THE CUT SETS, DECOMPOSITION THEOREMS AND REPRESENTATION THEOREMS ON R-FUZZY SETS

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 6, Number 1, Pages c 2010 Institute for Scientific Computing and Information 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 type- II 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 theorem 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 X}, R to stand for real numbers set, and R = R {, + }. For A, B P (X), A B = {x X x A or x B}, A B = {x X x A and x B}, and A c = {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,

2 62 J. LI AND H. LI A B A B and A B. For a, b R, a b = sup{a, b}, a b = inf{a, b}, and a c = a. It is easy to verify that ( R,,, c) is fuzzy lattice. We use T to 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: Definition 2.1 [1]. The generalized real function A : X R is called R-fuzzy set over X, and R X stands for the set of all R-fuzzy sets over X. For A, B R, we define A B, A B, and A c by the membership functions as follows: A B : X R x A(x) B(x); A B : X R x A(x) B(x); A c : X R x (A(x)) c. We also define A B by A(x) 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 cut sets for R-fuzzy sets can be obtained as follows: For A R X and a R, (1) (1) : A a = {x X A(x) a}, Aā = {x X A(x) > a}, (2) : A [a] = {x X a + A(x) 0}, A [ā] = {x X a + A(x) > 0}, (3) : A a = {x X A(x) a}, A ā = {x X A(x) < a}, (4) : A [a] = {x X a + A(x) 0}, A [ā] = {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. Property 3.1. For A R X and a R, we have (1)A [a] = A a c, A [ā] = A a c; (2)A [a] = A ac, A [ā] = A ac ; (3)A a = Aā c, A ā = A a c ; (4)A [a] = A a c c, A [ā] = A a c c. Proof. According to (1), the proof is obvious. Property 3.2. Let A, B R X and a R, then (1) Aā A a. (2) a b = A a A b, Aā A b. (3) A B = A a B a, Aā Bā. (4) (A c ) a = (A a c) c, (A c )ā = (A a c) c. (5) (A B) a = A a B a, (A B)ā = Aā Bā,(A B) a = A a B a, (A B)ā = Aā Bā. (6) (A t ) a ( A t ) a, (A t )ā = ( A t )ā, ( A t ) a = (A t ) a, ( A t )ā (A t )ā. (7) A a b = A a A b, A a b = Aā A, A a b = A a A b, A a b = Aā A b. b (8) Let a t R(t T ), a = a t and b = a t, then A b = A at, A A at, A a A at, Aā = A at. b (9) A = X, A + = Proof. (1)-(5) is obvious.

3 THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS 63 (6) (i) x (A t ) a, t T such that x (A t ) a. It follows that ( A t)(x) A t (x) a, that is x ( A t) a. Hence, (A t ) a ( A t ) a. (ii) x (A t )ā, t T such that x (A t )ā. It follows that ( A t)(x) A t (x) > a, that is x ( A t)ā. Hence, (A t )ā ( A t )ā. On the other hand, for x ( A t)ā, we have ( A t)(x) > a. It follows that there exits a t T such that A t (x) > a, namely, x (A t )ā (A t )ā. So we have (A t)ā = ( A t)ā. Similarly, we can prove ( A t) a = (A t) a and ( A t)ā (A t)ā. (7) It is easy to verify that A a b = {x A(x) (a b)} = {x A(x) a and A(x) b} = A a A b and A a b = {x A(x) (a b)} = {x A(x) a or A(x) b}a a A b. Similarly, we can prove A a b = Aā A and A a b = Aā A b. (8)It is easy to verify that b A b = {x A(x) a t } = A at, Aā = A at. Since A(x) > a t ( for t T ) A(x) > a t and A(x) a t there exists t T such thata(x) a t, we have A A at, A a A at. b (9) is obvious. It is easy to get the following properties by Property 3.1 and Property 3.2. Property 3.3. Let A, B R X and a R, then (1) A ā A a. (2) a b = A a A b, A ā A b. (3) A B = A a B a, A ā B ā. (4)(A c ) a = (A ac ) c, (A c ) ā = (A ac ) c. (5) (A B) a = A a B a, (A B) ā = A ā B ā,(a B) a = A a B a, (A B) ā = A ā B ā. (6) ( (A t )) a = A a t, ( (A t )) ā A ā t, ( A t ) a (A t ) a, ( A t ) ā = (A t ) ā. (7) A a b = A a A b, A a b = A ā A b, Aa b = A a A b, A a b = A ā A b. (8) Let a t R(t T ), a = a t and b = a t, then A a = A a t, A ā A a t, A b A a t, A b = A a t. (9) A =, A + = X. Property 3.4. Let A, B R X and a R, then (1) A [ā] A [a]. (2) a b = A [a] A [b], A [ā] A [b ]. (3) A B = A [a] B [a], A [ā] B [ā]. (4) (A c ) [a] = (A [a c ]) c, (A c ) [ā] = (A [a c ]) c.

