is
∑∑= =
−+−=
3
1
3
1
)()()()(
2
|)()(||)()(|),(
j i
iGiFiGiFHIFSS
uuuud jjjj εεεε γγµµ
ϖω
=2
|)1.02.0||8.08.0(||)4.05.0||5.05.0(||)2.03.0||7.06.0(| −+−+−+−+−+−
+2
|)2.02.0||7.06.0(||)3.02.0||7.07.0(||)2.01.0||8.08.0(| −+−+−+−+−+−
+2
|)2.01.0||7.07.0(||)2.02.0||6.06.0(||)3.04.0||6.05.0(| −+−+−+−+−+−
=0.5.
Similarly, we have the following:
),( ϖωnHIFSSd =
33),(
×ϖωH
IFSSd =0.056.
∑∑= =
−+−=
3
1
3
1
2)()(
2)()(
2
))()(())()((),(
j i
iGiFiGiFEIFSS
uuuud jjjj εεεε γγµµ
ϖω =0.224.
33),(),(
×= ϖωϖω
EIFSSnE
IFSS
dd =0.075.
3.2. Entropy on intuitionistic fuzzy soft sets
At first, we introduce some new basic notions for intuitionistic fuzzy soft sets in order to define the entropy of intuitionistic fuzzy soft sets.
Definition 9. Let E be a set of parameters. Suppose that F, E and G, E are two
9
intuitionistic fuzzy soft sets over U, we say that F, A ≼ G, B if and only if for all x U and E,
F( )(x) G( )(x) and F( )(x) G( )(x).
Definition 10. Let E={e1, e2, …, en} be a parameter set. The not set of E denoted by ⌉E is defined by ⌉E={ e1, e2, …, en} where ei not ei. It should be noted that we assume that ei=ei (i.e., not not ei=ei) for any ei E.
Definition 11. The complement of an intuitionistic fuzzy soft set = F, E =[aij] is denoted by F, E C (or C, [aij]C) and is defined by F, E C= FC, ⌉E , where
FC: ⌉E IF(U) is a mapping given by FC( )={ x, ( F( )(x), F( )(x)) | x U}
={ x, ( F( )(x), F( )(x)) | x U} for all ⌉E.
Example 3. For Example 1, by Definition 11 we can compute FC( 1)(u1) as follows:
FC( 1)(u1)=( )( 1εγ ¬¬F (u1), )( 1εµ ¬¬F (u1))
=( )( 1εγ F (u1), )( 1εµF (u1))
=(0.3, 0.6).
Similarly, we have the following:
FC( 2)(u1)=(0.1, 0.8), FC( 3)(u1)=(0.4, 0.5), FC( 1)(u2)=(0.5, 0.5), FC( 2)(u2)=(0.2, 0.7), FC( 3)(u2)=(0.2, 0.6), FC( 1)(u3)=(0.2, 0.8), FC( 2)(u3)=(0.2, 0.6), and FC( 3)(u3)=(0.1, 0.7).
Therefore, we can obtain the following complement F, E C (i.e., FC, ⌉E ) of the intuitionistic
fuzzy soft set F, E :
FC( 1) = FC(not bright) = (0.3, 0.6)/u1+(0.5, 0.5)/u2+(0.2, 0.8)/u3, FC( 2) = FC(not colorful) = (0.1, 0.8)/u1+(0.2, 0.7)/u2+(0.2, 0.6)/u3, FC( 3) = FC(not fade) = (0.4, 0.5)/u1+(0.2, 0.6)/u2+(0.1, 0.7)/u3.
Furthermore, we get an intuitionistic fuzzy soft matrix of F, E C as follows:
[aij]C=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)7.0,1.0()6.0,2.0()8.0,2.0()6.0,2.0()7.0,2.0()5.0,5.0()5.0,4.0()8.0,1.0()6.0,3.0(
.
Definition 12. The union of two intuitionistic fuzzy soft sets F, E and G, E over U is an intuitionistic fuzzy soft set H, E , where E, x U,
H( )(x)=max{ F( )(x), G( )(x)} and H( )(x)=min{ F( )(x), G( )(x)}.
We denote it by F, E ⋒ G, E = H, E .
Definition 13. The intersection of two intuitionistic fuzzy soft sets F, E and G, E over U is an intuitionistic fuzzy soft set H, E , where E, x U,
H( )(x)=min{ F( )(x), G( )(x)} and H( )(x)=max{ F( )(x), G( )(x)}.
10
We denote it by F, E ⋓ G, E = H, E .
Definition 14. Let = F, E be any intuitionistic fuzzy soft set over U, i.e., IFSS(U). We call is completely intuitionistic if F( )(x)= F( )(x)=0, E, x U.
Definition 15. To every f [0, 1][0, 1] [0, 1] we can associate a mapping f of IFSS(U) in FSS(U), with [0, 1], given by
f : IFSS(U) FSS(U), F, E f ( F, E )= F , E , where F is defined as follows: let F( )={ x, F( )(x), F( )(x) | x U}, E, F ( )=f (F( ))
=f ({ x, F( )(x), F( )(x) | x U}) ={ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U}.
Obviously, this operator f is an extension of the operator f given by Burillo and Bustince in [9]. That is, this operator f defined in Definition 15 is to assign an intuitionistic fuzzy soft set to a fuzzy soft set, however, the operator f given in [9] is to assign an intuitionistic fuzzy set to a fuzzy set.
Example 4. For Example 1, let =0.5. By Definition 15, we can compute F ( 1) as follows:
F ( 1)=f (F( 1)) =f ({ u1, 0.6, 0.3 , u2, 0.5, 0.5 , u3, 0.8, 0.2 }) ={ u1, 0.6+0.5 (1 0.6 0.3), 1 0.6 0.5 (1 0.6 0.3) , u2, 0.5+0.5 (1 0.5 0.5),
1 0.5 0.5 (1 0.5 0.5) , u1, 0.8+0.5 (1 0.8 0.2), 1 0.8 0.5 (1 0.8 0.2) } ={ u1, 0.65, 0.35 , u2, 0.5, 0.5 , u3, 0.8, 0.2 }.
Similarly, we have the following:
F ( 2)=f (F( 2))=f ({ u1, 0.8, 0.1 , u2, 0.7, 0.2 , u3, 0.6, 0.2 }) ={ u1, 0.85, 0.15 , u2, 0.75, 0.25 , u3, 0.7, 0.3 };
F ( 3)=f (F( 3))=f ({ u1, 0.5, 0.4 , u2, 0.6, 0.2 , u3, 0.7, 0.1 }) ={ u1, 0.55, 0.45 , u2, 0.7, 0.3 , u3, 0.8, 0.2 }.
Thus, we get the following fuzzy soft matrix:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)20.0,80.0()30.0,70.0()20.0,80.0()30.0,70.0()25.0,75.0()50.0,50.0()45.0,55.0()15.0,85.0()35.0,65.0(
.
Clearly, a fuzzy soft set (resp., matrix) is a special case of an intuitionistic fuzzy soft set (resp., matrix). That is to say, in a fuzzy soft set (or matrix) we have )( jF εµ (ui)+ )( jF εγ (ui)=1 for
any j E and ui U; however, in an intuitionistic fuzzy soft set (or matrix) we have 0 )( jF εµ (ui)+
)( jF εγ (ui) 1 for any j E and ui U.
Theorem 1. For all , [0 ,1], , IFSS(U), we have the following properties:
(1) if , then f ( )⋐f ( ); (2) if ⋐ , then f ( )⋐f ( ); (3) f (f ( ))=f ( );
11
(4) (f ( C))C=f1 ( ).
