Roughness Measures of Intuitionistic Fuzzy Sets

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Contents. Introduction. Motivation. Basic notions. Our results. Summary. Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu. Roughness Measures of Intuitionistic Fuzzy Sets ...
Roughness Measures of Intuitionistic Fuzzy Sets Lei Zhou1 1 Institute

Wen-Xiu Zhang1

Wei-Zhi Wu2

for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, China

2 School

of Mathematics, Physics and Information Science, Zhejiang Ocean University, China

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Speaker: Lei Zhou

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Contents

Introduction Motivation Basic notions Our results Summary

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Introduction

Rough set, as a new maths tool to describe uncertainty and incompleteness, has played a useful role in the study of intelligent systems. Since it was proposed by Pawlak in 1982, there have been many improvements in theories and applications for itself, and meanwhile, it has been connected with other uncertainty theories, such as, fuzzy set theory, probability theory, evidence theory, random set theory et.al. Now the theory of rough set is rapidly developing in order to satisfy more practical requirements, a number of documents have focused on this study (see literature listed in this paper).

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Intuitionistic fuzzy sets (IF sets for short), which was originated by Atanassov, is in fact a general form of fuzzy set. An IF set gives both a membership degree and a nonmembership degree and is thus more precise to describe the vagueness and uncertainty than a fuzzy set. Some concepts and operators of fuzzy sets are similarly introduced into IF sets that construct the basis of IF set theories. More recently, rough approximations of IF sets have attracted some researchers’ sight, several authors have made some progress in this field (see literature listed in this paper).

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Motivation

In this paper, we first introduce rough approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space. Then we examine basic properties of intuitionistic fuzzy rough sets. Finally, roughness measures of intuitionistic fuzzy sets are characterized by absolute roughness measure and relative roughness measure, and furthermore we analyze their properties.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Basic notions

Definition 1. Let a set U be fixed. An IF set A in U is an object having the form A = {hx, µA (x), γA (x)i | x ∈ U}, where µA : U → [0, 1] and γA : U → [0, 1] satisfy 0 ≤ µA (x) + γA (x) ≤ 1 for all x ∈ U, and µA (x) and γA (x) are respectively called the degree of membership and the degree of nonmembership of the element x ∈ U to A. The family of all IF subsets in U is denoted by IF(U). The complement of an IF set A is denoted by ∼ A = {hx, γA (x), µA (x)i | x ∈ U}.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Definition 2. An IF relation on U is an IF subset of U × U, that is, R = {h(x, y ), µR (x, y ), γR (x, y )i | x, y ∈ U}, where µR , γR : U ×U → [0, 1] satisfy 0 ≤ µR (x, y )+γR (x, y ) ≤ 1 for all (x, y ) ∈ U × U. We denote the family of all IF relations on U by IFR(U × U).

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Definition 3. Let R ∈ IFR(U × U), we say that R is • reflexive if µR (x, x) = 1 and γR (x, x) = 0 for all x ∈ U, • symmetric if for any (x, y ) ∈ U × U, µR (x, y ) = µR (y , x) and γR (x, y ) = γR (y , x). • transitive if R ≥ R

∨,∧ J

R, i.e., for any (x, z) ∈ U × U,

∧,∨

µR (x, z) ≥ ∨y ∈U [µR (x, y ) ∧ µR (y , z)] and γR (x, z) ≤ ∧y ∈U [γR (x, y ) ∨ γR (y , z)].

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Definition 4. Let A ∈ IF(U) and α, β ∈ [0, 1], the α−level bottom cut set of A, denoted by Aα , and the β−level top cut set of A, denoted by Aβ , are respectively defined as follows: Aα = {x ∈ U|µA (x) ≥ α},

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Aβ = {x ∈ U|γA (x) ≤ β}.

Roughness Measures of Intuitionistic Fuzzy Sets

Our Results Definition 5. Let U be a nonempty and finite universe of discourse and R ∈ IFR(U × U), the pair (U, R) is called an IF approximation space. For any A ∈ IF(U), the upper and lower approximations of A w.r.t. (U, R), denoted by R(A) and R(A), are two IF sets and are respectively defined as follows: 

R(A) = x, µR(A) (x), γR(A) (x) | x ∈ U , R(A) =



x, µR(A) (x), γR(A) (x) | x ∈ U ,

where µR(A) (x) = ∨y ∈U [µR (x, y ) ∧ µA (y )], γR(A) (x) = ∧y ∈U [γR (x, y ) ∨ γA (y )]; µR(A) (x) = ∧y ∈U [γR (x, y ) ∨ µA (y )], γR(A) (x) = ∨y ∈U [µR (x, y ) ∧ γA (y )]. Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

R(A) and R(A) are respectively called the upper and lower IF rough approximations of A w.r.t. (U, R). The pair (R(A), R(A)) is called the IF rough set of A w.r.t. (U, R), and R, R : IF(U) → IF(U) are referred to as upper and lower IF rough approximation operators respectively.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Theorem 1. Let (U, R) be an IF approximation space. Then the upper and lower approximation operators satisfy the following properties: ∀A, B ∈ IF(U), α, β ∈ [0, 1] with α + β ≤ 1, (IL1) R(A) =∼ R(∼ A), (IL2) R(1∼ ) = 1∼ , (IL3) R(A ∩ B) = R(A) ∩ R(B), (IL4) A ⊆ B =⇒ R(A) ⊆ R(B), (IL5) R(A ∪ B) ⊇ R(A) ∪ R(B), \ \ (IL6) R(A ∪ (α, β)) = R(A) ∪ (α, β), Moreover, if R is an IF reflexive relation, then (IRL7) R(A) ⊇ A.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