4 64 J. LI AND H. LI (5) (A B) [a] = A [a] B [a], (A B) [ā] = A [ā] B [ā],(a B) [a] = A [a] B [a], (A B) [ā] = A [ā] B [ā]. (6) (A t ) [a] ( A t ) [a], (A t ) [ā] = ( A t ) [ā], ( A t ) [a] = (A t ) [a], ( A t) [ā] (A t) [ā]. (7) A [a b] = A [a] A [b], A [a b] = A [ā] A, A [b ] [a b] = A [a] A [b], A [a b] = A [ā] A. [b ] (8) Let a t R(t T ), a = a t and b = a t, then A [a] = A [at ], A [ā] A [at ], A [b] A [at ], A = [b ] A [at ]. (9)A [ ] =, A [+ ] = X. Property 3.5. Let A, B R X and a R, then (1) A [ā] A [a]. (2) a b = A [a] A [b], A [ā] A [b ]. (3) A B = B [a] A [a], B [ā] A [ā]. (4) (A c ) [a] = (A [ac] ) c, (A c ) [ā] = (A [ac] ) c. (5) (A B) [a] = A [a] B [a], (A B) [ā] = A [ā] B [ā],(a B) [a] = A [a] B [a], (A B) [ā] = A [ā] B [ā]. (6) ( (A t )) [a] = A [a] t, ( (A t )) [ā] A [ā] t, ( A t ) [a] (A t ) [a], ( A t) [ā] = (A t) [ā]. (7) A [a b] = A [a] A [b], A [a b] = A [ā] A [b ], A [a b] = A [a] A [b], A [a b] = A [ā] A [b ]. (8) Let a t R(t T ), a = a t and b = a t, then A [b] = A [at], A [b ] A [at], A [a] A [at], A [ā] = A [at]. (9)A [ ] = X, A [+ ] =. 4. The decomposition theorems of R-fuzzy sets In this section, the decomposition theorems of R-fuzzy sets are obtained. For A P (X) and a R, we define aa, a A R X as follow: { (aa)(x) = Theorem 4.1 (1)A = aa a (2)A = aaā a, x A;, x / A, A c = a c A a ; A c = a c Aā; { (a A)(x) = (3)Let H : R P (X) such that Aā H(a) A a, then (i) A = ah(a) A c = a c H(a) (ii) a < b(a, b R), = H(a) H(b) a, x A; +, x / A.

5 THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS 65 (iii) A a = H(α), Aā = H(α) α<a α>a Proof. (1) We have ( aa a )(x) = [aa a (x)] = ( ( a c A a )(x) = (a c A a (x)) = A(x) a a c = A(x) a A(x) a a = A(x). [aa a (x)] = A(x) and With the same reason, we can prove (2). (3)(i) From (1), (2) and Aā H(a) A a, (i) of (3) is obvious. (ii)since a < b and Aā H(a) A a, we have H(a) Aā A b H(b). (iii) From (8) of Property 3.2 and Aā H(a) A a, we have A a = A α α<a α<a same reason, we can prove Aā = α>a H(α). Similarly, we can obtain the following theorems: Theorem 4.2 (1)A = a A a, A c = a c A a (2)A = a A ā, A c = a c A ā (3)Let H : R P (X) such that A ā H(a) A a then (i)a = a H(a), A c = a c H(a) (ii)a < b(a, b R), = H(a) H(b) (iii)a a = H(α) H(α), A ā = a<α α>a Theorem 4.3 1)A = a c A [a] A c = a A [a] ; (2)A = a c A [ā] A c = a A [ā]; (3)Let H : R P (X) such that A [ā] H(a) A [a], then (i) A = a c H(a) A c = a H(a) (ii) a < b(a, b R), = H(a) H(b) (iii) A [a] = H(α), Aā = H(α) α>a α<a Theorem 4.4 (1)A = a c A [a], A c = (2)A = a c A [ā], A c = aa [a] aa [ā] (3)Let H : R P (X) such that A [ā] H(a) A [a] then (i)a = a c H(a), A c = ah(a) (ii)a < b(a, b R), = H(a) H(b) (iii)a [a] = H(α) a<α H(α), A [ā] = α<a H(α) α<a A α = A a. With