Proof. Let = F, E =[aij]m n and = G, E =[bij]m n. Let f ( )=f ( F, E )= F , E , where F ( )=f (F( ))=f ({ x, F( )(x), F( )(x) | x U})={ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U}, E.
Let f ( )=f ( F, E )= F , E , where F ( )=f (F( ))=f ({ x, F( )(x), F( )(x) | x U})={ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U}, E.
(1) Obviously, E E. In the following, we will prove that for all x U and E, F ( )(x) F ( )(x). Since , then we have ( F( )(x)+ F( )(x)) ( F( )(x)+ F( )(x)). Since F ( )(x)= F( )(x)+ F( )(x) and
F ( )(x)= F( )(x)+ F( )(x), thus, for all x U and E, we have the following
F ( )(x) F ( )(x).
Therefore, f ( )⋐f ( ).
(2) Let f ( )=f ( G, E )= G , E , where G ( )=f (G( ))=f ({ x, G( )(x), G( )(x) | x U})={ x, G( )(x)+ G( )(x), 1 G( )(x) G( )(x) | x U}, E.
Obviously, E E. In the following, we will prove that for all x U and E, F ( )(x) G ( )(x). Since F( )(x)=1 F( )(x) F( )(x) and G( )(x)=1 G( )(x) G( )(x), then we have the following
F( )(x)+ F( )(x)= F( )(x)+ (1 F( )(x) F( )(x)) = +(1 ) F( )(x) F( )(x), and
G( )(x)+ G( )(x)= G( )(x)+ (1 G( )(x) G( )(x)) = +(1 ) G( )(x) G( )(x).
Since ⋐ , then we have F( )(x) G( )(x) and F( )(x) G( )(x) for all x U and A. Thus,
(1 ) F( )(x) (1 ) G( )(x) and F( )(x) G( )(x).
Hence, ( +(1 ) F( )(x) F( )(x)) ( +(1 ) G( )(x) G( )(x)). That is, we have the following
( F( )(x)+ F( )(x)) ( G( )(x)+ G( )(x)).
Since F ( )(x)= F( )(x)+ F( )(x) and G ( )(x)= G( )(x)+ G( )(x), thus, F ( )(x) G ( )(x). Therefore, f ( )⋐f ( ).
(3) Let f (f ( ))=f ( F , E )= (F ) , E , where (F ) ( )=f (F ( ))=f (f (F( ))), E. Obviously, E=E. In the following, we will prove f (f (F( )))=f (F( )).
Since f (F( ))={ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U}, then for any E we have the following
f (f (F( )))=f ({ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U}) ={ x, ( F( )(x)+ F( )(x))+ (1 ( F( )(x)+ F( )(x)) (1 F( )(x) F( )(x))), 1
(( F( )(x)+ F( )(x))+ (1 ( F( )(x)+ F( )(x)) (1 F( )(x) F( )(x)))) | x U} ={ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U} =f (F( )).
Therefore, f (f ( ))=f ( ).
(4) Let C= F, E C= FC, ⌉E , where FC( )={ x, F( )(x), F( )(x) | x U} for all ⌉E. Then we have the following
f ( C)=f ( F
C, ⌉E )= (FC) , ⌉E , where (FC) ( )=f (FC( ))=f ({ x, F( )(x), F( )(x) | x U}) ={ x,
12
F( )(x)+ (1 F( )(x) F( )(x)), 1 ( F( )(x)+ (1 F( )(x) F( )(x))) | x U}, ⌉E.
Thus, we have that (f ( C))C= (FC) , ⌉E C= ((FC) )
C, ⌉⌉E = ((FC) )C, E , where E,
((FC) )C( )={ x, 1 ( F( )(x)+ (1 F( )(x) F( )(x))), F( )(x)+ (1 F( )(x) F( )(x)) | x U}
={ x, 1 ( F( )(x)+ (1 F( )(x) F( )(x))), F( )(x)+ (1 F( )(x) F( )(x)) | x U} ={ x, 1 F( )(x) + F( )(x)+ F( )(x), F( )(x)+ F( )(x) F( )(x) | x U}
Since f1 ( )=f1 ( F, E )= F1 , E , where F1 ( )=f1 (F( ))=f1 ({ x, F( )(x), F( )(x) | x U})={ x, F( )(x)+(1 ) F( )(x), 1 F( )(x) (1 ) F( )(x) | x U}, E.
Obviously, E=E. In the following, we will prove ((FC) )C( )=F1 ( ).
Since F( )(x)+(1 ) F( )(x)= F( )(x)+(1 ) (1 F( )(x) F( )(x)) =1 F( )(x) + F( )(x)+ F( )(x), and 1 F( )(x) (1 ) F( )(x)=1 F( )(x) (1 ) (1 F( )(x) F( )(x)) = F( )(x)+ F( )(x) F( )(x),
Thus, ((FC) )C( )=F1 ( ).
Consequently, (f ( C))C=f1 ( ).
Let = F, E be an intuitionistic fuzzy soft set over U (i.e., IFSS(U)) and f be the operator given in Definition 15. The family of all fuzzy soft sets associated to by f will be denoted by {f ( )}={ } [0, 1].
One can verify the following property of { } [0, 1].
Theorem 2. For all = F, E IFSS(U), { } [0, 1], ⋐ is a totally ordered family of fuzzy soft sets.
Proof. Firstly, we prove { } [0, 1], ⋐ is a partial ordered family of fuzzy soft sets.
{ } [0, 1], =f ( )=f ( F, E )= F , E , where F ( )=f (F( ))=f ({ x, F( )(x), F( )(x) | x U})={ x, F( )(x)+ F( )(x), 1 F( )(x) F( )(x) | x U}, E.
Clearly, E E and ( F( )(x)+ F( )(x)) ( F( )(x)+ F( )(x)). Thus, ⋐ . That is, { } [0, 1], ⋐ satisfies reflexivity.
, { } [0, 1], if ⋐ and ⋐ , by the definition of fuzzy soft equal (see Section 2) we have = . Thus, { } [0, 1], ⋐ satisfies antisymmetry.
, , { } [0, 1], if ⋐ and ⋐ , by Definition 15, it is easy to verify ⋐ . Hence, { } [0, 1], ⋐ satisfies transitivity.
Therefore, { } [0, 1], ⋐ is a partial ordered family of fuzzy soft sets.
, { } [0, 1], without loss of generality, we assume that . By Property (1) of Theorem 1, we have ⋐ . That is, any two elements of { } [0, 1] are comparable.
Consequently, { } [0, 1], ⋐ is a totally ordered family of fuzzy soft sets.
Now we give the axiom definition of entropy of intuitionistic fuzzy soft sets. Intuitively, to measure the degree of fuzziness of an intuitionistic fuzzy soft set, the
conditions required for an intuitionistic entropy of an intuitionistic fuzzy soft set are as follows:
(i) will be null when the intuitionistic fuzzy soft set is a fuzzy soft set; (ii) will be maximum if the intuitionistic fuzzy soft set is completely intuitionistic;
13
(iii) the entropy of an intuitionistic fuzzy soft set will be equal to its complement; (iv) if the degree of membership and the degree of non-membership of each element increase,
the sum will do so as well, and therefore, this intuitionistic fuzzy soft set becomes more fuzzy, and therefore the entropy should decrease.
Taking into account the previous considerations, we give the following definition of intuitionistic entropy of an intuitionistic fuzzy soft set formally.
Definition 16. A real function I: IFSS(U) R+ is called an intuitionistic entropy (entropy for short) on IFSS(U), if I has the following properties:
(IP1) I( )=0 if and only if FSS(U); (IP2) let = F, E =[aij]m n, I( )=mn if and only if F( )(x)= F( )(x)=0, E, x U; (IP3) I( )=I( C) for all IFSS(U);
(IP4) if ≼ , then I( ) I( ), where = F, E and = G, E .
