﻿ 二型直觉模糊粗糙集
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Type-2 intuitionistic fuzzy rough sets
WANG Jinying, HAN Xiaobing, WANG Yanping
School of science, Liaoning university of technology, Jinzhou Liaoning 121001, China
Abstract: In this study, we integrate the theories of type-2 intuitionistic fuzzy sets and rough sets to construct a type-2-intuitionistic-fuzzy-and-rough-sets model. First, in type-2 intuitionistic fuzzy approximation space, we define a pair of type-2 intuitionistic fuzzy upper and lower approximation operators. We then discuss specific changes in these upper and lower approximation operators for a situation in which the type-2 intuitionistic fuzzy relations degenerate into common type-2-fuzzy and general-equivalence relations. Next, we generalize the definition of the inclusion relation between general type-2 fuzzy sets as type-2 intuitionistic fuzzy sets. On this basis, we then explored the properties of type-2-intuitionistic-fuzzy upper and lower approximation operators. We then defined reflexive, symmetric, and transitive type-2 intuitionistic fuzzy relations. Finally, we discuss the relations between these three special type-2 intuitionistic fuzzy relations and the characteristics of their approximation operators. Conclusions drawn in this study further enrich the theories of type-2 intuitionistic fuzzy and rough sets and establish a good theoretical basis for the application of the type-2 intuitionistic fuzzy information system.
Key words: intuitionistic fuzzy sets     rough sets     type-2 fuzzy sets     type-2 intuitionistic fuzzy sets     type-2 intuitionistic fuzzy rough sets     approximation operators

1 二型直觉模糊集的基本理论

μA(x)表示xA的隶属程度，vA(x)表示xA的非隶属程度。

1)A∩B={〈x,μAB(x),vAB(x)〉|xU}，

2)AB={〈x,μAB(x),vAB(x)〉|xU}，

3)Ac={〈x,vA(x),μA(x)〉|xU}。

1)交换律,AB=BAAB=BA

2)结合律,A∪(BC)=(AB)∪CA∩(B∩C)=(A∩B)∩C

3)幂等律,AA=AAA=A

4)对合律,(Ac)c=A

5)德摩根律,(AB)c=AcBc，(A∩B)c=AcBc

2 二型直觉模糊粗糙集及其性质

1)当R退化为U×U上的普通二型模糊关系，A退化为U上的普通二型模糊集时，定义4中的二型直觉模糊粗糙集退化为文献[10]中的二型模糊粗糙集。

2)当R退化为U×U上的等价关系，AU上的二型直觉模糊集时，定义4中的二型直觉模糊粗糙集退化为如下形式：

vA(x)vB(x)，其中序关系,定义为

2)若A⊆B，即 ∀x∈UμA(x) μB(x)且vA(x) vB(x)，则有

3)若R1R2，即 ∀ x∈UμR1(x,y) μR2(x,y)且vR1(x,y) vR2(x,y)，那么有

1)R(A∪B)=R(A)∪R(B)，

R(A∩B)=R(A)∩R(B)；

2)R(A∩B)R(A)∩R(B)，

R(A∪B)R(A)∪R(B)。

4)因为A∩B⊆A,B，由定理3的性质(2)有

3 二型直觉模糊关系与近似算子的特征联系

1)R是自反的⇔x∈U,μR(x,x)=1/1且vR(x,x)=1/0。

2)R是对称的⇔x,y∈U,μR(x,y)=μR(y,x)且vR(x,y)=vR(y,x)

3)R是传递的⇔ ∀x,y,zU,∇y∈UR(x,y)ΔμRy,z)]μR(x,z)

1)若R是自反的，则R(A)⊆AR(A)。

2)若R是对称的，则 ∀x,yU,μR(1x)(y)=μR(1y)(x),vR(1x)(y)=vR(1y)(x),μR(1U－y)(x)=μR(1U－x)(y),vR(1U－y)(x)=vR(1X－x)(y)。

3)若R是传递的，则R(R(A))⊆R(A),R(A)⊆R(R(A))。

R(A)⊆A⊆R(A)。

2)若R是对称的，则 ∀x,yU,μR(x,y)=μR(y,x)vR(x,y)=vR(y,x)

3)若R是传递的，则 ∀x,y,zU,∇y∈U[μR(x,yμR(y,z)μR(x,z)

4 结束语

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DOI: 10.11992/tis.201412013

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#### 文章信息

WANG Jinying, HAN Xiaobing, WANG Yanping

Type-2 intuitionistic fuzzy rough sets

CAAI Transactions on Intelligent Systems, 2015, 10(6): 943-948.
DOI: 10.11992/tis.201412013