THE CUT SETS, DECOMPOSITION THEOREMS AND

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INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 6, Number 1, Pages 61–71

THE CUT SETS, DECOMPOSITION THEOREMS AND ¯ REPRESENTATION THEOREMS ON R-FUZZY SETS JINQUAN LI AND HONGXING LI ¯ Abstract. In this paper, some kinds of cut sets for R-fuzzy sets are introduced and their properties are given. Moreover, the decomposition theorems and ¯ representation theorems on R-fuzzy sets are obtained based on different cut ¯ sets. Finally, the axiomatic definition of cut set for R-fuzzy set is introduced which characterizes the common properties of cut set. ¯ Key Words. R-fuzzy sets, Cut sets, Decomposition theorem, Representation theorem.

1. Introduction Since the fuzzy set was introduced by Zadeh [4], many new approaches and theories treating imprecision and uncertainty have been proposed, such as typeII fuzzy sets [6], interval valued fuzzy sets [5] and so on. L−-fuzzy sets are also expansion of ordinary fuzzy sets which membership function takes value in L [7]. ¯ R-fuzzy set [1], as a special L-fuzzy sets, is introduced by Li et al. in the study of fuzzy systems, which plays an important role in constructing fuzzy system. In theory of fuzzy systems, the cut sets of fuzzy sets play an important role [8–13], which reveals the relationship between fuzzy fuzzy sets and classical sets. Furthermore, decomposition theorems and representation theorems can be obtained based on the cut sets. In [2], the cut set of fuzzy set can also be described by neighborhood relation of fuzzy point and fuzzy set, which has many applications in fuzzy topology and fuzzy algebra. To best of our knowledge, the decomposition theorem and representation the¯ orem of R-fuzzy sets are not obtained in the literature. In this paper, we first ¯ introduced the cut sets of R−fuzzy sets based on neighborhood relation of fuzzy point and fuzzy set. Then, we obtain decomposition theorems and representation ¯ theorems. Finally, we put forward the axiomatic definition of cut set for R-fuzzy sets and discuss its properties. The organization of this paper is as follows. In section 2, we give some notations ¯ and definitions. In section 3, the properties of cut sets of R-fuzzy sets are given. In section 4, the decomposition theorems are obtained. In section 5, the representation ¯ theorems are gained. In section 6, the axiomatic definition of cut set for R-fuzzy sets is put forward and the properties are also given. 2. Preliminaries Throughout this paper, we use X to denote the universal set, P (X) = {A|A ⊆ ¯ = R ∪ {−∞, +∞}. For A, B ∈ P (X), X}, R to stand for real numbers set, and R A ∪ B = {x ∈ X|x ∈ A or x ∈ B}, A ∩ B = {x ∈ X|x ∈ A and x ∈ B}, and Ac = {x ∈ X|x ∈ / A}. A ⊆ B ⇐⇒ for x ∈ X, x ∈ A =⇒ x ∈ B, and Received by the editors July 11, 2008 and, in revised form, October 28, 2008. 61

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J. LI AND H. LI

¯ a ∨ b = sup{a, b}, a ∧ b = inf{a, b}, A ⊂ B ⇐⇒ A ⊆ B and A 6= B. For a, b ∈ R, ¯ ∨, ∧, c) is fuzzy lattice. We use T to and ac = −a. It is easy to verify that (R, stand for the index set. ¯ The R-fuzzy set [1] is introduced by Li et al. in the study of generalized fuzzy systems and its definition is as follows: ¯ is called R-fuzzy ¯ Definition 2.1 [1]. The generalized real function A : X → R ¯ X stands for the set of all R-fuzzy ¯ set over X, and R sets over X. ¯ we define A ∩ B, A ∪ B, and Ac by the membership functions as For A, B ∈ R, follows: ¯ ¯ ¯ A∩B : X →R A∪B : X →R Ac : X → R x → A(x) ∧ B(x); x → A(x) ∨ B(x); x → (A(x))c . We also define A ⊆ B by A(x) 6 B(x) for all x ∈ X. In [2] , Yuan has introduced some kinds of cut sets for ordinary fuzzy sets based on the relation of fuzzy point and fuzzy set. Similarly, some different kinds of the ¯ ¯ X and a ∈ R, ¯ cut sets for R-fuzzy sets can be obtained as follows: For A ∈ R (1) : Aa = {x ∈ X|A(x) > a},