Clearly, Definition 16 (intuitionistic entropy of an intuitionistic fuzzy soft set) is an extension of intuitionistic entropy for an intuitionistic fuzzy set given in [9].
Similarly as the entropy of an intuitionistic fuzzy set [9], from (IP2) of Definition 16, we can deduce the following property.
Theorem 3. I( )=Maximum if and only if = F, E =[aij]m n=[0]m n, that is, aij=( F( j)(ui), F( j)(ui))=(0, 0), i.e., F( j)(ui)=0 and F( j)(ui)=0 for all j E, xi U, where 0 i m and 0 j n.
Proof. “ ”. Let = F, E =[0]m n. Let = G, E be any intuitionistic fuzzy soft set. Since G( j)(ui) 0
and G( j)(ui) 0 for all j E, xi U, where 0 i m and 0 j n, by Definition 9, we have that ≼ . Thus, I( ) I( )
by Property (IP4) of Definition 16 for all , then I( )=Maximum. “ ”. Let I( )=Maximum. We assume that = F, E [0]m n, then there is j E and xi U (where 0 i m
and 0 j n) such that F( j)(ui) 0 (or F( j)(ui) 0). We construct the following intuitionistic fuzzy soft set = G, E with G( j)(ui)= F( j)(ui)/2 and G( j)(ui)= F( j)(ui) for all j E and xi U, then by Definition 9, we
have that ≼ . Thus, by Property (IP4) of Definition 16 we have I( ) I( ) which contradicts the
hypothesis I( )=Maximum. Therefore, =[0]m n.
Our goal is give an expression which allows us to create entropies for intuitionistic fuzzy soft sets. It is the same as the approach to create entropies for intuitionistic fuzzy sets [9], let us take the following set given in [9]:
D={(x, y) [0, 1] [0, 1]| x+y 1}
and with it let us construct D: D [0, 1], which satisfies the following conditions:
(1) D(x, y)=1 if and only if x+y=1; (2) D(x, y)=0 if and only if x=0=y; (3) D(x, y)= D(y, x); (4) if x x and y y then D(x, y) D(x , y ).
From here we will use the following notation D= .
14
In the following, we will give an expression which allows us to construct different intuitionistic entropies for intuitionistic fuzzy soft sets by extending the expression of intuitionistic entropies for intuitionistic fuzzy sets given in [9].
Theorem 4. Let I: IFSS(U) R+ and = F, E =[aij]m n IFSS(U). If
I( )= ∑∑= =
−n
j
m
iiFiF uuΦ
jj1 1
)()( )))(),((1( εε γµ ,
where verifies the previous conditions (1) (4), then I is an intuitionistic entropy.
Proof. I( )=0 iff ∑∑= =
−n
j
m
iiFiF uuΦ
jj1 1
)()( )))(),((1( εε γµ =0 iff ( F( j)(ui), F( j)(ui))=1 for all j E, xi U
iff F( j)(ui)+ F( j)(ui))=1 for all j E, xi U iff belonging to FSS(U). Thus, I satisfies Property (IP1) of Definition 16.
I( )=mn iff ∑∑= =
−n
j
m
iiFiF uuΦ
jj1 1
)()( )))(),((1( εε γµ =mn iff ( F( j)(ui), F( j)(ui))=0 for all j E, xi U iff
F( j)(ui)= F( j)(ui)=0 for all j E, xi U. Hence, I satisfies Property (IP2) of Definition 16.
Since C= F, E C= FC, ⌉E where FC( )={ ui, F( j)(ui), F( j)(ui) | ui U} for all j ⌉E, then we have
I( )= ∑∑= =
−n
j
m
iiFiF uuΦ
jj1 1
)()( )))(),((1( εε γµ = ∑∑= =
−n
j
m
iiFiF uuΦ
jj1 1
)()( )))(),((1( εε µγ =I( C). Thus, I satisfies
Property (IP3) of Definition 16.
Let = G, E =[bij]m n. If ≼ , then we have F( j)(ui) G( j)(ui) and F( )(ui) G( )(ui) for all ui U and j E.
Thus, we have ( F( j)(ui), F( j)(ui)) ( G( j)(ui), G( j)(ui)). Thereby, 1 ( F( j)(ui), F( j)(ui)) 1 ( G( j)(ui),
G( j)(ui)) for all ui U and j E. Hence, ∑∑= =
−n
j
m
iiFiF uuΦ
jj1 1
)()( )))(),((1( εε γµ
∑∑= =
−n
j
m
iiGiG uuΦ
jj1 1
)()( )))(),((1( εε γµ . That is, I( ) I( ).Thus, I satisfies Property (IP4) of Definition 16.
Therefore, I is an intuitionistic entropy.
In [9], Burillo and Bustince give some expressions for intuitionistic entropy of intuitionistic fuzzy sets. By extending these expressions, we can obtain some examples for intuitionistic entropy of intuitionistic fuzzy soft sets.
Example 5. Let = F, E =[aij]m n IFSS(U). It is easy to verify the following expressions are the entropies of :
(1) I( )= ∑∑= =
+−n
j
m
iiFiF uu
jj1 1
)()( )))()((1( εε γµ ;
(2) I( )= ,...3,2,)))()((1(1 1
)()( =+−∑∑= =
nuun
j
m
i
niFiF jj εε γµ ;
(3) I( )= ∑∑= =
+−⋅+−n
j
m
i
uu
iFiFijFijF
jjeuu
1 1
))()((1
)()( )))()((1( )()( εε γµ
εε γµ ;
(4) I( )= ∑∑= =
+⋅+−n
j
m
iiFiFiFiF uuuu
jjjj1 1
)()()()( ))))()()(2/sin(())()((1( εεεε γµπγµ .
In the following, we will introduce a function from IFSS(U) to R+, which is an extension of
15
the I , -function from IFS(U) to R+ given in [9].
Definition 17. Let , : [0, 1] [0, 1] be such that if x+y 1, then (x)+ (y) 1, with x, y [0, 1]. We will call I , -function of the intuitionistic fuzzy soft set = F, E =[aij]m n IFSS(U) to R+
I , ( )=mn ∑∑= =
+n
j
m
iiFiF uu
jj1 1
)()( ))(('))(( εε γϕµϕ .
Obviously, 0 I , ( ) mn for all =[aij]m n belonging to IFSS(U). Towards the I , -function, we have the following property.
Theorem 5. For all = F, E =[aij]m n, = G, E =[bij]m n IFSS(U), we have the following
I , ( ⋒ )+I , ( ⋓ )=I , ( )+I , ( ).
Proof. By Definition 17, we have the following
I , ( ⋒ )=mn ∑∑= =
+n
j
m
iiGiFiGiF uuuu
jjjj1 1
)()()()( )})(),((min{')})(),((max{ εεεε γγϕµµϕ , and
I , ( ⋓ )= mn ∑∑= =
+n
j
m
iiGiFiGiF uuuu
jjjj1 1
)()()()( )})(),((max{')})(),((min{ εεεε γγϕµµϕ .
Thus, we have
I , ( ⋒ )+I , ( ⋓ )
=2 mn ∑∑= =
+n
j
m
iijFijGijF uuu
1 1)()()( ),((min{)})(),((max{ εεε µϕµµϕ +)}()( ijG uεµ ),((min{' )( iF u
jεγϕ
)})(),((max{')}( )()()( iGiFiG uuujjj εεε γγϕγ +
=mn ∑∑= =
+n
j
m
iiFiF uu
jj1 1
)()( ))(('))(( εε γϕµϕ +mn ∑∑= =
+n
j
m
iiGiG uu
jj1 1
)()( ))(('))(( εε γϕµϕ
= I , ( )+I , ( ).
