Interval-valued Intuitionistic Fuzzy Rough Sets - Rough Set Theory ...

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Interval-valued . . . 

Rough Sets 



Approximation . . . 

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Interval-valued intuitionistic 

fuzzy-rough sets 

Yanhua Wu, Kedian Li 

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Department of Mathematics and Information 

Science, Zhangzhou Normal University, Zhangzhou 

363000, P. R. China 

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Interval-valued intuitionistic fuzzy-rough 

sets 

Introduction 

Interval-valued Intuitionistic Fuzzy Sets 


Interval-valued Intuitionistic Fuzzy-Rough Sets Model 

Interval-valued Intuitionistic Fuzzy Sets and Hamming Distance 

Approximation Roughness 

References 

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Introduction 






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1 Introduction 

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The present paper is inspired by the ideology of [16]. This paper combines 

interval-valued intuitionistic fuzzy sets and rough sets. It studies rougheness in 

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interval-valued intuitionistic fuzzy sets and proposes one kind of interval-valued 

intuitionistic fuzzy-rough sets models under the equivalence relation in crisp 

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sets. That extends the classical rough set defined by Pawlak. All the properties 

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are established over finite fields in this paper. 

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Introduction 






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2 Interval-valued Intuitionistic Fuzzy Sets 

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Definition 1.[2] Let a set X = {x 1 , x 2 , · · · , x n } be fixed. An intervalvalued 

intuitionistic fuzzy set A in X is an object having the form 

A = {〈x, µ A (x), ν A (x)〉|x ∈ X} 

where µ A (x) ⊂ [0, 1] and ν A (x) ⊂ [0, 1] satisfy 

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supµ A (x) + supν A (x) ≤ 1 

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for all x ∈ X, and µ A (x) and ν A (x) are, respectively, called the degree 

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of membership and the degree of non-membership of the element x ∈ 

X to A. If supµ A (x)=infµ A (x) and supν A (x)=infν A (x), then A will 

degenerate to be a standard intuitionistic fuzzy set. 

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Definition 2.[2] Let 

A = {〈x, µ A (x), ν A (x)〉|x ∈ X}, 

A 1 = {〈x, µ A1 (x), ν A1 (x)〉|x ∈ X}, 

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A 2 = {〈x, µ A2 (x), ν A2 (x)〉|x ∈ X} 

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be interval-valued intuitionistic fuzzy sets. Denote, 

µ LA1 ,A 2 

(x) = {inf µ A1 (x), inf µ A2 (x)}, µ UA1 ,A 2 

(x) = {sup µ A1 (x), sup µ A2 (x)}; 

ν LA1 ,A 2 

(x) = {inf ν A1 (x), inf ν A2 (x)}, ν UA1 ,A 2 

(x) = {sup ν A1 (x), sup ν A2 (x)}, 

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then 

⋃ 

A 1 A2 = {〈x, [max µ LA1 ,A 2 

(x), max µ UA1 ,A 2 

(x)], 

[min ν LA1 ,A 2 

(x), min ν UA1 ,A 2 

(x)]〉|x ∈ X}; 

⋂ 

A 1 A2 = {〈x, [min µ LA1 ,A 2 

(x), min µ UA1 ,A 2 

(x)], 

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[max ν LA1 ,A 2 

(x), max ν UA1 ,A 2 

(x)]〉|x ∈ X}; 

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A c = {〈x, ν A (x), µ A (x)〉|x ∈ X}. 

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Definition 3.[9] Let F L (x) be the set of all interval-valued intuitionistic 

fuzzy sets of X, then 

A 1 ⊆ A 2 , 

we have 

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inf µ A1 (x) ≤ inf µ A2 (x), sup µ A1 (x) ≤ sup µ A2 (x); 

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inf ν A1 (x) ≥ inf ν A2 (x), sup ν A1 (x) ≥ sup ν A2 (x). 

