The article provides a definition and properties of inclusion degree in type-two intuitionistic fuzzy sets and, on the basis of inclusion degree, gives the definition and attributes of type two intuitionistic fuzzy rough sets based on inclusion degree. At the same time, uncertainty measurement factors such as approximate accuracy, roughness, and importance are provided.
In 1975, Professor Zadeh1 extended the membership functions in traditional fuzzy set to type-one fuzzy set. On the basis, the notion of type-two fuzzy set was proposed, which essentially fuzzies the fuzzy set again, so that the generated type-two fuzzy set can represent deeper uncertainty problems, greatly enhancing the ability to handle and characterize uncertainty problems in the objective world. Although the notion of type-two fuzzy set has been put forward for nearly 50 years, it has not attracted the attention of scholars at home and abroad for a long time due to its complexity in mathematical representation, geometric presentation, and computational efficiency. Until 2000, with the strong promotion of renowned scholar Mendel and his students2–8 at the University of Southern California in the United States, the research on type-two fuzzy theory gradually became a hot topic in uncertain system research. Especially in the past 25 years, accompanied by the rapid development of type-two fuzzy systems and word calculation theory research, the theory of type two fuzzy decision-making has attracted increasing attention from scholars and has achieved a large number of excellent results in information fusion,9–17 preference relationship theory,18–24 measurement theory,25–34 and decision-making methods.35–42 This article provides the definition and properties of type two intuitionistic fuzzy rough sets based on inclusion degree.
Basic knowledge
Type-two fuzzy set
Zadeh1 provided a notion of a type-n fuzzy set in 1975.
1 A fuzzy set is an n type. If its membership function takes the value of n-1 fuzzy set, then the membership function range of one type of fuzzy set is [0,1].
Type-one fuzzy set: . A type-one fuzzy set is a special case when the fuzzy set domain is [0,1], which can be called a unit fuzzy set
8 Assuming a non-empty domain, a type-two fuzzy set can be characterized by a type-two membership function ,
wherein .
If the domain of discourse U is a finite or countable set, can be delimited as
If the domain of discourse U is an infinite uncountable set, can be delimited as
Among them, represents the summation of all possible and .
Type-two intuitionistic fuzzy set
43 Assuming W is a domain, is called a type-two intuitionistic fuzzy set on W. Among them,
43 Suppose there are two type-two intuitionistic fuzzy sets G and H on the domain of universe U, suppose
among
The following operation can be defined:
(1)
(2)
(3)
43 If S, M, and Q are three type-two intuitionistic fuzzy sets on the domain W, then the following equation holds:
(1) Law of commutation
(2) Law of associativity: ,
(3) Law of idempotency:
(4) Law of double complementation:
(5) De Morgan Law: ,
43 Suppose W and V are two non-empty finite fields, and the type-two intuitionistic fuzzy subset defined on the direct product space is said to be the type-two intuitionistic fuzzy relationship from W to V, recorded as
for satisfy .
44 If G and H are two type-two intuitionistic fuzzy sets on the domain W,
The order relationship is defined as follows:
44 If E, F, and Q are three type two intuitionistic fuzzy sets on the domain W if , this is
The following equation holds:
Type-two fuzzy rough set
45 Suppose R is a type-two fuzzy relationship on , where is a type-two fuzzy approximation space, E is a type-two fuzzy set on the domain W, E’s upper and lower approximations of the approximation space are the type-two fuzzy sets defined on W, as follows
is called a type-two fuzzy rough set of E with respect to the approximate space .
45 Suppose and are the upper and lower approximation operators in definition 2.7, for any and the following properties hold:
(1) ,
(2) if for , thus , then for ,
thus , and
(3)
(4) ,
(5) ,
(6) .
Type-two intuitionistic fuzzy rough set
45 Suppose R is a type-two intuitionistic fuzzy relationship on , where is a type-two intuitionistic fuzzy approximation space, S is a type-two intuitionistic fuzzy set on the domain U, and S’s upper and lower approximations to the approximation space are the type two intuitionistic fuzzy sets defined on U, as follows
among
is called a type-two intuitionistic fuzzy rough set of S with respect to the approximate space .
45 Suppose be the lower and upper approximations in Definition 1.8, for any , then:
(1) , ,
(2) ,
(3) ,
(4) , , and
(5) , .