Similarly as the I , -function from IFS(U) to R+ given in [9], there are I , -functions that are not entropies. For example,
I , ( )=mn ∑∑= =
+n
j
m
i
iFiF uujj
1 1
)()(
2
)()( εε γµ, where = F, E =[aij]m n IFSS(U),
is an I , -function. However, it is easy to verify it is not an intuitionistic entropy. On the other hand, there are entropies that are not I , -functions. For instance,
I , ( )=mn ∑∑= =
+n
j
m
iiFiF uu
jj1 1
2)()( ))()(( εε γµ , where = F, E =[aij]m n IFSS(U),
is an intuitionistic entropy. However, it is easy to verify it is not an I , -function. Of course, there are entropies that are also I , -functions. For example,
I , ( )=mn ∑∑= =
+−n
j
m
iiFiF uu
jj1 1
)()( ))()((1 εε γµ , where = F, E =[aij]m n IFSS(U).
Now we introduce a property.
Theorem 6 [9]. If : [0, 1] [0, 1] satisfies
16
(1) is increasing, (2) (x)=0 if and only if x=0, (3) (x)+ (y)=1 if and only if x+y=1,
then (x)+ (y) satisfies the conditions (1) (4) of the function defined previously.
We will denote the I , -functions such that = , as I , -function. The following theorem characterizes the intuitionistic entropies of intuitionistic fuzzy soft
sets in a general way.
Theorem 7. Let I: IFSS(U) R+, : [0, 1] [0, 1] and = F, E =[aij]m n IFSS(U). I is an
intuitionistic entropy and an I , -function if and only if I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ ,
where satisfies the conditions (1) (3) of Theorem 6.
Proof. It is similar to the proof of the theorem in [9] (see Page 313 of Reference [9]). “ ”. Let us consider : [0, 1] [0, 1] [0, 1] such that (x, y)= (x)+ (y) and let us take the
following set D={(x, y) [0, 1] [0, 1]| x+y 1}. We will restrict ourselves to the function D: D [0, 1], which we denote by to simplify the notation.
By Theorems 6 and 4, we have the following
I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ is an intuitionistic entropy.
Let , [0, 1] and + 1. In the following we will prove ( )+ ( ) 1. We construct the following intuitionistic fuzzy soft set:
[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)0,1(...)0,1(),(
)0,1(...)0,1(),()0,1(...)0,1(),(
βα
βαβα
MMM.
Thus, I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ
=mn m( ( )+ ( )) m(n 1)( (1)+ (0)).
By Property (3) of Theorem 6, we have that (1)+ (0)=1. Then we have the following
I( )=mn m( ( )+ ( )) m(n 1)=m(1 ( ( )+ ( ))).
Since I( ) is an intuitionistic entropy, then I( ) 0. Thus, m(1 ( ( )+ ( ))) 0. Then we have 1 ( ( )+ ( )) 0. Hence, ( )+ ( ) 1.
Therefore, I is an intuitionistic entropy and an I , -function. “ ”. Let I be an intuitionistic entropy and an I , -function. Since I is an I , -function, then I has
the following form
I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ .
In the following, we will prove that satisfies the conditions (1) (3) of Theorem 6.
(1) Let , , [0, 1]. We construct the following intuitionistic fuzzy soft sets:
17
= F, E =[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)0,0(...)0,0()0,(
)0,0(...)0,0()0,()0,0(...)0,0()0,(
α
αα
MMM, and
= G, E =[bij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)0,0(...)0,0()0,(
)0,0(...)0,0()0,()0,0(...)0,0()0,(
β
ββ
MMM.
Thus, I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ
=mn m( ( )+ (0)) m(n 1)( (0)+ (0)), and
I( )= ∑∑= =
+−n
j
m
iiG u
j1 1
)( ))(((1( εµϕ ))))(( )( iG ujεγϕ
=mn m( ( )+ (0)) m(n 1)( (0)+ (0)).
Since , then we know ≼ . Since I is an intuitionistic entropy, then I( ) I( ). Hence,
mn m( ( )+ (0)) m(n 1)( (0)+ (0)) mn m( ( )+ (0)) m(n 1)( (0)+ (0)).
Then, ( ) ( ). Thus, is increasing.
(2) Now we prove that ( )=0 if and only if =0. If =0, we construct the following intuitionistic fuzzy soft set:
= F, E =[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)0,0(...)0,0()0,0(
)0,0(...)0,0()0,0()0,0(...)0,0()0,0(
MMM.
Thus, I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ
=mn mn( (0)+ (0)).
By Property (IP2) of Definition 16, we know that I( )=mn. Thus, we have the following
mn mn( (0)+ (0))=mn.
Then, (0)+ (0)=0. Hence, (0)=0, i.e., ( )=0. If ( )=0, we construct the following intuitionistic fuzzy soft set:
= F, E =[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)0,0(...)0,0()0,(
)0,0(...)0,0()0,()0,0(...)0,0()0,(
α
αα
MMM.
Thus, I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ
=mn m( ( )+ (0)) m(n 1)( (0)+ (0)).
18
By the previous part we know that (0)=0. Since ( )=0, then we have that I( )=mn. By Property (IP2) of Definition 16, we know that =0.
(3) Now we prove that ( )+ ( )=1 if and only if + =1. If + =1, we take the following intuitionistic fuzzy soft set:
= F, E =[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
),(...),(),(
),(...),(),(),(...),(),(
βαβαβα
βαβαβαβαβαβα
MMM.
By Property (IP1) of Definition 16, we know that I( )=0. Thus,
I( )= ∑∑= =
+−n
j
m
i1 1
)((1( αϕ )))(βϕ =0.
Hence, ( )+ ( )=1. If ( )+ ( )=1, three things may occur:
(i) + <1. We construct the following intuitionistic fuzzy soft set:
= F, E =[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
)0,1(...)0,1(),(
)0,1(...)0,1(),()0,1(...)0,1(),(
βα
βαβα
MMM.
Thus, I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ
=mn m( ( )+ ( )) m(n 1)( (1)+ (0)).
Applying the previous part we have that (1)+ (0)=1. Then we have the following
I( )=mn m( ( )+ ( )) m(n 1)( (1)+ (0))=0.
By Property (IP1) of Definition 16, we know that FSS(U). Thus, + =1, contradicting the hypothesis.
(ii) + >1. We know that 1 +1 =2 ( + ) 1. Since +1 =1 and +1 =1, then we have that ( )+ (1 )=1 and ( )+ (1 )=1. Thus,
( )+ ( )+ (1 )+ (1 )=2. Then we have that (1 )+ (1 )=1. Now we construct the following intuitionistic fuzzy soft set:
= F, E =[aij]m n=⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−−−
)0,1(...)0,1()1,1(
)0,1(...)0,1()1,1()0,1(...)0,1()1,1(
βα
βαβα
MMM.
Thus, I( )= ∑∑= =
+−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(( )( iF ujεγϕ
=mn m( (1 )+ (1 )) m(n 1)( (1)+ (0))=0.
By Property (IP1) of Definition 16, we know that FSS(U). Thus, 1 +1 =1, i.e., + =1, contradicting the hypothesis.
19
Therefore, the only possibility is (iii) if ( )+ ( )=1, then we have that + =1.