6 66 J. LI AND H. LI 5. The representation theorems of R-fuzzy sets Let U (X) = {H H is inverse order nested set}. The operations,, and c on U (X) are defined as follows: (2) H t : ( H t )(λ) = H t (λ) H t : ( H t )(λ) = H t (λ) H c : H c (λ) = (H( λ)) c Then (U (X),,, c ) is fuzzy lattice. We can get the representation theorems as follows: Theorem 5.1 Let T : U (X) R X, H T (H) = λh(λ). Then, (1) T (H) λ H(λ) T (H) λ. (2) T (H) λ = H(a), T (H) λ = H(a) λ<a (3) T is a surjective homomorphism from U (X) to R X, that is T is surjection and T ( H t ) = T (H t ), T ( H λ ) = T (H λ ), T (H c ) = (T (H)) c. λ R Proof. (1) For x T (H) λ, we have T (H)(x) = λ R [λh(λ)(x)] > λ. It follows that λ 0 > λ such that x H(λ 0 ) H(λ). On the other hand, H(λ) T (H) λ is obvious. So we have T (H) λ H(λ) T (H) λ. (2) From (8) of Property 3.2 and (1), we have T (H) λ = (T (H) a ) H(a) (T (H)) a = T (H) λ. With the same reason, we have T (H) λ = λ<a H(a). (3) For A R X, let H A : R P (X) be defined by H A (a) = A a. known by (1) of Property 3.2 and (1) of Theorem 3.1, H A is inverse order nested set and T (H A ) = aa a = ah A (a). Hence T is surjection. T ( H t ) = λ( H t (λ)) = λ R (λh t (λ)) = λ R λ R (λh t (λ)) = T (H t ). With the same reason, we have T ( H λ ) = T (H λ ) and T (H c ) = (T (H)) c. Similarly, we can get the following theorems. Let V (X) = {H is order nested set}. The operations,, and c are same as (2), then (V (X),,, c ) is also fuzzy lattice. We also can get the following theorem. Theorem 5.2 Let T : V (X) R X, H T (H) = λ c H(λ). Then, (1) T (H) [λ] H(λ) T (H) [λ]. (2) T (H) [λ] = H(a), T (H) [λ] = H(a) λ>a (3) T is a surjective homomorphism from V (X) to R X, that is T is surjection and T ( H t ) = T (H t ), T ( H λ ) = T (H λ ), T (H c ) = (T (H)) c. λ R

7 THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS 67 (3) However, If the operations,, and c on V (x) is defined as H t : ( H t )(λ) = H t (λ) H t : ( H t )(λ) = H t (λ) H c : H c (λ) = (H( λ)) c, then, (V (X),,, c ) is also a fuzzy lattice, and we can get the following theorem: Theorem 5.3 Let T : V (X) R X, H T (H) = λ H(λ). Then, (1) T (H) λ H(λ) T (H) λ. (2) T (H) λ = H(a), T (H) λ = H(a) λ>a (3) T is a surjective homomorphism from V (X) to R X, that is T is surjection and T ( H t ) = T (H t ), T ( H λ ) = T (H λ ), T (H c ) = (T (H)) c. If the operations,, and c on U (X) are same as (3), then (U (X),,, c ) is also a fuzzy lattice and we have the following theorem. Theorem 5.4 Let T : U (X) R X, H T (H) = λ c H(λ). Then, (1) T (H) [λ] H(λ) T (H) [λ]. (2) T (H) [λ] = H(a), T (H) [λ] = H(a) λ<a (3) T is a surjective homomorphism from U (X) to R X, that is T is surjection and T ( H t ) = T (H t ), T ( H λ ) = T (H λ ), T (H c ) = (T (H)) c. λ R λ R 6. The axiomatic definition of the cut sets The axiomatic definition of cut sets for ordinary fuzzy sets is introduced by Yuan and Wu [3] and the common properties of cut sets are discussed. Similarly, we give axiomatic definition of R-fuzzy sets and discuss its properties. For A R X and λ R, we define λa, λ A, λ A, λ A R X as follows: For x X, (λa)(x) = λ A(x), λ A(x) = λ A(x); (λ A)(x) = λ (A(x)) c, (λ A)(x) = λ (A(x)) c. Definition 6.1. The function f : R R X P (X), (a, A) f(a, A) is called f-cut sets of A. Theorem 6.1. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A, { for A P (X) and λ < + ;, a λ; (3) f(λ, aa) = for A f(λ, A), a > λ. then, f(λ, A) = A λ for λ R and A R X.