(1)

Aa ¯ (2) : A[a] = {x ∈ X|a + A(x) > 0}, (3) : Aa = {x ∈ X|A(x) 6 a}, Aa¯ (4) : A[a] = {x ∈ X|a + A(x) 6 0},

= {x ∈ X|A(x) > a}, A[a] = {x ∈ X|a + A(x) > 0}, ¯ = {x ∈ X|A(x) < a}, A[a¯] = {x ∈ X|a + A(x) < 0}.

¯ 3. The properties of cut sets of R-fuzzy sets ¯ In this section, we discuss the properties of cut sets for R-fuzzy sets. We obtain the following properties. ¯ X and a ∈ R, ¯ we have Property 3.1. For A ∈ R c c (1)A[a] = Aac , A[a] = Aac ; (2)A[a] = Aa , A[a¯] = Aa ; ¯ (3)Aa = Aa c , Aa¯ = Aa c ; (4)A[a] = Aac c , A[a¯] = Aac c . ¯ Proof. According to (1), the proof is obvious. ¯ X and a ∈ R, ¯ then Property 3.2. Let A, B ∈ R (1) Aa ⊆ Aa . (2) a ¯6 b =⇒ Aa ⊇ Ab , Aa ⊇ Ab . ¯ ¯ . (3) A ⊆ B =⇒ Aa ⊆ Ba , Aa ⊆ B a ¯ c c c (4) (A )a = (Aac ) , (A )a = (Aa¯c )c . (5) (A ∪ B)a = Aa ∪ Ba , ¯(A ∪ B)a = Aa ∪ Ba ,(A ∩ B)a = Aa ∩ Ba , (A ∩ B)a = ¯ ¯ ¯ ¯ Aa ∩ Ba . [ [ [ [ \ \ \ ¯ ¯ (6) (At )a ⊆ ( At )a , (At )a = ( At )a , ( At )a = (At )a , ( At )a ⊆ ¯ ¯ t∈T ¯ t∈T t∈T t∈T t∈T t∈T \ t∈T (At )a . ¯ t∈T (7) Aa∨b = Aa ∩ Ab , Aa∨b = Aa ∩ Ab , Aa∧b = Aa ∪ Ab , Aa∧b = Aa ∪ Ab . ^¯ _ ¯ ¯ ¯ ¯ ∈ T ), a = (8) Let at ∈ R(t at and b = at , then

t∈T

Ab =

\

t∈T

\

[ [ Aat , Ab ⊆ Aat , Aa ⊇ Aa t , A a = Aat . ¯ t∈T ¯ t∈T t∈T t∈T

(9) A−∞ = X, A+∞ = ∅ Proof. (1)-(5) is obvious.

THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS

(6) (i) ∀x ∈

[

63

S (At )a , ∃t ∈ T such that x ∈ (At )a . It follows that ( t∈T At )(x) >

t∈T

At (x) > a, that is x ∈ (

S t∈T

At )a . Hence,

[

(At )a ⊆ (

t∈T

[

[

At )a .