Let = F, E =[aij]m n IFSS(U). Among all the possible intuitionistic entropies for intuitionistic fuzzy soft sets characterized in Example 5, we choose the following
I( )= ∑∑= =
+−n
j
m
iiFiF uu
jj1 1
)()( )))()((1( εε γµ = ∑∑= =
n
j
m
iiF u
j1 1
)( )(επ .
Clearly, the above expression for the entropy of an intuitionistic fuzzy soft set is an extension of the expression for the entropy of an intuitionistic fuzzy set given in [9].
Finally, we give entropies of intuitionistic fuzzy soft sets by using the distance of intuitionistic fuzzy soft sets (see Section 3.1).
Theorem 8. Let = F, E =[aij]m n IFSS(U) and {f ( )}={ } [0, 1] be the family of fuzzy soft sets associated to by the operator f defined in Definition 15. Then
(1) I( )=2 ),( αωωHIFSSd ;
(2) I( )= ),( 10 ωωHIFSSd ;
(3) ),( βα ωωHIFSSd =( ) I( ) with , [0, 1], .
Proof. (1) I( )= ∑∑= =
n
j
m
iiF u
j1 1
)( )(επ
=2 ∑∑= =
⋅−−−+⋅−−n
j
m
i
iFiFiFiFiFiF uuuuuujjjjjj
1 1
)()()()()()(
2
|)()1()()(||)()()(| εεεεεε παγγπαµµ
=2 ),( αωωHIFSSd .
(2) ),( 10 ωωHIFSSd =
∑∑= =
−−−−++−n
j
m
i
iFiFiFiFiFiF uuuuuujjjjjj
1 1
)()()()()()(
2
|))()(1()(1||))()(()(| εεεεεε πµµπµµ
= ∑∑= =
+n
j
m
i
iFiF uujj
1 1
)()(
2
|))(||))(| εε ππ= ∑∑
= =
n
j
m
iiF u
j1 1
)( )(επ =I( ).
(3) ),( βα ωωHIFSSd =
∑∑= =
⋅−−−⋅−−+⋅+−⋅+n
j
m
i
iFiFiFiFiFiFiFiF uuuuuuuujjjjjjjj
1 1
)()()()()()()()(
2|))()(1()()(1||))()(()()(| εεεεεεεε πβµπαµπβµπαµ
=( ) ∑∑= =
n
j
m
iiF u
j1 1
)( )(επ =( ) I( ).
Observing the consequences of Theorem 8, it is deduced that the intuitionistic entropy for an intuitionistic fuzzy soft set has the same form as the one given by Burillo and Bustince [9] to define the intuitionistic entropy for an intuitionistic fuzzy set. 4. Entropy on interval-valued fuzzy soft sets
In this section we give the structure of entropy on interval-valued fuzzy soft sets. At first, we discuss the relationship between intuitionistic fuzzy soft sets [42][44][47] and interval-valued fuzzy soft sets [65].
It is well-known that Atanassov and Gargov [6] using
(a) the map f assigns to every interval-valued fuzzy set A (i.e, A IVF(U)) an intuitionistic fuzzy set B (i.e., B IF(U)), B=f(A) given by
20
B(x)= A (x) and B(x)=1 A+(x).
(b) the map g assigns to every intuitionistic fuzzy set B (i.e, B IF(U)) an interval-valued fuzzy set A (i.e., A IVF(U)), A=g(B) given by
A(x)=[ B(x), 1 B(x)].
proved that IF(U) and IVF(U) are equipollent generalizations of the notion of F(U) [9][12].
Similarly, we can prove that IFSS(U) and IVFSS(U) are equipollent generalizations of the notion of FSS(U) using the following mappings:
(a) the map f assigns to every interval-valued fuzzy soft set = F, E (i.e, IVFSS(U)) an intuitionistic fuzzy soft set = G, E (i.e., IFSS(U)), =f( ) given by
G( )(x)= F( ) (x) and G( )(x)=1 F( )
+(x) for all x U and E.
(b) the map g assigns to every intuitionistic fuzzy soft set = G, E (i.e, IFSS(U)) an interval-valued fuzzy soft set = F, E (i.e., IVFSS(U)), =g( ) given by
F( )(x)=[ G( )(x), 1 G( )(x)] for all x U and E.
Therefore, we can give the structure of entropy on interval-valued fuzzy soft sets by transforming the structure of entropy on intuitionistic fuzzy soft sets presented in Section 3.
Now we introduce some new basic notions for interval-valued fuzzy soft sets in order to give the structure of entropy on interval-valued fuzzy soft sets.
Definition 18. Let E be a set of parameters. Suppose that F, E and G, E are two
interval-valued fuzzy soft sets over U, we say that F, A ≼ G, B if and only if for all x U and E,
F( ) (x) G( )
(x) and F( )+(x) G( )
+(x).
Example 6. Let = F, E and = G, E be two interval-valued fuzzy soft sets over U as follows, where U={u1, u2, u3}, E={ 1, 2, 3}:
F( 1)(u1)=[0.5, 0.9], F( 2)(u1)=[0.7, 0.9], F( 3)(u1)=[0.5, 0.7], F( 1)(u2)=[0.4, 0.6], F( 2)(u2)=[0.6, 0.9], F( 3)(u2)=[0.6, 0.8], F( 1)(u3)=[0.7, 0.8], F( 2)(u3)=[0.5, 0.8], F( 3)(u3)=[0.6, 0.9].
G( 1)(u1)=[0.6, 0.7], G( 2)(u1)=[0.8, 0.9], G( 3)(u1)=[0.5, 0.6], G( 1)(u2)=[0.5, 0.5], G( 2)(u2)=[0.7, 0.8], G( 3)(u2)=[0.6, 0.8], G( 1)(u3)=[0.8, 0.8], G( 2)(u3)=[0.6, 0.8], G( 3)(u3)=[0.7, 0.9].
By Definition 18, we have ≼ .
Obviously, it is similar to the definition of intuitionistic fuzzy soft matrix (see Definition 7), we also can define an m n interval-valued fuzzy soft matrix of an interval-valued fuzzy soft set. The following is an example.
Example 7. Assume that two interval-valued fuzzy soft sets = F, E and = G, E are defined as in Example 6. Then we can get the interval-valued fuzzy soft matrices of and are as follows,
21
respectively:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
]9.0,6.0[]8.0,5.0[]8.0,7.0[]8.0,6.0[]9.0,6.0[]6.0,4.0[]7.0,5.0[]9.0,7.0[]9.0,5.0[
and ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
]9.0,7.0[]8.0,6.0[]8.0,8.0[]8.0,6.0[]8.0,7.0[]5.0,5.0[]6.0,5.0[]9.0,8.0[]7.0,6.0[
.
Definition 19. The complement of an interval-valued fuzzy soft set = F, E =[aij] (where aij= )( jF εµ (ui)=[ )( jF εµ (ui), )( jF εµ +(ui)]) is denoted by F, E C (or C, [aij]C) and is defined by F,
E C= FC, ⌉E , where FC: ⌉E IVF(U) is a mapping given by
FC( )={ x, [1 F( )+(x), 1 F( )
(x)] | x U} ={ x, [1 F( )
+(x), 1 F( ) (x)] | x U} for all ⌉E.
Example 8. For Example 6, by Definition 19 we can compute FC( 1)(u1) as follows:
FC( 1)(u1)=[1 )( 1εµ ¬¬F+(u1), 1 )( 1εµ ¬¬F
(u1)]
=[1 )( 1εµF+(u1), 1 )( 1εµF
(u1)]
=[0.1, 0.5].