8 68 J. LI AND H. LI Proof. Case 1: λ < +. We have f(λ, A) = f(λ, aaā) = = Aā = A f(λ, aaā) = a = A λ. f(λ, Aā) Case 2: λ = +. Let a = +, then it is easy to verify that f(λ, A) = = A λ. From case 1 and case 2, we have f(λ, A) = A λ for λ R, A R X. Theorem 6.2. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A, for { A P (X) and λ > ; X, a λ; (3) f(λ, a A) = for A f(λ, A), a < λ. then, f(λ, A) = A λ for λ R and A R X. Proof. Case 1: λ >. We have f(λ, A) = f(λ, a A a ) = f(λ, A a ) = A a = A a f(λ, a A a ) = = A λ. Case 2: λ =. Let a =, then it is easy to verify that f(λ, A) = f(λ, a A) = X = A λ. Theorem 6.3. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A, { for A P (X) and λ > ;, a λ; (3) f(λ, aa) = for A f(λ, A), a > λ. then, f(λ, A) = A [λ] for λ R and A R X. Proof. Case 1: λ >. We have f(λ, A) = f(λ, a c A [ā]) = f(λ, a c A [ā]) = f(λ, A [ā]) = λ>a A [ā] = A [ a] = A [λ]. Case 2: λ =. Let a = +, then it is easy to verify that f(λ, A) = = A [λ]. From case 1 and case 2, we have f(λ, A) = A [λ] for λ R, A R X. Theorem 6.4. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A, for { A P (X) and λ < + ; X, a λ; (3) f(λ, a A) = for A f(λ, A), a < λ. then, f(λ, A) = A [λ] for λ R and A R X.

9 THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS 69 Proof. Case 1: λ < +. We have f(λ, A) = f(λ, a c A [a] ) = f(λ, a c A [a] ) = = λ<a A [a] = A [ a] = A [λ]. a c < λ f(λ, A [a] ) Case 2: λ = +. Let a =, then it is easy to verify that f(λ, A) = f(λ, a A) = X = A [λ]. From case 1 and case 2, we have f(λ, A) = A [λ] for λ R, A R X. Theorem 6.5. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A c,{ for A P (X) and λ > ;, a λ; (3) f(λ, a A) = f(λ, A c for A ), a < λ. then, f(λ, A) = A λ for λ R and A R X. Proof. Case 1: λ >. We have f(λ, A) = f(λ, a A ā ) = f(λ, (A ā ) c ) = A ā = A a f(λ, a A ā ) = = A λ. Case 2: λ =. Let a =, then it is easy to verify that f(λ, A) = = A λ. From case 1 and case 2, we have f(λ, A) = A λ for λ R, A R X. Theorem 6.6. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A c, for { A P (X) and λ < + ; X, a λ; (3) f(λ, a A) = f(λ, A c for A ), a > λ. then, f(λ, A) = A λ for λ R and A R X. Proof. Case 1: λ < +. We have f(λ, A) = f(λ, a A ā ) = f(λ, (A a ) c ) = A a = A a f(λ, a A ā ) = = A λ. Case 2: λ = +. Let a = +, then it is easy to verify that f(λ, A) = X = A λ. From case 1 and case 2, we have f(λ, A) = A λ for λ R, A R X. Theorem 6.7. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A c,{ for A P (X) and λ < + ;, a λ; (3) f(λ, a A) = f(λ, A c for A ), a < λ. then, f(λ, A) = A [λ] for λ R and A R X.