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Introduction 






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3 Rough Sets 

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Definition 4. [8] Let U be a set called the universe and R an equivalence 

relation of U, we call (U, R) a Pawlak approximation space. Any 

subset X ⊆ U, can be characterized with respect to the relation R . 

The equivalence class of element x by the relation R will be denoted 

by [x] R , that is, [x i ] R = {x j |(x i , x j ) ∈ R}. 

For all X ⊆ U,the subsets R(X) = {x i |[x i ] R ⊆ X}, R(X) = 

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{x i |[x i ] R 

⋂ X ≠ ∅}, are called the R-lower approximations of Xand 

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the R-upper approximations of X respectively. If R(X) = R(X), we 

say that X is definable. Any pair of the form R(X) = (R(X), R(X)) 

is called a rough set of is called a rough set of X. 

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Theorem 1. [14] Let (U, R) be a approximation space, For all X ⊆ 

U,denote 

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R ′ (X) = ⋃ {[x i ] R |[x i ] R ⊆ X}, 

R ′ (X) = ⋃ {[x i ] R |[x i ] R 

⋂ X ≠ ∅}, 

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then 

R ′ (X) = R(X), R ′ (X) = R(X). 

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Theorem 2. [14] Let (U, R) be an approximation space. For X ⊆ U, 

R(X) and R(X) are the R-lower and R-upper approximations of X, 

respectively. Then we have 

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(1) R(U) = R(U) = U, R(∅) = R(∅) = ∅; 

(2) R(X) ⊆ X ⊆ R(X); 

(3) R(X ⋂ Y ) = R(X) ⋂ R(Y ), R(X ⋃ Y ) = R(X) ⋃ R(Y ); 

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(4) R(X C ) = (R(X)) C , R(X C ) = (R(X)) C . 

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4 Interval-valued Intuitionistic Fuzzy- 

Rough Sets Model 

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Definition 5.[16] Let (U, R) be a Pawlak approximation space, for 

fuzzy set A on U . Denote 

R(A)(x) = min{A(y)|y ∈ [x] R }, 

R(A)(x) = max{A(y)|y ∈ [x] R }. 

Then R(A) and R(A) are called the lower and upper approximations 

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of fuzzy sets A with respect to approximation space (U, R), respectively. 

We call (R(A), R(A)) the rough fuzzy set of A with respect to 

(U, R). 

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Definition 6. Let (U, R) be a Pawlak approximation space, for an 

interval-valued intuitionistic fuzzy set A = {〈x, µ A (x), ν A (x)〉|x ∈ X} 

denote, 

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A R = {〈x, [ inf 

y∈[x] R 

{inf µ A (y)}, inf 

y∈[x] R 

{sup µ A (y)}], 

[ sup 

y∈[x] R 

{inf ν A (y)}, sup 

y∈[x] R 

{sup ν A (y)}]〉|x ∈ U}, 

A R = {〈x, [ sup 

y∈[x] R 

{inf µ A (y)}, sup 

y∈[x] R 

{sup µ A (y)}], 

[ inf 

y∈[x] R 

{inf ν A (y)}, inf 

y∈[x] R 

{sup ν A (y)}]〉|x ∈ U}, 

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and 

It is easy to see that 

[ inf 

y∈[x] R 

{inf µ A (y)}, inf 

y∈[x] R 

{sup µ A (y)}] ⊂ [0, 1], 

[ sup 

y∈[x] R 

{inf ν A (y)}, sup 

y∈[x] R 

{sup ν A (y)}] ⊂ [0, 1], 

inf {sup µ A (y)} + sup {sup ν A (y)} ≤ 1, 

y∈[x] R y∈[x] R 

then A R is an interval-valued intuitionistic fuzzy set. 