Similarly, we have the following:
FC( 2)(u1)=[0.1, 0.3], FC( 3)(u1)=[0.3, 0.5], FC( 1)(u2)=[0.4, 0.6], FC( 2)(u2)=[0.1, 0.4], FC( 3)(u2)=[0.2, 0.4], FC( 1)(u3)=[0.2, 0.3], FC( 2)(u3)=[0.2, 0.5], and FC( 3)(u3)=[0.1, 0.4].
Therefore, we get an interval-valued fuzzy soft matrix of F, E C as follows:
[aij]C=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
]4.0,1.0[]5.0,2.0[]3.0,2.0[]4.0,2.0[]4.0,1.0[]6.0,4.0[]5.0,3.0[]3.0,1.0[]5.0,1.0[
.
Definition 20. To every K [0, 1]D[0, 1] (where D[0, 1] denotes the set of all the closed subintervals of the interval [0, 1]) we can associate a mapping K of IVFSS(U) in FSS(U), with [0, 1], given by
K : IFSS(U) FSS(U), F, E K ( F, E )= F , E , where F is defined as follows: let F( )={ x, [ F( )
(x), F( )+(x)] | x U},
E, F ( )=K (F( ))
=K ({ x, [ F( ) (x), F( )
+(x)] | x U}) ={ x, F( )
(x)+ WF( )(x), 1 F( ) (x) WF( )(x) | x U}.
Example 9. For Example 6, let =0.5. By Definition 20, we can compute F ( 1) as follows:
F ( 1)=K (F( 1)) =K ({ u1, [0.5, 0.9] , u2, [0.4, 0.6] , u3, [0.7, 0.8] }) ={ u1, 0.5+0.5 (0.9 0.5), 1 0.5 0.5 (0.9 0.5) , u2, 0.4+0.5 (0.6 0.4), 1 0.4
0.5 (0.6 0.4) , u1, 0.7+0.5 (0.8 0.7), 1 0.7 0.5 (0.8 0.7) } ={ u1, 0.7, 0.3 , u2, 0.5, 0.5 , u3, 0.75, 0.25 }.
22
Similarly, we have the following:
F ( 2)=K (F( 2))=K ({ u1, [0.7, 0.9] , u2, [0.6, 0.9] , u3, [0.5, 0.8] }) ={ u1, 0.8, 0.2 , u2, 0.75, 0.25 , u3, 0.65, 0.35 };
F ( 3)=K (F( 3))=K ({ u1, [0.5, 0.7] , u2, [0.6, 0.8] , u3, [0.6, 0.9] }) ={ u1, 0.6, 0.4 , u2, 0.7, 0.3 , u3, 0.75, 0.25 }.
Thus, we get the following fuzzy soft matrix:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)25.0,75.0()35.0,65.0()25.0,75.0()30.0,70.0()25.0,75.0()50.0,50.0()40.0,60.0()20.0,80.0()30.0,70.0(
.
We may define some distance measures between two interval-valued fuzzy soft sets = F, E and = G, E over U. For example, the Hamming distance and the normalized Hamming distance are defined as follows.
Definition 21. Let = F, E =[aij]m n and = G, E =[bij]m n be two interval-valued fuzzy soft sets over U. We define the Hamming distance between and as follows:
∑∑= =
++−− −+−=
n
j
m
i
iGiFiGiFHIVFSS
uuuud jjjj
1 1
)()()()(
2
|)()(||)()(|),( εεεε µµµµ
ϖω .
We define the normalized Hamming distance between and by the following formula:
mn
ddHIVFSSnH
IVFSS
),(),( ϖωϖω = .
Example 10. Assume that two interval-valued fuzzy soft sets = F, E and = G, E are defined as in Example 6.
By Definition 21, the Hamming distance between and is
∑ ∑= =
++−− −+−=
3
1
3
1
)()()()(
2
|)()(||)()(|),(
j i
ijGijFijGijFHIVFSS
uuuud
εεεε µµµµϖω
=2
|)8.08.0||8.07.0(||)5.06.0||5.04.0(||)7.09.0||6.05.0(| −+−+−+−+−+−+
2|)8.08.0||6.05.0(||)8.09.0||7.06.0(||)9.09.0||8.07.0(| −+−+−+−+−+−
+
2|)9.09.0||7.06.0(||)8.08.0||6.06.0(||)6.07.0||5.05.0(| −+−+−+−+−+−
=0.6.
The normalized Hamming distance between and is as follows:
33),(),(
×= ϖωϖω
HIVFSSnH
IVFSSdd =
96.0
=0.067.
By using the relationship between intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets, we can obtain the following definition and theorem which characterizes the entropies for interval-valued fuzzy soft sets.
23
Definition 22. Let , : [0, 1] [0, 1] be such that if x+y 1, then (x)+ (y) 1, with x, y [0, 1]. We will call I , -function of the interval-valued fuzzy soft set = F, E =[aij]m n IVFSS(U) to R+
I , ( )=mn ∑∑= =
+− −+n
j
m
iiFiF uu
jj1 1
)()( ))(1('))(( εε µϕµϕ .
Theorem 9. Let I: IVFSS(U) R+, : [0, 1] [0, 1] and = F, E =[aij]m n IVFSS(U). I is an entropy
and an I , -function if and only if I( )= ∑∑= =
− +−n
j
m
iiF u
j1 1
)( ))(((1( εµϕ ))))(1( )( iF uj
+− εµϕ , where satisfies
the conditions (1) (3) of Theorem 6.
Let = F, E =[aij]m n IVFSS(U). We may give entropy of an interval-valued fuzzy soft set as follows:
I( )= ∑∑= =
+− −+−n
j
m
iiFiF uu
jj1 1
)()( )))(1)((1( εε µµ = ∑∑= =
n
j
m
iiF uW
j1 1
)( )(ε .
Similarly as intuitionistic fuzzy soft sets, we also may give entropies of interval-valued fuzzy soft sets by using the distance of interval-valued fuzzy soft sets as follows.
Theorem 10. Let = F, E =[aij]m n IVFSS(U) and {K ( )} [0, 1] be the family of fuzzy soft sets associated to by the operator K defined in Definition 20. Then
(1) I( )=2 ))(,( ωω αKd HIVFSS ;
(2) I( )= ))(),(( 10 ωω KKd HIVFSS ;
(3) ))(),(( ωω βα KKd HIVFSS =( ) I( ) with , [0, 1], .
Remark. The entropies presented in this paper describe the fuzziness degree of intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets. Naturally, we should consider some real-life applications to demonstrate the usefulness of this proposed new entropies. It is well-known that entropy can be used to estimate similarity for various fuzzy sets [38][60], thus, we will investigate the relationship between similarity measure and entropy of intuitionistic fuzzy soft sets and use the similarity measure to solve real-life problems such as pattern recognition and decision making. 5. Related work
The work described in this paper covers soft sets that have been studied in the literature. In this section, we will roughly discuss these related works.