t∈T

S (At )a , ∃t ∈ T such that x ∈ (At )a . It follows that ( t∈T At )(x) > ¯ ¯ t∈T [ [ S At (x) > a, that is x ∈ ( t∈T At )a . Hence, (At )a ⊆ ( At )a . On the other ¯ ¯ ¯ t∈T t∈T S W hand, for ∀x ∈ ( t∈T At )a , we have ( t∈T At )(x) > a. It follows that there [ ¯ exits a t ∈ T such that At (x) > a, namely, x ∈ (At )a ⊆ (At )a . So we have ¯ ¯ t∈T S S T T ( t∈T At )a . Similarly, we can prove ( t∈T At )a = t∈T (At )a and Tt∈T (At )a = T ( t∈T At )¯a ⊆ t∈T (At )a¯. (7) It is¯easy to verify ¯that Aa∨b = {x|A(x) > (a∨b)} = {x|A(x) > a and A(x) > b} = Aa ∩ Ab and Aa∧b = {x|A(x) > (a ∧ b)} = {x|A(x) > a or A(x) > b}Aa ∪ Ab . Similarly, we can prove Aa∨b = Aa ∩ Ab and Aa∧b = Aa ∪ Ab . ¯ ¯ ¯ ¯ (8)It is easy to verify that _ \ [ Ab = {x|A(x) > at } = Aat , Aa = Aa t . ¯ t∈T t∈T t∈T V S Since A(x) > at ( for t ∈ T ) ; A(x) > t∈T at and A(x) > t∈T at ; there exists t ∈ T such thatA(x) > at , we have \ [ Ab ⊆ Aat , Aa ⊇ Aa t . ¯ t∈T t∈T (ii) ∀x ∈

(9) is obvious. It is easy to get the following properties by Property 3.1 and Property 3.2. ¯ X and a ∈ R, ¯ then Property 3.3. Let A, B ∈ R a a (1) A¯ ⊆ A . (2) a 6 b =⇒ Aa ⊆ Ab , Aa¯ ⊆ Ab ¯. (3) A ⊆ B =⇒ Aa ⊆ B a , Aa ⊆ B a¯ . ¯ c c (4)(Ac )a = (Aa )c , (Ac )a¯ = (Aa )c . (5) (A ∪ B)a = Aa ∩ B a , (A ∪ B)a¯ = Aa¯ ∩ B a¯ ,(A ∩ B)a = Aa ∪ B a , (A ∩ B)a¯ = a A¯ ∪ B a¯[ . \ [ \ \ [ \ (6) ( (At ))a = At a , ( (At ))a¯ ⊆ At a¯ , ( At )a ⊇ (At )a , ( At )a¯ = [

t∈T

t∈T

(At )a¯ .

t∈T

t∈T

t∈T

t∈T

t∈T

t∈T

a∧b a b a∧b (7) Aa∨b = Aa ∪ Ab , Aa∨b =^Aa¯ ∪ Ab = Aa¯ ∩ Ab ¯ , A _= A ∩ A , A ¯. ¯ (8) Let at ∈ R(t ∈ T ), a = at and b = at , then

a

A =

\ t∈T

t∈T

A , Aa¯ ⊆ at

\ t∈T

t∈T at

b

A , A ⊇

[ t∈T

(9) A−∞ = ∅, A+∞ = X. ¯ X and a ∈ R, ¯ then Property 3.4. Let A, B ∈ R (1) A[a] ⊆ A[a] . ¯ b =⇒ A ⊆ A , A ⊆ A . (2) a 6 [a] [b] [a] [b] ¯ (3) A ⊆ B =⇒ A[a] ⊆ B[a] , A[a] ⊆ B¯[a] . (4) (Ac )[a] = (A[ac ] )c , (Ac )[a] ¯= (A[a¯c ] )c . ¯

Aa t , A b ¯=

[ t∈T

Aat .

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J. LI AND H. LI

(5) (A ∪ B)[a] = A[a] ∪ B[a] , (A ∪ B)[a] = A[a] ∪ B[a] ,(A ∩ B)[a] = A[a] ∩ B[a] , ¯ ¯ ¯ (A ∩ B)[a] = A[a] ∩ B[a] . [ \ [¯ [ \ [¯ ¯ (At )[a] ⊆ ( At )[a] , (At )[a] = ( At )[a] , ( At )[a] = (At )[a] , (6) ¯ ¯ t∈T t∈T t∈T t∈T t∈T t∈T T T ( t∈T At )[a] ⊆ t∈T (At )[a] . ¯ = A ∪ A¯ , A (7) A[a∨b] [a] [b] [a∨b] = A[a] ∪ A[b] , A[a∧b] = A[a] ∩ A[b] , A[a∧b] = ¯ ¯ A[a] ∩ A[b] . ^ _ ¯ ¯ ¯ ∈ T ), a = (8) Let at ∈ R(t at and b = at , then t∈T