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and 

Similarly, we have 

[ sup 

y∈[x] R 

{inf µ A (y)}, sup 

y∈[x] R 

{sup µ A (y)}] ⊂ [0, 1], 

[ inf 

y∈[x] R 

{inf ν A (y)}, inf 

y∈[x] R 

{sup ν A (y)}] ⊂ [0, 1], 

sup {sup µ A (y)} + inf {sup ν A (y)} ≤ 1, 

y∈[x] R 

y∈[x] R 

then A R is also an interval-valued intuitionistic fuzzy set. 

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If A R = A R , then A is a definable set, otherwise A is an intervalvalued 

intuitionistic fuzzy rough set. A R and A R are called the lower 

and upper approximations of interval-valued intuitionistic fuzzy set 

with respect to approximation space , respectively. and are simply denoted 

by A with respect to approximation space (U, R), respectively. 

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A R and A R are simply denoted by A and A hereafter. 

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Theorem 3. Let A, B be interval-valued intuitionistic fuzzy sets and 

A and A the lower and upper approximations of interval-valued intuitionistic 

fuzzy set A with respect to approximation space (U, R), respectively. 

B and B be the lower and upper approximations of intervalvalued 

intuitionistic fuzzy set Bwith respect to approximation space 

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(U, R), respectively. Then we have 

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(1) A ⊆ A ⊆ A; 

(2) A ⋃ B = A ⋃ B, A ⋂ B = A ⋂ B; 

(3) A ⋃ B ⊆ A ⋃ B, A ⋂ B ⊆ A ⋂ B; 

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(4) (A) = (A) = A, (A) = (A) = A; 

(5) U = U, ∅ = ∅; 

(6) If A ⊆ B, then A ⊆ B and A ⊆ B; 

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(7) A C = (A) C , A C = (A) C . 

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Remark 1. “ sup ” is equal to “ max ”§“ inf ” is equal to “ min ” over 

finite fields. 

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Definition 7. Let (U, R) be a Pawlak approximation space, and A and 

B two interval-valued intuitionistic fuzzy sets over U . 

If A = B, then A and B are called interval-valued intuitionistic fuzzy 

lower rough equal. 

If A = B, then A and B are called interval-valued intuitionistic fuzzy 

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upper rough equal. 

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If A = B,A = B, then A and B are called interval-valued intuitionistic 

fuzzy rough equal. 

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Theorem 4. Let (U, R) be a Pawlak approximation space, and A and 

B two interval-valued intuitionistic fuzzy sets over U. Then, 

(1) A = B ⇔ A ⋂ B = A, A ⋂ B = B; 

(2) A = B ⇔ A ⋃ B = A, A ⋃ B = B; 

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(3) If A = A ′ and B = B ′ , then A ⋃ B = A ′ ⋃ B ′ ; 

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(4) If A = A ′ and B = B ′ , then A ⋂ B = A ′ ⋂ B 

′; 

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(5) If A ⊆ B and B = ∅, then A = ∅; 

(6) If A ⊆ B and B = U, then A = U; 

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(7) If A = ∅ or B = ∅, then A ⋂ B = ∅; 

(8) If A = U or B = U, then A ⋃ B = U; 

(9) A = U ⇔ A = U;(10)A = ∅ ⇔ A = ∅; 

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5 Interval-valued Intuitionistic Fuzzy Sets 

and Hamming Distance 

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Definition 8. [12] LetA 1 and A 2 be interval-valued intuitionistic fuzzy 

sets over U then the standard hamming distance of A 1 and A 2 is defined 

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as follows 

d(A 1 , A 2 ) = 1 

4n 

n∑ 

(|µ L A 1 

(x j ) − µ L A 2 

(x j )| + |µ U A 1 

(x j ) − µ U A 2 

(x j )| 

j=1 

+|ν L A 1 

(x j ) − ν L A 2 

(x j )| + |ν U A 1 

(x j ) − ν U A 2 

(x j )|) 

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And we can obtain the standard hamming distance of A and A from 

