A new specific property of Schweizer-Sklard operators is discovered that Schweizer-Sklard operations are continued fuzzy operators. The relation between Schweizer-Sklard operators and other Fuzzy operators is analysis. The conclusion that Schweizer-Sklard operations could partly replace other fuzzy operations most in use is given. At the same time the relation between Schweizer-Sklard operations and Zadeh operations as well as probability operators is given

WANG, Chunli

ZHANG, Shiqiang

English

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Studies in Mathematical SciencesVol. 3, No. 1, 2011, pp. 52-57DOI: 10.3968/j.sms.1923845220110301.1Z482ISSN 1923-8444 [Print]ISSN 1923-8452 [Online]www.cscanada.netwww.cscanada.orgA New Specific Property of Schweizer-Sklard OperatorsWANG Chunli1; ZHANG Shiqiang1,∗1Department of mathematics; Lab. of forensic medicine and biomedicine information, Chongqing Medical University,China, 400016∗Corresponding author. E-mail: math808@sohu.comReceived 8 June 2011; accepted 31 July 2011Abstract: A new specific property of Schweizer-Sklard operators is discovered that Schweizer-Sklard op-erations are continued fuzzy operators. The relation between Schweizer-Sklard operators and other Fuzzyoperators is analysis. The conclusion that Schweizer-Sklard operations could partly replace other fuzzyoperations most in use is given. At the same time the relation between Schweizer-Sklard operations andZadeh operations as well as probability operators is given.Keywords: Operator; Fuzzy operator; Schweizer-Sklard operator; Continued fuzzy operatorWANG Chunli, ZHANG Shiqiang (2011). A New Specific Property of Schweizer-Sklard Operator. Studies in Mathe-matical Sciences, 3(1), 52-57. Available from: URL: http://www.cscanada.net/index.php/sms/article/view/j.sms.1923845220110301.1Z482. DOI: http://dx.doi.org/10.3968/j.sms.1923845220110301.1Z482.INTRODUCTIONSince Zadeh established Fuzzy set theorem in 1965, he introduced a pair of Fuzzy operators. They arenamed Zadeh operators[1]. There is a lot of discussion about the fuzzy operators[2,3,4,5,6,7,8,9,10], but thereis little discussion about the Schweizer-Sklard operators. This paper discussed membership relation inSchweizer-Sklard, Zadeh operator and other generalized operator in common use.1. PRIMARY DEFINITIONS OF SCHWEIZER-SKLARD OPER-ATORSWe introduce a pair of important operators, Schweizer-Sklard operators, whose primary definition are asfollows[4].Definition 1.1 ∀A˜, B˜, C˜ ∈ P(U),∀u ∈ U,p ∈(£∞,+∞). Schweizer-Sklard product C˜(u) of A˜(u) andB˜(u),denotedC˜(u) = A˜(u)•ss B˜(u), is defined byA˜(u)•ss B˜(u) =0 i f p > 0, A˜(u) = 0 or B˜(u) = 0[A˜(u)−p + B˜(u)−p − 1]−1/p i f p > 0, A˜(u) • B˜(u) , 0A˜(u) • B˜(u) i f p = 0[A˜(u)−p + B˜(u)−p − 1]−1/p i f p < 0, A˜(u)−p + B˜(u)−p > 10 i f p < 0, A˜(u)−p + B˜(u)−p ≤ 1(1.1)52WANG Chunli; ZHANG Shiqiang/Studies in Mathematical Sciences Vol.3 No.1, 2011Definition 1.2 ∀A˜, B˜, C˜ ∈ P(U),∀u ∈ U,p ∈(-∞,+∞). Schweizer-Sklard sum C˜(u) of A˜(u) andB˜(u), denotedC˜(u) = A˜(u)+ss B˜(u), is defined byA˜(u)+ss B˜(u) =1 i f p > 0, A˜(u) = 1 or B˜(u) = 11 − {[1 − A˜(u)]−p + [1 − B˜(u)]−p − 1}−1/p i f p > 0, A˜(u) • B˜(u) , 1A˜(u) + B˜(u) − A˜(u) • B˜(u) i f p = 01 − {[1 − A˜(u)]−p + [1 − B˜(u)]−p − 1}−1/p i f p < 0, [1 − A˜(u)]−p + [1 − B˜(u)]−p > 10 i f p < 0, [1 − A˜(u)]−p + [1 − B˜(u)]−p ≤ 1(1.2)The pair of operators (•ss,+ss) is called Schweizer-Sklard operators.2. A NEW IMPORTANT SPECIFIC PROPERTY OF SCHWEIZER-SKLARD OPERATORSOn property of Schweizer-Sklard operators, the paper[4] has proved that formula (1.1) and (1.2) are monotonefunctions for variables A˜(u) orB˜(u).But a new important specific property of Schweizer-Sklard operators has not mentioned up to now.We give a new conclusion as follows:Theorem 2.1 When p >0, formula (1.1) is monotone increasing function for the parameter p.Proof. ∀u ∈ U, if p >0, A˜(u) · B˜(u) ,0, letF(p) = [A˜(u)−p + B˜(u)−p − 1]−1/p = exp{ ln[A˜(u)−p + B˜(u)−p − 1]−p }then∂∂pF(p) = F(p) • ∂∂p{ ln[A˜(u)−p + B˜(u)−p − 1]−p } = F(p) •G(p)G(p) =[A˜(u)−p + B˜(u)−p − 1] ln[A˜(u)−p + B˜(u)−p − 1] − A˜(u)−p ln A˜(u)−p − B˜(u)−p ln B˜(u)−pp2[A˜(u)−p + B˜(u)−p − 1] .Let x = A˜(u)−p, y = B˜(u)−p thenG(p) =(x + y − 1) ln(x + y − 1) − x ln x − y ln yp2(x + y − 1) =1p2ln[(x + y − 1)x1− 1x + yx y1− 1y + xy ].Because A˜(u) ∈(0,1] and B˜(u) ∈(0,1],then x = A˜(u)−p ∈[1,+∞) and y = B˜(u)−p ∈[1,+∞).this means x + y-1≥1, y/x >0, x/y >0, 1-1/x ≥0, 1-1/y ≥0. We have1-1/x + y/x >0, 1-1/y + x/y >0thenx1−1x +yx ≥ 1, y1− 1y + xy ≥ 1This means G(P) ≥0. Because F(p) >0,we have∂∂pF(p) = F(p) • ∂∂p{ ln[A˜(u)−p + B˜(u)−p − 1]−p } = F(p) •G(p) ≥ 0This means that when p >0, formula (1.1) is monotone increasing function for the parameter p.53WANG Chunli; ZHANG Shiqiang/Studies in Mathematical Sciences Vol.3 No.1, 2011Theorem 2.2 When p <0, formula (1.1) is monotone decreasing function for the parameter p.Proof. ∀u ∈ U, if p <0, A˜(u)−p + B˜(u)−p >1, letF(p) = [A˜(u)−p + B˜(u)−p − 1]−1/p = exp{ ln[A˜(u)−p + B˜(u)−p − 1]−p }then∂∂pF(p) = F(p) • ∂∂p{ ln[A˜(u)−p + B˜(u)−p − 1]−p } = F(p) •G(p)G(p) =[A˜(u)−p + B˜(u)−p − 1] ln[A˜(u)−p + B˜(u)−p − 1] − A˜(u)−p ln A˜(u)−p − B˜(u)−p ln B˜(u)−pp2[A˜(u)−p + B˜(u)−p − 1] .Let x = A˜(u)−p,y = B˜(u)−p thenG(p) =(x + y − 1) ln(x + y − 1) − x ln x − y ln yp2(x + y − 1) =1p2ln[(x + y − 1)x1− 1x + yx y1− 1y + xy ].Because A˜(u) ∈(0,1] and B˜(u) ∈(0,1],then x = A˜(u)−p ∈(0,1] and y = B˜(u)−p ∈(0,1].We know x + y >1, then0< x + y-1<1.Because y ≤1 then y-1≤0, that is 0< x + y-1< x.This means0 < x+y−1x ≤ 1, that is0 < x1−1x +yx ≤ 1.Similarly we have0 < x+y−1y ≤ 1.that is0 < y1−1y +xy ≤ 1.This means G(P) ≤0. Because F(p) >0, we can introduce∂∂pF(p) = F(p) • ∂∂p{ ln[A˜(u)−p + B˜(u)−p − 1]−p } = F(p) •G(p) ≤ 0This means that when p <0, formula (1.1) is monotone decreasing function for the parameter p.Theorem 2.3 When p >0, formula (1.2) is monotone decreasing function for the parameter p.Proof. Omitted.This means that when p >0, formula (1.2) is monotone decreasing function for the parameter p.Theorem 2.4 When p <0, formula (1.2) is monotone increasing function for the parameter p.Proof. Omitted.This means that when p <0, formula (1.2) is monotone increasing function for the parameter p.The formula (1.1) and the formula (1.2) have defined infinite fuzzy operators. It is quite evident that from thetheorem 2.1 to the theorem 2.4 have given a sort of continued Fuzzy operators[7,8]. This is a new importantproperty of Schweizer-Sklard operators.3. THE RELATION OF THE SCHWEIZER-SKLARD OPERATIONSAND OTHER FUZZY OPERATIONS MOST IN USETo any fuzzy setA˜, B˜ ∈ P(U), ∀u ∈ U, Zadeh operator and three other common generalized operators(probability operators, boundary operators and infinite operators) have relation[9,10]:A˜(u)∞ˆB˜(u) ⊆ A˜(u) ⊗ B˜(u) ⊆ A˜(u)•ˆB˜(u) ⊆ A˜(u) ∧ B˜(u)54WANG Chunli; ZHANG Shiqiang/Studies in Mathematical Sciences Vol.3 No.1, 2011⊆ A˜(u) ∨ B˜(u) ⊆ A˜(u)+ˆB˜(u) ⊆ A˜(u) ⊕ B˜(u) ⊆ A˜(u) _∞B˜(u)For the Schweizer-Sklard operators(•ss,+ss), we give a new conclusion as follows:Theorem 3.1 ∀ A˜, B˜ ∈ P(U),∀u ∈ U, For the parameter p → +∞, the continued Fuzzy operators, that is,Schweizer-Sklard operators (•ss,+ss) and Zadeh operators (∧,∨) have relation as follows:[A˜(u)•ss B˜(u)]+∞ = A˜(u) ∧ B˜(u) ⊆ A˜(u) ∨ B˜(u) = [A˜(u) +ss B˜(u)]+∞ (3.1)Proof. If the parameter p → +∞, then ∀u ∈ U, the C˜(u) in definition 2.1 is denoted [A˜(u) •ss B˜(u)]+∞ and[A˜(u)+ss B˜(u)]+∞ as follows:limp→+∞[A˜(u)•ss B˜(u)] = [A˜(u)•ss B˜(u)]+∞, limp→+∞[A˜(u)+ss B˜(u)] = [A˜(u)+ss B˜(u)]+∞.Zadeh operators (∧, ∨Å) are denoted byA˜(u) ∧ B˜(u) = min[A˜(u), B˜(u)], A˜(u) ∨ B˜(u) = max[A˜(u), B˜(u)].We subdivide proof into two steps:(a) ∀ A˜, B˜ ∈ P(U), ∀u ∈ U, when the parameter p >0,if A˜(u)B˜(u)=0, thenlimp→+∞[A˜(u)•ss B˜(u)] = 0 = min[A˜(u), B˜(u)] = A˜(u) ∧ B˜(u).If A˜(u)B˜(u) ,0, let A˜(u) ≥ B˜(u) thenlimp→+∞[A˜(u)•ss B˜(u)] = limp→+∞[A˜(u)−p + B˜(u)−p − 1]−1/p= limp→+∞ expln[A˜(u)−p + B˜(u)−p − 1]−p= limp→+∞ expA˜(u)−p ln A˜(u) + B˜(u)−p ln B˜(u)A˜(u)−p + B˜(u)−p − 1= B˜(u) = A˜(u) ∧ B˜(u)(b) ∀ A˜, B˜ ∈ P(U), ∀u ∈ U, when the parameter p >0, if A˜(u)=1 or B˜(u)=1, thenlimp→+∞[A˜(u)+ss B˜(u)] = 1 = max[A˜(u), B˜(u)] = A˜(u) ∨ B˜(u)If A˜(u)B˜(u) ,1, let A˜(u) ≥ B˜(u) thenlimp→+∞[A˜(u)+ss B˜(u)] = limp→+∞{1 − {[1 − A˜(u)]−p + [1 − B˜(u)]−p − 1}−1/p}= 1 − limp→+∞ expln{[1 − A˜(u)]−p + [1 − B˜(u)]−p − 1}−p= 1 − limp→+∞ exp[1 − A˜(u)]−p ln[1 − A˜(u)] + [1 − B˜(u)]−p ln[1 − B˜(u)][1 − A˜(u)]−p + [1 − B˜(u)]−p − 1= A˜(u) = A˜(u) ∨ B˜(u)It is quite evident that (a) and (b) implies (3.1).Theorem 3.2 ∀ A˜, B˜ ∈ P(U), ∀u ∈ U, for the parameter p →+0, the continued Fuzzy operators, that is,Schweizer-Sklard operators (•ss,+ss) and probability operators (•ˆ,+ˆ) have relation as follows:A˜(u)•ˆB˜(u) = [A˜(u) •ss B˜(u)]+0 ⊆ [A˜(u) +ss