A[a] =

\

t∈T

\

[ [ A[at ] , A[a] ⊆ A[at ] , A[b] ⊇ A[at ] , A[b] = A[at ] . ¯ ¯ t∈T t∈T t∈T t∈T

(9)A[−∞] = ∅, A[+∞] = X. ¯ X and a ∈ R, ¯ then Property 3.5. Let A, B ∈ R [a] [a] (1) A ¯ ⊆ A . (2) a 6 b =⇒ A[a] ⊇ A[b] , A[a¯] ⊇ A[b ¯ ]. (3) A ⊆ B =⇒ B [a] ⊆ A[a] , B [a¯] ⊆ A[a¯] . c c (4) (Ac )[a] = (A[a ] )c , (Ac )[a¯] = (A[a ] )c . (5) (A ∪ B)[a] = A[a] ∩ B [a] , (A ∪ B)[a¯] = A[a¯] ∩ B [a¯] ,(A ∩ B)[a] = A[a] ∪ B [a] , [a] [a] [a] (A ∩ B)[ ¯ = A ¯ ∪ B\ ¯. [ \ \ [ [a] (6) ( (At )) = At [a] , ( (At ))[a¯] ⊆ At [a¯] , ( At )[a] ⊇ (At )[a] , t∈T t∈T t∈T t∈T t∈T t∈T S T ( t∈T At )[a¯] = t∈T (At )[a¯] . ] [a∧b] = A[a] ∪ A[b] , A[a∧b] = (7) A[a∨b] = A[a] ∩ A[b] , A[a∨b] = A[a¯] ∩ A[b ¯, A [a] [b] A¯ ∪A¯ . ^ _ ¯ ∈ T ), a = (8) Let at ∈ R(t at and b = at , then t∈T

A[b] =

\

] A[at ] , A[b ¯ ⊆

t∈T

\

t∈T

A[at ] , A[a] ⊇

t∈T

[

A[at ] , A[a¯] =

t∈T

[

A[at ] .

t∈T

(9)A[−∞] = X, A[+∞] = ∅. ¯ 4. The decomposition theorems of R-fuzzy sets ¯ In this section, the decomposition theorems of R-fuzzy sets are obtained. ¯ we define aA, a • A ∈ R ¯ X as follow: For A ∈ P (X) and a ∈ R, ½ (aA)(x) = Theorem [ 4.1 (1)A = aAa

Ac =

a, x ∈ A; −∞, x ∈ / A, \

½ (a • A)(x) =

a, x ∈ A; +∞, x ∈ / A.

ac • Aa ;

¯ a∈R

[

¯

¯ a∈R

¯ a∈R

a∈R \ aAa Ac = ac • Aa ; ¯ ¯ ¯ ¯ a∈R a∈R ¯ → P (X) such that Aa ⊆ H(a) ⊆ Aa , then (3)Let H : R [ \ ¯ (i) A = aH(a) Ac = ac • H(a)

(2)A =

¯ =⇒ H(a) ⊇ H(b) (ii) a < b(a, b ∈ R),

THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS

65

[ H(α), Aa = H(α) ¯ α>a αa

ac =

_

A(x)>a

a = A(x).

