Fuzzy Degree and Similarity Measure of Fuzzy Set ... - Semantic Scholar
Fuzzy Degree and Similarity Measure of Fuzzy Set with Three Kinds of Negations Lei Yang and Zhenghua Pan School of Science, Jiangnan University, Wuxi, China [email protected]
Abstract. Negation in information processing is an important notion, especially in fuzzy information processing. How to cognize and deal with various negations of fuzzy information, a new fuzzy set FSCOM with contradictory negation, opposite negation and medium negation was presented in [1]. This paper defines fuzzy degree and similarity measure of fuzzy set FSCOM, gives some properties and calculation formulas of the fuzzy degree and similarity measure, and also discusses their applications. Keywords: Three kinds of negations of fuzzy set, fuzzy degree, similarity measure.
1
Introduction
Negation in information processing is an important notion, especially in fuzzy information processing. Some scholars suggested that the different negations should be applied in knowledge processing in recent years [2][3]. Zhenghua Pan proposed that there are five kinds of contradictory negative relations and opposite negative relations in concept of knowledge [4], and came up with a fuzzy set FSCOM which has three kinds of negations [1]. Contradictory relation and opposite relation in knowledge information are not distinguished in traditional mathematical and Zadeh fuzzy set, but they are differentiated in fuzzy set FSCOM. In this paper, we define fuzzy degree and similarity measure of the fuzzy set FSCOM, and give the calculation formulas of fuzzy degree and similarity measures, and also discuss their applications.
2
Basic Concept of FSCOM
Definition 1 [5]. Let U be domain. Mapping ΨA : U → [0, 1] confirmed a fuzzy subset A on U (i.e. A∈P(U)), where mapping ΨA is called the membership function of A, ΨA (x) is the degree of membership of x in A, for short A(x). {A(x) | x∈U} for short A(U). H. Deng et al. (Eds.): AICI 2011, Part I, LNAI 7002, pp. 543–550, 2011. © Springer-Verlag Berlin Heidelberg 2011
544
L. Yang and Z. Pan
Definition 2 [1]. Let A ∈ P(U), λ ∈ (0, 1). (1) If mapping Ψ ╕: A(U) → [0, 1] ╕
╕
satisfies Ψ (A(x)) = 1− A(x), Ψ confirmed a fuzzy subset A╕ on U, A╕(x) = Ψ (A(x)). A╕ is called opposite negation set of A. when (2) If mapping Ψ ~: ╕
A(U) → [0, 1] ~
Satisfies Ψ (A(x)) = 2λ − 1 (A(x) −λ)+1−λ, when λ∈[½, 1) and A(x)∈(λ, 1] 1− λ 2λ − 1 A(x)+1−λ, when λ∈[½, 1) and A(x)∈[0, 1−λ) 1− λ 1 − 2λ A(x)+λ, when λ∈(0, ½] and A(x)∈[0, λ)
(1) (2) (3)
λ 1 − 2λ (A(x)+λ −1)+λ, when λ∈(0, ½] and A(x)∈(1−λ, 1] λ
(4)
0.5, when A(x) = 0.5
(5)
~
~
~
~
~
Ψ confirmed a fuzzy subset A on U, A (x) = Ψ (A(x)). A is called medium negation set of A. (3) If mapping Ψ ¬ : A(U) → [0, 1] ¬
satisfies Ψ (A(x)) = Max (A (x), A (x)), Ψ¬ confirmed a fuzzy subset on U, written as A¬, A¬(x) = Ψ ¬(A(x)). A¬ is called contradictory negation set of A. This fuzzy set on domain U is defined by the definition 1 and 2, we call it “Fuzzy Set with Contradictory negation, Opposite negation and Medium negation”, for short FSCOM.
3
╕
~
Fuzzy Degree of FSCOM
A fuzzy concept can be expressed by a fuzzy set, and the membership function is able to potray a fuzzy set. Fuzzy concepts are different, so the corresponding fuzzy sets are different. The question of ambiguity of a fuzzy set is aware and studied by foreign scholars at first. DeLuca and Termini [6] proposed fuzzy metric theory in 1972. Definition 3. If D(A,B) exists following conditions: (1) D(A,B)≥0 , D(A,B)=0 if and only if A=B; (2) D(A,B)=D(B,A); (3) D(A,B)≤D(A,C)+ D(B,C). D(A,B) is called the distance between A and B.