Definition 8, 

d(A, A) = 1 

4n 

n∑ 

(|µ L A (x j) − µ L(x A j)| + |µ U A (x j) − µ U(x A j)| 

j=1 

+|ν L A (x j) − ν L A (x j)| + |ν U A (x j) − ν U A (x j)|) 

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where 

µ L A (x j) = inf 

y∈[x] R 

{inf µ A (y)},µ U A (x j) = inf 

y∈[x] R 

{sup µ A (y)}, 

ν L A (x j) = sup 

y∈[x] R 

{inf ν A (y)},ν U A (x j) = sup 

y∈[x] R 

{sup ν A (y)}, 

µ L A (x j) = sup 

y∈[x] R 

{inf µ A (y)},µ U A (x j) = sup 

y∈[x] R 

{sup µ A (y)}, 

ν L A (x j) = inf 

y∈[x] R 

{inf ν A (y)},ν U A (x j) = inf 

y∈[x] R 

{sup ν A (y)}, 

y = x j , (j = 1, 2 · · · n). 

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Theorem 5. Let (U, R) be approximation space, A be an intervalvalued 

intuitionistic fuzzy set over U . Then 

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(1) If d(A, A) = 0, then A is a definable set. 

(2) If 0 < d(A, A) < 1,then A is an interval-valued intuitionistic 

fuzzy rough set. 

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Theorem 6. Let (U, R) be a Pawlak approximation space, and A and 

B two interval-valued intuitionistic fuzzy sets over U. Then 

(1) d(A, A) ≥ d(A, A) and d(A, A) ≥ d(A, A); 

(2) d(A ⋃ B, A ⋃ B) = 0, d(A ⋂ B, A ⋂ B) = 0; 

(3) d(A ⋃ B, A ⋃ B) ≥ d(A ⋃ B, A ⋃ B) 

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and 

d(A ⋃ B, A ⋃ B) ≥ d(A ⋃ B, A ⋃ B); 

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and d(A ⋂ B, A ⋂ B) ≥ d(A ⋂ B, A ⋂ B) 

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and 

d(A ⋂ B, A ⋂ B) ≥ d(A ⋂ B, A ⋂ B); 

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(4) d((A), (A)) = 0, d((A), A) = 0, d((A), A) = 0; 

d((A), (A)) = 0, d((A), A) = 0, d((A), A) = 0; 

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(5) d(U, U) = 0, d(∅, ∅) = 0 

(6) If A ⊆ B, then d(A, B) ≥ d(A, B) and d(A, B) ≥ d(B, B); 

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d(A, B) ≥ d(A, A) and d(A, B) ≥ d(A, B); 

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(7) d(A C , (A) C ) = 0, d(A C , (A) C ) = 0; 

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6 Approximation Roughness 

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Definition 9. Let F L (X) be the set of all interval-valued intuitionistic 

fuzzy sets of X. For A ∈ F L (X), x ∈ X, X ⊆ U, U a nonempty 

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universe, R an equivalence relation on U, [x] R the equivalence class of 

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element x by the relation R, then its cardinality |A| is defined as follows 

|A| = sup(( ⋃ µ A (y)) ⋃ ( ⋃ ν A (y)))−inf(( ⋃ µ A (y)) ⋃ ( ⋃ ν A (y))). 

y∈[x] R y∈[x] R y∈[x] R y∈[x] R 

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Definition 10. Let (U, R) be a Pawlak approximation space, and A 

an interval-valued intuitionistic fuzzy set over U . The approximation 

roughness ρ R (A) of A with respect to approximation space (U, R) is 

defined by ρ R (A) = 1 − |A| 

|A| . If |A| = 0, we have ρ R(A) = 0 . The approximation 

precision η R (A) of A with respect to approximation space 

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(U, R) is defined by η R (A) = |A| 

|A| . 

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It is clear that 0 ≤ ρ R (A) ≤ 1, 0 ≤ η R (A) ≤ 1. 

If A is a definable set, we have ρ R (A) = 0, η R (A) = 1. 