A(x)>a

With the same reason, we can prove (2). (3)(i) From (1), (2) and Aa ⊆ H(a) ⊆ Aa , (i) of (3) is obvious. (ii)Since a < b and Aa ⊆ H(a) ⊆ Aa , we have¯ H(a) ⊇ Aa ⊇ Ab ⊇ H(b). (iii) From (8) of Property \ \ ¯ ¯\ 3.2 and Aa ⊆ H(a) ⊆ Aa , we have Aa = Aα ⊇ H(α) ⊇ Aα = Aa . With ¯ α λ; ¯ X and a, λ ∈ R. ¯ (3) f (λ, a ◦ A) = for ∀A ∈ R f (λ, A), a < λ. ¯ and A ∈ R ¯X . then, f (λ, A) = Aλ for ∀λ ∈ R Proof. Case 1: λ > −∞. We have \ \ [ f (λ, A) = f (λ, a ◦ Aa ) = f (λ, a ◦ Aa ) = f (λ, Aa ) =

[ a −∞. We have [ [ f (λ, A) = f (λ, ac A[a] ) = f (λ, ac A[a] ) ¯ ¯ ¯ ¯ a∈R a∈R [ [ = f (λ, A[a] ) = A[a] = A _ = A[λ] . [ a] ¯ ¯ aa a −λ; ¯ X and a, λ ∈ R. ¯ (3) f (λ, a ◦ A) = for ∀A ∈ R f (λ, A), a < −λ. ¯ and A ∈ R ¯X . then, f (λ, A) = A[λ] for ∀λ ∈ R

THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS

69

Proof. Case 1: λ < +∞. We have \ \ \ f (λ, A) = f (λ, ac ◦ A[a] ) = f (λ, ac ◦ A[a] ) = f (λ, A[a] ) =

¯ a∈R

\

¯ a∈R

A[a] = A ^ [

λ −∞; ∅, a > λ; ¯ X and a, λ ∈ R. ¯ for ∀A ∈ R (3) f (λ, a • A) = f (λ, Ac ), a < λ. ¯ and A ∈ R ¯X . then, f (λ, A) = Aλ for ∀λ ∈ R Proof. Case 1: λ > −∞. We have \ [ [ f (λ, A) = f (λ, a • Aa¯ ) = f (λ, a • Aa¯ ) = f (λ, (Aa¯ )c ) =

¯ a∈R

[ aλ

a

a

a>λ

Case 2: λ = +∞. Let a = +∞, then it is easy to verify that f (λ, A) = X = Aλ . ¯ A∈R ¯X . From case 1 and case 2, we have f (λ, A) = Aλ for ∀λ ∈ R, Theorem \6.7. If the [ f -cut set f satisfies the following conditions (1) f (λ, At ) = f (λ, At ); t∈T

t∈T

(2) f (λ, A) = Ac , ½for A ∈ P (X) and λ < +∞; ∅, a > −λ; ¯ X and a, λ ∈ R. ¯ (3) f (λ, a • A) = for ∀A ∈ R f (λ, Ac ), a < −λ. ¯ and A ∈ R ¯X . then, f (λ, A) = A[λ] for ∀λ ∈ R

70

J. LI AND H. LI

Proof. Case 1: λ < +∞. We have \ [ [ f (λ, A) = f (λ, ac • A[a¯] ) = f (λ, ac • A[a¯] ) = f (λ, (A[a¯] )c ) =

¯ a∈R

[

¯ a∈R

a>λ

[a]

a>λ

A ¯ = A ^ = A[λ] . a] [ a>λ

Case 2: λ = +∞. Let a = −∞, then it is easy to verify that f (λ, A) = ∅ = A[λ] . ¯ A∈R ¯X . From case 1 and case 2, we have f (λ, A) = A[λ] for ∀λ ∈ R, Theorem \ f -cut set f satisfies the following conditions [6.8. If the At ) = f (λ, At ); (1) f (λ, t∈T

t∈T

(2) f (λ, A) = Ac , for ½ A ∈ P (X) and λ > −∞; X, a 6 −λ; ¯ X and a, λ ∈ R. ¯ (3) f (λ, a M A) = for ∀A ∈ R f (λ, Ac ), a > −λ. ¯ and A ∈ R ¯X . then, f (λ, A) = A[λ] for ∀λ ∈ R Proof. Case 1: λ > −∞. We have \ \ \ f (λ, A) = f (λ, ac M A[a] ) = f (λ, ac M A[a] ) = f (λ, (A[a] )c ) =

\ a