10 70 J. LI AND H. LI Proof. Case 1: λ < +. We have f(λ, A) = f(λ, a c A [ā] ) = = A [ā] = A [ a] f(λ, a c A [ā] ) = = A [λ]. f(λ, (A [ā] ) c ) Case 2: λ = +. Let a =, then it is easy to verify that f(λ, A) = = A [λ]. From case 1 and case 2, we have f(λ, A) = A [λ] for λ R, A R X. Theorem 6.8. If the f-cut set f satisfies the following conditions (1) f(λ, A t ) = (2) f(λ, A) = A c, for { A P (X) and λ > ; X, a λ; (3) f(λ, a A) = f(λ, A c for A ), a > λ. then, f(λ, A) = A [λ] for λ R and A R X. Proof. Case 1: λ >. We have f(λ, A) = f(λ, a c A [a] ) = f(λ, (A [a] ) c ) = A [a] = A [ a] f(λ, a c A [a] ) = = A [λ]. Case 2: λ =. Let a = +, then it is easy to verify that f(λ, A) = X = A [λ]. From case 1 and case 2, we have f(λ, A) = A [λ] for λ R, A R X. Remark: From discussion above, we know that every kind of cut sets has three important properties, which is the same as the respective cut sets for R-fuzzy sets in section Conclusion In this paper, we first introduce some kinds of cut sets for R-fuzzy sets and give their properties, respectively. Next, we obtain decomposition theorems and representation theorems of R-fuzzy sets based these cut sets and their properties. At last, we give the axiomatic definition of cut sets for R-fuzzy sets and discuss its properties. Acknowledgments The author thanks professor Xuehai Yuan for his help in preparing this paper, and the anonymous referees for their helpful suggestions. This research was supported by National Natural Science Foundation of China ( ) and the National 863 High-Tech Program of China (No.2006AA04Z163). References [1] Hongxing Li, Xuehai Yuan, Jiayin Wang, Cheng Zhang, Kaibiao Sun, Degang Wang, R-fuzzy sets and their applications in construting fuzzy systems, to appear. [2] Xuehai Yuan, (1997) New cut sets and their applications, Fuzzy Systems and Mathematics, Vol. 11, pp [3] Xuehai Yuan, Zhihua Wu, (1999) The axiomatic definition of cut sets for fuzzy sets, Journal of Liaoning Normal University (Natural Science), Vol. 22, pp [4] L.A. Zadeh, (1965) Fuzzy sets, Information and Control, Vol. 8, pp [5] L.A. Zadeh, (1975) The concept of a linguistic variable and its application to approximate reasoning-i, Information Sciences, Vol. 8, pp

11 THE CUT SETS, DECOMPOSITION AND REPRESENTATION THEOREMS 71 [6] J.M. Mendel, (2007) Advances in type-2 fuzzy sets and systems, Information Sciences, Vol. 177, pp [7] J.A. Goguen, (1967) L-fuzzy sets, J Math Anal Appl, Vol. 18, pp [8] C. Bertoluzza, M. Solci, M.L. Caodieci, (2001) Measure of a fuzzy set: The α-cut approach in the finite case, Fuzzy Sets and Systems, Vol. 123, pp [9] J.N. Garcia, Z. Kutalik, K.H. Cho et al, (2003) Level sets and the minimum volume sets of probability density function, International Journal of Approximate Reasoning, Vol. 34, pp [10] E. Pap, D. Surla, (2000) Lebesgue measure of α-cut sets a pproach for finding the height of the membership function, Fuzzy Sets and Systems, Vol. 111, pp [11] M.C. Florea, A. L. Jousselme, D. Crenier et al, (2008) Approximation techniques for the transformation of the fuzzy sets into random sets, Fuzzy Sets and Systems, Vol. 159, pp [12] D. Dubios, E. Hullermeier, H. Prade, (2003) On the representation of fuzzy rules in terms of crisp rules, Information Sciences, Vol. 151, pp [13] B. Seselja, A. Tepavcevic, (2003) Completion of ordered structures by cuts of fuzzy sets: an over view, Fuzzy Sets and Systems, Vol. 136, pp School of Mathematical Sciences, Beijing Normal University, Beijing , P.R. China ljq 1023@mail.bnu.edu.cn(Jinquan Li) School of Electronic and Information Engineering, Dalian University of Technology, Dalian , P.R. China lihx@dlut.edu.cn(hongxing Li)