Fuzzy Degree and Similarity Measure of Fuzzy Set with Three Kinds of Negations
545
Definition 4. Let X be domain, A is a fuzzy set in FSCOM, ∀x∈X. B={(x1, A(x1)), (x2, A(x2)),…, (xn, A(xn))},(A (xi)≥0.5,i=1,2,…,n) C={(y1, A(y1)), (y2, A(y2)),…, (ym, A(ym))},(A (yj)
Abstract. Negation in information processing is an important notion, especially in fuzzy information processing. How to cognize and deal with various negations of fuzzy information, a new fuzzy set FSCOM with contradictory negation, opposite negation and medium negation was presented in [1]. This paper defines fuzzy degree and similarity measure of fuzzy set FSCOM, gives some properties and calculation formulas of the fuzzy degree and similarity measure, and also discusses their applications. Keywords: Three kinds of negations of fuzzy set, fuzzy degree, similarity measure.
1
Introduction
Negation in information processing is an important notion, especially in fuzzy information processing. Some scholars suggested that the different negations should be applied in knowledge processing in recent years [2][3]. Zhenghua Pan proposed that there are five kinds of contradictory negative relations and opposite negative relations in concept of knowledge [4], and came up with a fuzzy set FSCOM which has three kinds of negations [1]. Contradictory relation and opposite relation in knowledge information are not distinguished in traditional mathematical and Zadeh fuzzy set, but they are differentiated in fuzzy set FSCOM. In this paper, we define fuzzy degree and similarity measure of the fuzzy set FSCOM, and give the calculation formulas of fuzzy degree and similarity measures, and also discuss their applications.
2
Basic Concept of FSCOM
Definition 1 [5]. Let U be domain. Mapping ΨA : U → [0, 1] confirmed a fuzzy subset A on U (i.e. A∈P(U)), where mapping ΨA is called the membership function of A, ΨA (x) is the degree of membership of x in A, for short A(x). {A(x) | x∈U} for short A(U). H. Deng et al. (Eds.): AICI 2011, Part I, LNAI 7002, pp. 543–550, 2011. © Springer-Verlag Berlin Heidelberg 2011
544
L. Yang and Z. Pan
Definition 2 [1]. Let A ∈ P(U), λ ∈ (0, 1). (1) If mapping Ψ ╕: A(U) → [0, 1] ╕
╕
satisfies Ψ (A(x)) = 1− A(x), Ψ confirmed a fuzzy subset A╕ on U, A╕(x) = Ψ (A(x)). A╕ is called opposite negation set of A. when (2) If mapping Ψ ~: ╕
A(U) → [0, 1] ~
Satisfies Ψ (A(x)) = 2λ − 1 (A(x) −λ)+1−λ, when λ∈[½, 1) and A(x)∈(λ, 1] 1− λ 2λ − 1 A(x)+1−λ, when λ∈[½, 1) and A(x)∈[0, 1−λ) 1− λ 1 − 2λ A(x)+λ, when λ∈(0, ½] and A(x)∈[0, λ)
(1) (2) (3)
λ 1 − 2λ (A(x)+λ −1)+λ, when λ∈(0, ½] and A(x)∈(1−λ, 1] λ
(4)
0.5, when A(x) = 0.5
(5)
~
~
~
~
~
Ψ confirmed a fuzzy subset A on U, A (x) = Ψ (A(x)). A is called medium negation set of A. (3) If mapping Ψ ¬ : A(U) → [0, 1] ¬
satisfies Ψ (A(x)) = Max (A (x), A (x)), Ψ¬ confirmed a fuzzy subset on U, written as A¬, A¬(x) = Ψ ¬(A(x)). A¬ is called contradictory negation set of A. This fuzzy set on domain U is defined by the definition 1 and 2, we call it “Fuzzy Set with Contradictory negation, Opposite negation and Medium negation”, for short FSCOM.
3
╕
~
Fuzzy Degree of FSCOM
A fuzzy concept can be expressed by a fuzzy set, and the membership function is able to potray a fuzzy set. Fuzzy concepts are different, so the corresponding fuzzy sets are different. The question of ambiguity of a fuzzy set is aware and studied by foreign scholars at first. DeLuca and Termini [6] proposed fuzzy metric theory in 1972. Definition 3. If D(A,B) exists following conditions: (1) D(A,B)≥0 , D(A,B)=0 if and only if A=B; (2) D(A,B)=D(B,A); (3) D(A,B)≤D(A,C)+ D(B,C). D(A,B) is called the distance between A and B.
Fuzzy Degree and Similarity Measure of Fuzzy Set with Three Kinds of Negations
545
Definition 4. Let X be domain, A is a fuzzy set in FSCOM, ∀x∈X. B={(x1, A(x1)), (x2, A(x2)),…, (xn, A(xn))},(A (xi)≥0.5,i=1,2,…,n) C={(y1, A(y1)), (y2, A(y2)),…, (ym, A(ym))},(A (yj)