The concept and basic properties of soft set theory were presented in [45][51]. Concretely, Maji et al. [45] introduced several algebraic operations in soft set theory and published a detail theoretical study on soft sets. Based on the analysis of several operations on soft sets introduced in [45], Ali et al. [3] presented some new algebraic operations for soft sets and proved that certain De Morgan’s laws hold in soft set theory w.r.t. these new definitions. In order to extend the expressive power of soft sets, Jiang et al. [23] used the concepts of Description Logics (DLs) [26] to act as the parameters of soft sets. That is, an extended soft set theory based on DLs was presented in [23]. Aktas and Cagman [2] introduced the basic properties of soft sets, compared soft sets to the related concepts of fuzzy sets and rough sets and gave a definition of soft groups. Acar et al. [1] introduced the concepts of soft rings. Feng et al. [17] defined soft semirings and several related notions to establish a connection between soft sets and semirings. Jun et al. [28] applied the notion of soft sets to ordered semigroups and introduced the notions of (trivial, whole) soft ordered
24
semigroup, soft ordered subsemigroup, soft left (right) ideal, and left (right) idealistic soft ordered semigroup. Jun [27] applied soft sets to the theory of BCK/BCI-algebras, and introduced the concept of soft BCK/BCI-algebras. Jun and Park [32] and Jun et al. [30][31] reported the applications of soft sets in ideal theory of BCK/BCI-algebras and d-algebras. Zhan and Jun [70] introduced the notions of soft BL-algebras based on fuzzy sets. Xiao et al. [61] proposed the notion of exclusive disjunctive soft sets and studied some of its operations. Qin and Hong [54] deal with the algebraic structure of soft sets. Majumdar and Samanta [49] proposed two types of similarity measure between soft sets and made a comparative study of these two techniques. Kharal [35] introduced set operations based measures for soft sets and presented some properties of the new measures. Moreover, the new similarity measures were applied to the problem of financial diagnosis of firms.
Maji et al. [43] initiated the study on hybrid structures involving both fuzzy sets and soft sets. In [43] the notion of fuzzy soft sets was introduced as a fuzzy generalization of classical soft sets and some basic properties were discussed in detail. Afterwards, many researchers have worked on this concept.
Aygunoglu and Aygun [7] introduced the concept of fuzzy soft groups. Jun et al. [29] applied fuzzy soft sets to deal with several kinds of theories in BCK/BCI-algebras. The notions of fuzzy soft BCK/BCI-algebras, (closed) fuzzy soft ideals and fuzzy soft p-ideals were introduced [29]. Feng et al. [18] investigated the problem of combining soft sets with fuzzy sets and rough sets. In general, three different types of hybrid models were presented, which were called rough soft sets, soft rough sets and soft rough fuzzy sets, respectively. Maji et al. [42][44][47] and Xu et al. [63] extended (classical) soft sets to intuitionistic fuzzy soft sets and vague soft sets, respectively. Yang et al. [65] presented the concept of the interval-valued fuzzy soft sets by combining the interval-valued fuzzy set and soft set models. Jiang et al. [24] combined the interval-valued intuitionistic fuzzy sets and soft sets, from which a new soft set model, i.e., interval-valued intuitionistic fuzzy soft set theory, was obtained. Majumdar and Samanta [48] further generalized the concept of fuzzy soft sets as introduced by Maji et al. [43], that is, the concept of generalised fuzzy soft sets was presented. Moreover, they studied the similarity between two generalised fuzzy soft sets, and an application of this similarity measure in medical diagnosis was shown. In this paper, we investigated entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets. To the best of our knowledge, this is the first attempt in this direction. 6. Conclusion
In this paper, we investigate entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets. Concretely, we define the distance measures between intuitionistic fuzzy soft sets and give an axiom definition of intuitionistic entropy for an intuitionistic fuzzy soft set and a theorem which characterizes it. Furthermore, we discuss the relationship between intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets and transform the structure of entropy for intuitionistic fuzzy soft sets to the interval-valued fuzzy soft sets. Our work in this paper is completely theoretical.
As far as future directions are concerned, these will include the entropy on interval-valued intuitionistic fuzzy soft sets. It is also desirable to further explore the applications of using the entropy of intuitionistic fuzzy soft sets, in particular, we will investigate the relationship between similarity measure and entropy of intuitionistic fuzzy soft sets.
25
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments as well as helpful suggestions from Associate Editor and Editor-in-Chief which greatly improved the exposition of the paper. The works described in this paper are supported by The National Natural Science Foundation of China under Grant Nos. 61272066 and 61272067; The Natural Science Foundation of Guangdong Province of China under Grant Nos. S2012030006242 and 10151063101000031; The Program for New Century Excellent Talents in University in China. References
[1] U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Computers & Mathematics with Applications 59 (11)
(2010) 3458-3463.
[2] H. Aktas, N. Cagman, Soft sets and soft groups, Information Sciences 177 (13) (2007) 2726-2735.
[3] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Computers &
Mathematics with Applications 57 (9) (2009) 1547-1553.
[4] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1) (1986) 87-96.
[5] K. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg/New York, 1999.
[6] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (3) (1989)
343-349.
[7] A. Aygunoglu, H. Aygun, Introduction to fuzzy soft groups, Computers & Mathematics with Applications 58
(6) (2009) 1279-1286.
[8] K. V. Babitha, J. J. Sunil, Transitive closures and orderings on soft sets, Computers & Mathematics with
Applications 62 (5) (2011) 2235-2239.
[9] P. Burillo, H. Bustince, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and
Systems 78 (3) (1996) 305-316.
[10] N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Computers & Mathematics with
Applications 59 (10) (2010) 3308-3314.
[11] N. Cagman, S. Enginoglu, Soft set theory and uni–int decision making, European Journal of Operational
Research 207 (2) (2010) 848-855.
[12] G. Deschrijver, E. E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and
Systems 133 (2) (2003) 227-235.
[13] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk, H. Prade. Terminological difficulties in fuzzy set theory The
case of “Intuitionistic Fuzzy Sets”, Fuzzy Sets and Systems 156 (3) (2005) 485-491.
[14] D. Dubois, H. Prade, Fuzzy Set and Systems: Theory and Applications, Academic Press, New York, 1980.
[15] J. L. Fan, Y. L. Ma, Some new fuzzy entropy formulas, Fuzzy Sets and Systems 128 (2) (2002) 277-284.
[16] F. Feng, Y. B. Jun, X. Liu, L. Li, An adjustable approach to fuzzy soft set based decision making, Journal of
Computational and Applied Mathematics 234 (1) (2010) 10-20.
[17] F. Feng, Y. B. Jun, X. Zhao, Soft semirings, Computers & Mathematics with Applications 56 (10) (2008)
2621-2628.
[18] F. Feng, C. Li, B. Davvaz, M. I. Ali , Soft sets combined with fuzzy sets and rough sets: A tentative approach,
Soft Computing 14 (9) (2010) 899-911.
[19] M. B. Gorzalczany, A method of inference in approximate reasoning based on interval valued fuzzy sets,
Fuzzy Sets and Systems 21 (1) (1987) 1-17.
26
[20] M. Higashi, G. J. Klir, On measure of fuzziness and fuzzy complements, International Journal of General
Systems 8 (3) (1982) 169-180.
[21] W. L. Hung, M. S. Yang, Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent
Systems 21 (4) (2006) 443-451.
[22] Y. Jiang, Y. Tang, Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making,
Applied Mathematical Modelling 35 (2) (2011) 824-836.
[23] Y. Jiang, Y. Tang, Q. Chen, J. Wang, S. Tang, Extending soft sets with description logics, Computers &
Mathematics with Applications 59 (6) (2010) 2087-2096.
[24] Y. Jiang, Y. Tang, Q. Chen, H. Liu, J. Tang, Interval-valued intuitionistic fuzzy soft sets and their properties,
Computers & Mathematics with Applications 60 (3) (2010) 906-918.
[25] Y. Jiang, Y. Tang, Q. Chen, H. Liu, J. Tang, Extending fuzzy soft sets with fuzzy description logics,
Knowledge-Based Systems 24 (7) (2011) 1096-1107.
[26] Y. Jiang, Y. Tang, J. Wang, P. Deng, S. Tang, Expressive fuzzy description logics over lattices,
Knowledge-Based Systems 23 (2) (2010) 150-161.
[27] Y. B. Jun, Soft BCK/BCI-algebras, Computers & Mathematics with Applications 56 (5) (2008) 1408-1413.