B˜(u)]+0 = A˜(u)+ˆB˜(u) (3.2)55WANG Chunli; ZHANG Shiqiang/Studies in Mathematical Sciences Vol.3 No.1, 2011Proof. If the parameter p →+0, then ∀u ∈ U, the C˜(u) in definition 1.1 is denoted [A˜(u) •ss B˜(u)]+0 and[A˜(u)+ss B˜(u)]+0 as follows:limp→+0[A˜(u)•ss B˜(u)] = [A˜(u)•ss B˜(u)]+0, limp→+0[A˜(u)+ss B˜(u)] = [A˜(u)+ss B˜(u)]+0.Zadeh operators (•ˆ,+ˆ) are denoted byA˜(u)•ˆB˜(u) = A˜(u)B˜(u), A˜(u)+ˆB˜(u) = A˜(u) + B˜(u) − A˜(u)B˜(u).We subdivide proof into two steps:(a) ∀ A˜, B˜ ∈ P(U), ∀u ∈ U, when the parameter p >0,if A˜(u)B˜(u)=0, thenlimp→+∞[A˜(u)•ss B˜(u)] = 0 = A˜(u)B˜(u) = A˜(u)•ˆB˜(u)If A˜(u)B˜(u) ,0, let A˜(u) ≥ B˜(u) thenlimp→+0[A˜(u)•ss B˜(u)] = limp→+0[A˜(u)−p + B˜(u)−p − 1]−1/p= limp→+0expln[A˜(u)−p + B˜(u)−p − 1]−p= limp→+0expA˜(u)−p ln A˜(u) + B˜(u)−p ln B˜(u)A˜(u)−p + B˜(u)−p − 1 = A˜(u)B˜(u) = A˜(u)•ˆB˜(u)(b) ∀ A˜, B˜ ∈ P(U), ∀u ∈ U, when the parameter p >0, if A˜(u)=1 or B˜(u)=1, let A˜(u)=1 thenlimp→+0[A˜(u)+ss B˜(u)] = 1 = 1 + B˜(u) − 1 • B˜(u) = A˜(u) + B˜(u) − A˜(u)B˜(u) = A˜(u)+ˆB˜(u)If A˜(u)B˜(u) ,1, let A˜(u) ≥ B˜(u) thenlimp→+0[A˜(u)+ss B˜(u)] = limp→+0{1 − {[1 − A˜(u)]−p + [1 − B˜(u)]−p − 1}−1/p}= 1 − limp→+0expln{[1 − A˜(u)]−p + [1 − B˜(u)]−p − 1}−p= 1 − limp→+0exp[1 − A˜(u)]−p ln[1 − A˜(u)] + [1 − B˜(u)]−p ln[1 − B˜(u)][1 − A˜(u)]−p + [1 − B˜(u)]−p − 1= A˜(u) + B˜(u) − A˜(u)B˜(u)= A˜(u)+ˆB˜(u)It is quite evident that (a) and (b) implies (3.2).From theorem 2.1 to theorem 3.2 implies that continued Fuzzy operators, Schweizer-Sklard operations,could replace other fuzzy operations most in use. At the same time, when the parameter p >0, fromtheorem 2.1 to theorem 3.2 implies that continued Fuzzy operators, Schweizer-Sklard operations, and Zadehoperations as well as probability operators have relation as follows:A˜(u)•ˆB˜(u) ⊆ A˜(u) •ss B˜(u) ⊆ A˜(u) ∧ B˜(u) ⊆ A˜(u) ∨ B˜(u) ⊆ A˜(u) +ss B˜(u) ⊆ A˜(u)+ˆB˜(u) (3.3)CONCLUSIONFrom theorem 2.1 to theorem 3.2 implies that Schweizer-Sklard operations are continued Fuzzy operatorsand that Schweizer-Sklard operations could partly replace other fuzzy operations most in use.56WANG Chunli; ZHANG Shiqiang/Studies in Mathematical Sciences Vol.3 No.1, 2011REFERENCES[1] L. A. Zadeh (1965). Fuzzy Sets, Information and Control, (8), 338-359.[2] LI H. X., WANG P. Z. (1994). Fuzzy Mathematics. Beijing: National Defence Industriy Press, 128-129.[3] CHENG Y. Y. (1984). Fuzzy Mathematics. Wuhan: Huazhong Institute of Technology Press, 30-36.[4] LOU C. Z. (1989). Introductory Fuzzy Sets. Beijing: Beijing Normal University Press, 44-56.[5] SU Z. F. 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International Conference on Computer, Mechatronics,Control and Electronic Engineering(CMCE2010), IEEE, 1, 144-147.57

A New Specific Property of Schweizer-Sklard Operators

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A New Specific Property of Schweizer-Sklard Operators

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