(IU1) R(A) =∼ R(∼ A), (IU2) R(0∼ ) = 0∼ , (IU3) R(A ∪ B) = R(A) ∪ R(B), (IU4) A ⊆ B =⇒ R(A) ⊆ R(B), (IU5) R(A ∩ B) ⊆ R(A) ∩ R(B), \ \ (IU6) R(A ∩ (α, β)) = R(A) ∩ (α, β), Moreover, if R is an IF reflexive relation, then (IRU7) R(A) ⊇ A.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Definition 6. Let (U, R) be an IF approximation space and A, B ∈ IF(U), A and B are called IF lower R−equal if R(A) = R(B), denote by A∼B; A and B are called IF upper R−equal if R(A) = R(B), denote by A ' B; A and B are called IF R−equal if R(A) = R(B) and R(A) = R(B), denote by A ≈ B.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Theorem 2. Let (U, R) be an IF approximation space. Then in IF(U), we have (1) A∼B iff A ∩ B∼A and A ∩ B∼B; (2) A ' B iff A ∪ B ' A and A ∪ B ' B; (3) If A∼A0 and B∼B 0 , then A ∩ B∼A0 ∩ B 0 ; (4) If A ' A0 and B ' B 0 , then A ∪ B ' A0 ∪ B 0 ; (5) If A ⊆ B and A∼U, then B∼U; (6) If A ⊆ B and B ' ∅, then A ' ∅; (7) If A∼∅ or B∼∅, then A ∩ B∼∅; (8) If A ' U or B ' U, then A ∪ B ' U.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Definition 7. Let (U, R) be an IF approximation space. A ∈ IF(U), 0 < β ≤ α ≤ 1, an absolute roughness measure of A w.r.t. parameters α, β and the approximation space (U, R) is defined by |[R(A)]α | ρα,β , R (A) = 1 − |[R(A)]β | meanwhile, a relative roughness measure of A w.r.t. parameters α, β and the approximation space (U, R) is defined by |[R(A)]1−α | . %α,β (A) = 1 − R |[R(A)]1−β | Specially, we think that ρα,β R (A) = 0 when |[R(A)]β | = 0, and 1−β | = 0, where |X | is the cardinality of %α,β (A) = 0 when |[R(A)] R the crisp set X .

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Remark. Absolute roughness measure and relative roughness measure express the same essence from two different aspectsmembership and non-membership. This will help with the analysis of rough approximation qualities of IF sets in fuzzy data processing.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Theorem 3. Let (U, R) be an IF approximation space, A ∈ IF(U), and 0 < β ≤ α ≤ 1. Then α,β (1) 0 ≤ ρα,β R (A) ≤ 1, 0 ≤ %R (A) ≤ 1; α,β (2) If β is fixed, then ρα,β R (A) and %R (A) increase with α inα,β creasing; If α is fixed, then ρR (A) and %α,β R (A) decrease with β increasing. Theorem 4. Let (U, R) be an IF approximation space and 0 < β ≤ α ≤ 1. If A is a constant IF set, i.e., there exist two numbers [ δ, θ ∈ [0, 1] such that A = (δ, θ), then (1) if 0 < β < δ < α ≤ 1, then ρα,β R (A) = 1; and for 0 < δ < β ≤ α ≤ 1 or 0 < β ≤ α ≤ δ ≤ 1, we have ρα,β R (A) = 0 α,β (2) if 0 < 1−α < θ < 1−β, then %R (A) = 1; for 0 < θ ≤ 1−α ≤ 1 − β ≤ 1 or 0 < 1 − α ≤ 1 − β < θ ≤ 1, we have %α,β R (A) = 0.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Theorem 5. Let (U, R) be an IF approximation space, A, B ∈ IF(U) and 0 < β ≤ α ≤ 1. If A ⊆ B, then α,β α,β (1) ρα,β R (A) ≥ ρR (B) when [R(A)]β = [R(B)]β , and ρR (A) ≤ ρα,β R (B) when [R(A)]α = [R(B)]α ; α,β 1−β = [R(B)]1−β , and %α,β (A) ≤ (2) %α,β R (A) ≥ %R (B) when [R(A)] R 1−α = [R(B)]1−α . %α,β (B) when [R(A)] R Theorem 6. Let (U, R) be an IF approximation space, A, B ∈ IF(U), and 0 < β ≤ α ≤ 1. Then α,β α,β α,β (1) A∼B =⇒ ρα,β R (A ∩ B) ≤ ρR (A), ρR (A ∩ B) ≤ ρR (B), α,β α,β α,β %α,β R (A ∩ B) ≤ %R (A), and %R (A ∩ B) ≤ %R (B); α,β α,β α,β (2) A ' B =⇒ ρα,β R (A ∪ B) ≤ ρR (A), ρR (A ∪ B) ≤ ρR (B), α,β α,β α,β %α,β R (A ∪ B) ≤ %R (A), and %R (A ∪ B) ≤ %R (B); α,β α,β α,β (3) A ≈ B =⇒ ρR (A) = ρR (B), %R (A) = %α,β R (B).

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Theorem 7. Let (U, R) and (U, S) be two IF approximation spaces, A ∈ IF(U), and 0 < β ≤ α ≤ 1. If S ⊆ R, then α,β α,β α,β ρα,β S (A) ≤ ρR (A) and %S (A) ≤ %R (A).

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Summary

In this paper we have introduced rough approximations of IF sets with respect to an IF approximation space and presented basic properties of IF rough approximation operators. We then introduced the concepts of absolute roughness measures and relative roughness measures of IF sets to describe roughness of the IF sets from membership degree and non-membership degree. For further study, the roughness measures of IF sets in rough data analysis are required.

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets

Thanks!

Lei Zhou, Wen-Xiu Zhang, Wei-Zhi Wu

Roughness Measures of Intuitionistic Fuzzy Sets