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Theorem 7. Let (U, R) be a Pawlak approximation space, and Aan 

interval-valued intuitionistic fuzzy set over U. 

(1) If d(A, A) = 0, then ρ R (A) = 0, η R (A) = 1. 

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(2) If 0 < d(A, A) < 1, then 0 < ρ R (A) < 1, 0 < η R (A) < 1. 

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Theorem 8. 

Let (U, R) be a Pawlak approximation space, A an 

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interval-valued intuitionistic fuzzy set over U, S an equivalence relation 

on U , and S ⊆ R , then A R ⊆ A S , A S ⊆ A R . 

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Theorem 9. Let S ⊆ R and A be an interval-valued intuitionistic 

fuzzy set over U . Then 

(1) η S (A) ≥ η R (A); 

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(2) ρ S (A) ≤ ρ R (A). 

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Remark 2. We can obtain from Theorem 9 that the more fineness partition 

is, the more approximation precision is, the less approximation 

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roughness is. 

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7 References 

[1] ATANASSOV K. Intuitionistic fuzzy sets [J]. Fuzzy Sets and Systems, 

1986, 20(1): 87-96. 

[2] ATANASSOV K, GARGOV G. Interval-valued intuitionistic fuzzy 

sets [J]. Fuzzy Sets and Systems, 1989, 31: 343-249. 

[3] BUSTINCE H, BURILLO P. Vague sets are intuitionistic fuzzy sets 

[J]. Fuzzy Sets and Systems, 1996, 79(3): 403-405. 

[4] CORNELIS C, COCK M D, KERRE E E. Intuitionistic fuzzy rough 

sets: at the crossroads of imperfect knowledge [J]. Expert Systems, 

2003, 20(5): 260õ270. 

[5] DUBOIS D, PRADE H. Rough fuzzy sets and fuzzy rough sets [J]. 

International Journal of General Systems, 1990, 17(2): 191õ209. 

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[6] GAU W L, BUEHRER D J. Vague sets [J]. IEEE Trans on Systems, 

Man, and Cybernetics, 1993, 23(2): 610-614. 

[7] GONG Z T, SUN B Z, CHEN D G.. Rough set theory for intervalvalued 

fuzzy information systems [J], Information Sciences, 2008, 

178(8): 1968õ1985. 

[8] PAWLAK Z. Rough sets [J]. International Journal of Computer and 

Information Sciences, 1982, 11: 341-356. 

[9] TAPAS K M, SAMANTA S K. Topology of interval-valued intuitionistic 

fuzzy sets [J]. Fuzzy Sets and Systems, 2001, 119(3): 483- 

494. 

[10] WU W-Z, MI J-S, ZHANG W-X. Generalized fuzzy rough sets 

[J]. Information Sciences, 2003, 151: 263õ282. 

[11] WU W-Z, ZHANG W-X. Constructive and axiomatic approaches 

of fuzzy approximation operators [J. Information Sciences, 2004, 159: 

233õ254. 

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[12] XU Z-S. The Theory and Application of Intuitionistic Fuzzy Information 

Integration [M]. Beijing: Science Press, 2008: 45-46, 89-93. 

[13] ZHANG Z-M. An interval-valued intuitionistic fuzzy rough set 

model [J]. Fundamenta Information, 2009, 97: 471õ498. 

[14] ZHANG W-X, QIU G.-F. Uncertain Decision Making Based on 

Rough Sets [M]. Beijing: Tsinghua University Press, 2005: 14-16. 

[15] ZHUO L, WU W-Z, ZHANG W-X. On intuitionistic fuzzy rough 

sets and their topological structures [J]. International Journal of General 

Systems, 2009, 38: 589õ616. 

[16] ZHANG Z-L, ZHANG J-L, XIAO Q-M. Fuzzy algebra and rough 

algebra [M]. Wuhan: Wuhan University Press, 2007: 116-135. 

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Thanks! 

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