Type-two intuitionistic fuzzy rough set based on inclusion degree
Suppose U is a non-empty finite set, where represents the entire set of classical sets in U, and represents the entire set of type-two intuitionistic fuzzy sets in U设 , if for any , there is a corresponding value of , and it satisfies:
(1) ,
(2) , , and
(3) , .
Then D is called the inclusion degree on .
On the basis of the above definition and the Euclidean distance formula between interval numbers, a new formula for calculating inclusion is given:
Suppose W is a non-empty finite set, for any ,
Then D is called the inclusion degree.
The following proves that this formula satisfies the above three conditions.
Proof: (1) Clearly established.
(2) Necessity: if , then and , so
Sufficiency: if
Then ,
Thus ,
So and , then .
(3) If , then , , so
Because
Thus .
For any , if , then
(1) .
(2) .
Proof: (1) if , then and , because
Also because
so .
(2) Similarly, it can be proven.
Suppose is a fuzzy inclusion approximation space, where is a weak fuzzy partition on W, and D is the inclusion degree on . For any , upper approximations and lower approximations of M to bases on the parameter for a type-two intuitionistic fuzzy rough set stem from inclusion degree are defined as follows:
We can define positive domain, negative domain, and boundary domain :
For parameters , rough sets in this definition can be divided into the following four categories:
(1) If , , then is said to be partially definable.
(2) If , , then is said to be internally definable.
(3) If , , then is said to be externally definable.
(4) If , , then is said to be completely undefinable.
Then, we can obtain the following properties.
Suppose , if D is the inclusion degree on , then the approximation operator based on inclusion degree satisfies the following properties:
(1) For any , then .
(2) For any and , then ,
.
(3) For any , ,
.
(4) For any , ,
.
(5) For any , if , , then ,
.
Proof: (1) According to Definition 3.1, it can be inferred that
(2) If , then and , so
This is , if , then ,
so .
The same as .
(3) Because , , by (2) we hold ,
, so .
Because , , then ,
, thus .
(4) Similar to (3) evidence method.
(5) If , when and , so .
The same as .
From (5) of Theorem 3.2, it can be seen that the positive field increases with the decrease of , the negative field increases with the increase of , and the boundary shrinks. Therefore, the following theorem holds:
Theorem 3.3. Suppose , then for any , we hold
(1) .
(2) .
Proof: (1) If , from the definitions of lower approximations, it can be inferred:
Because increases as decreases, so
If there exists , from the definitions of upper approximations, it can be inferred .
If , then , because , so , contradiction between and , so .
Then (1) holds.
(2) If , by the definitions of upper approximations, we hold
According to Theorem 3.2 of (5), decreases as increases, therefore
If there exists , so and .
Thus , for any hold, so . But contradictory to , so
Thus (2) holds.
Suppose , then
Proof: obviously .
Because , so when monotonically decreases towards and monotonically increases towards , the boundary domain monotonically decreases, so
if there exists ,
then , but .
When , then . For any , thus .
When , then . For any , thus .
So . This contradicts with ; therefore, the theorem is proven.
From the above theorem, it can be seen that as monotonically decreases towards and monotonically increases towards , the boundary domain gradually shrinks to:
Uncertainty metrics
If is the approximate precision of the set , and
Among them , represents the cardinality of set .
Obviously .
If is the roughness degree of the set , and
If is a weak fuzzy partition on U, is called the approximate classification accuracy of . If
If is a weak fuzzy partition on U, is called the approximate quality of classification of . If
If is a weak fuzzy partition on U, is called the granularity of , if
If is a weak fuzzy partition on U, is called the resolution , if
If A is the set of all attributes, , , then the importance of m to is denoted as , if
Suppose be a type-two intuitionistic fuzzy set on U, where is the degree of ambiguity of with respect to . If
Conclusion
This article provides the definition of inclusion degree in type-two intuitionistic fuzzy sets, a new calculation formula for inclusion degree, and verifies the feasibility of the formula. Furthermore, the definition and attributes of type two intuitionistic fuzzy rough sets based on inclusion degree are given, and at the same time, the type two fuzzy set was expanded.
Statements and declarations
Footnotes
Conflicting interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The authors received no financial support for the research, authorship, and/or publication of this article.
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