[28] Y. B. Jun, K. J. Lee, A. Khan, Soft ordered semigroups, Mathematical Logic Quarterly 56 (1) (2010) 42-50.
[29] Y. B. Jun, K. J. Lee, C. H. Park, Fuzzy soft set theory applied to BCK/BCI-algebras, Computers &
Mathematics with Applications 59 (9) (2010) 3180-3192.
[30] Y. B. Jun, K. J. Lee, C. H. Park, Soft set theory applied to ideals in d-algebras, Computers & Mathematics
with Applications 57 (3) (2009) 367-378.
[31] Y. B. Jun, K. J. Lee, J. Zhan, Soft p-ideals of soft BCI-algebras, Computers & Mathematics with
Applications 58 (10) (2009) 2060-2068.
[32] Y. B. Jun, C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Information Sciences
178 (11) (2008) 2466-2475.
[33] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press, New York, 1975.
[34] S. J. Kalayathankal, G. S. Singh, A fuzzy soft flood alarm model, Mathematics and Computers in Simulation
80 (5) (2010) 887-893.
[35] A. Kharal, Distance and similarity measures for soft sets, New Mathematics and Natural Computation 6 (3)
(2010) 312-334.
[36] Z. Kong, L. Gao, L. Wang, Comment on “A fuzzy soft set theoretic approach to decision making problems”,
Journal of Computational and Applied Mathematics 223 (2) (2009) 540-542.
[37] D. V. Kovkov , V. M. Kolbanov, D. A. Molodtsov, Soft sets theory-based optimization, Journal of Computer
and Systems Sciences International 46 (6) (2007) 872-880.
[38] J. Li, G. Deng, H. Li, W. Zeng, The relationship between similarity measure and entropy of intuitionistic
fuzzy sets, Information Sciences 188 (1) (2012) 314-321.
[39] X. Liu, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and
Systems 52 (3) (1992) 305-318.
[40] S. G. Loo, Measures of fuzziness, Cybernetica 20 (3) (1977) 201-210.
[41] A. D. Luca, S. Termini, A definition of a nonprobability entropy in the setting of fuzzy sets theory,
Information and Control 20 (1972) 301-312.
[42] P. K. Maji, More on intuitionistic fuzzy soft sets, in: H. Sakai, M. K. Chakraborty, A. E. Hassanien, D.
Slezak, W. Zhu (Eds.), Proceedings of the 12th International Conference on Rough Sets, Fuzzy Sets, Data
Mining and Granular Computing (RSFDGrC 2009), Lecture Notes in Computer Science, vol. 5908, Springer,
2009, pp. 231-240.
27
[43] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, Journal of Fuzzy Mathematics 9 (3) (2001) 589-602.
[44] P. K. Maji, R. Biswas, A. R. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9 (3) (2001)
677-692.
[45] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Computers & Mathematics with Applications 45 (4-5)
(2003) 555-562.
[46] P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Computers &
Mathematics with Applications 44 (8-9) (2002) 1077-1083.
[47] P. K. Maji, A. R. Roy, R. Biswas, On intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 12 (3)
(2004) 669-683.
[48] P. Majumdar, S. K. Samanta, Generalised fuzzy soft sets, Computers & Mathematics with Applications 59 (4)
(2010) 1425-1432.
[49] P. Majumdar, S. K. Samanta, Similarity measure of soft sets, New Mathematics and Natural Computation 4
(1) (2008) 1-12.
[50] J. S. Mi, Y. Leung, W. Z. Wu, An uncertainty measure in partition-based fuzzy rough sets, International
Journal of General Systems 34 (1) (2005) 77-90.
[51] D. Molodtsov, Soft set theory—First results, Computers & Mathematics with Applications 37 (4-5) (1999)
19-31.
[52] M. M. Mushrif, S. Sengupta, A. K. Ray, Texture classification using a novel, soft set theory based
classification algorithm, in: P. J. Narayanan S. K. Nayar, H. Y. Shum (Eds.), Proceedings of the 7th Asian
Conference on Computer Vision, Lecture Notes in Computer Science, vol. 3851, Spriger, 2006, pp. 246-254.
[53] J. H. Park, O. H. Kim, Y. C. Kwun, Some properties of equivalence soft set relations, Computers &
Mathematics with Applications 63 (6) (2012) 1079-1088.
[54] K. Qin, Z. Hong, On soft equality, Journal of Computational and Applied Mathematics 234 (5) (2010)
1347-1355.
[55] A. R. Roy, P. K. Maji, A fuzzy soft set theoretic approach to decision making problems, Journal of
Computational and Applied Mathematics 203 (2) (2007) 412-418.
[56] E. Szmidt, J. Kacprzyk, Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems 118 (3) (2001)
467-477.
[57] E. Trillas, T. Riera, Entropies in finite fuzzy sets, Information Sciences 15 (2) (1978) 159-168.
[58] I. B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (2) (1986)
191-210.
[59] I. K. Vlachos, G. D. Sergiadis, Subsethood, entropy, and cardinality for interval-valued fuzzy sets—An
algebraic derivation, Fuzzy Sets and Systems 158 (12) (2007) 1384-1396.
[60] C. Wei, P. Wang, Y. Zhang, Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their
applications, Information Sciences 181 (19) (2011) 4273-4286.
[61] Z. Xiao, K. Gong, S. Xia, Y. Zou, Exclusive disjunctive soft sets, Computers & Mathematics with
Applications 59 (6) (2010) 2128-2137.
[62] Z. Xiao, K. Gong, Y. Zou, A combined forecasting approach based on fuzzy soft sets, Journal of
Computational and Applied Mathematics 228 (1) (2009) 326-333.
[63] W. Xu, J. Ma, S. Wang, G. Hao, Vague soft sets and their properties, Computers & Mathematics with
Applications 59 (2) (2010) 787-794.
[64] H. Yang, Z. Guo, Kernels and closures of soft set relations, and soft set relation mappings, Computers &
Mathematics with Applications, 61 (3) (2011) 651-662.
[65] X. B. Yang, T. Y. Lin, J. Y. Yang, Y. Li, D. Yu, Combination of interval-valued fuzzy set and soft set,
28
Computers & Mathematics with Applications 58 (3) (2009) 521-527.
[66] X. B. Yang, D. J. Yu, J. Y. Yang, C. Wu, Generalization of soft set theory: From crisp to fuzzy case, in: B. Y.
Cao (Ed.), Proceeding of the Second International Conference on Fuzzy Information and Engineering,
Advance on Soft Computing, vol. 40, Springer-Verlag, 2007, pp. 345-354.
[67] L. A. Zadeh, Fuzzy sets and systems, in: Proceedings of the Symposium on Systems Theory, Polytechnic
Institute of Brooklyn, NY, 1965, pp. 29-37.
[68] L. A. Zadeh, Fuzzy sets, Information and Control 8 (3) (1965) 338-353.
[69] W. Zeng, H. Li, Relationship between similarity measure and entropy of interval valued fuzzy sets, Fuzzy
Sets and Systems 157 (11) (2006) 1477-1484.
[70] J. Zhan, Y. B. Jun, Soft BL-algebras based on fuzzy sets, Computers & Mathematics with Applications 59 (6)
(2010) 2037-2046.
[71] H. Zhang, W. Zhang, C. Mei, Entropy of interval-valued fuzzy sets based on distance and its relationship
with similarity measure, Knowledge-Based Systems 22 (6) (2009) 449-454.
[72] Y. Zou, Z. Xiao, Data analysis approaches of soft sets under incomplete information, Knowledge-Based
Systems 21 (8) (2008) 941-945.