Monotone Measures of Intuitionistic Fuzzy Sets - Ifigenia.org
Monotone Measures of Intuitionistic Fuzzy Sets Piotr Nowak
Systems Research Institute Polish Academy of Sciences ul. Newelska 6, 01-447 Warszawa, Poland [email protected]
Abstract Our main goal is to extend the domain of a monotone measure to a class of intuitionistic fuzzy sets. The extension is made by using of the Choquet integral. Keywords: Monotone Measure, Fuzzy Measure, Intuitionistic Fuzzy Set, Choquet Integral, Probability of an Intuitionistic Fuzzy Event
1 Introduction
Real- or complex-valued integrals can be treated as functionals defined on a certain space of functions. One of the fundamental properties of this functionals is that the integral of the characteristic function of a set is equal to its measure. Therefore it is obvious that an extension of a monotone measure to intuitionistic fuzzy sets is the first step to define an integral of imprecisely defined functions or random variables. The extension of the domain of a monotone measure to a class of intuitionistic fuzzy sets is the main goal of this paper. We also prove some basic properties of such extended measures. The proposed definition is in agreement with the notion of probability of intuitionistic fuzzy events, which was introduced and discussed by Gerstenkorn and Ma´nko (see [6]) and Szmidt and Kacprzyk ([8], [9]) as well as Grzegorzewski and Mrówka in [7]. Section 2 of the paper includes basic definitions and notations. Section 3 is devoted to the definition and basic properties of the proposed extension of a monotone measure to a class of intuitionistic fuzzy sets. In its third subsection we show that the probability of intuitionistic fuzzy events is an extended monotone measure.
In the real world we often meet imprecisely defined notions. Fuzzy sets were introduced to deal with them. The intuitionistic fuzzy sets, defined by Atanassov in [1], can be regarded as generalizations of fuzzy sets, because they are described by two functions: one expressing the degree of membership and the other one expressing the degree of nonmembership, which do not need to sum up to one. Classical measure theory and classical theory of integration do not describe all real world situations too. Therefore monotone measures, called fuzzy measures are considered. It is not possible to integrate with respect to them in the classical way, since their additivity is not assumed. Choquet integral, proposed by Vitali 2 Intuitionistic Fuzzy Sets ([10]) and Choquet ([4]), is one of the most important integrals defined without the assump- We first recall the definition of intuitionistic tion of the σ-additivity of the measure. fuzzy sets and introduce some basic notations.
Definition 1 Let Ω denote a universe of dis-
course. Then an intuitionistic fuzzy set Ω is a set of ordered triples
A in
3
Extension Measures
of
Monotone
3.1 Monotone Measures
A = {ω,µA (ω) ,νA (ω) : ω ∈ Ω} , Definition 4 Let Ω be an arbitrary set and let where µA ,ν A : Ω → [0, 1] are such that 0 ≤ F be a σ-algebra of its subsets. Then every µA (ω)+ν A (ω) ≤ 1 for each ω ∈ Ω. The family function µ : F → R+ = [0, ∞ ) satisfying the
of all intuitionistic fuzzy sets on Ω is denoted by IFS (Ω).
For each ω the numbers µA (ω) and ν A (ω) represent the degree of membership and degree of nonmembership of an element ω ∈ Ω to A, respectively. The function πA (ω) = 1 − µA (ω) − ν A (ω) is interpreted as hesitation margin.
properties: 1. , 2. for every ⊂ ∈ F ≤ is called a monotone measure or a fuzzy measure. We say that is continuous from below if for every sequence { n }n=1 ⊂ F and ∈ F
µ (∅) = 0
A B
µ (A) µ (B)
µ
C ∞ C Cn C ⇒ µ (Cn) µ (C ) . In general we do not assume that µ is con-
Definition 2 Let A,B ∈ IFS (Ω) with tinuous from below. If Ω is finite, it is often A = {ω,µA (ω) ,ν A (ω) : ω ∈ Ω} and B = assumed that F = 2Ω . {ω, µB (ω) ,ν B (ω) : ω ∈ Ω}. A ⊆ B ⇔ ∀ω ∈ Ω µA (ω) ≤ µB (ω) and ν A (ω) ≥ ν B (ω). Definition 5 A monotone measure Consequently, A = B ⇔ A ⊆ B and B ⊆ A. µ : F → R+ is supermodular, if for every
∅ denotes the intuitionistic fuzzy set with A,B ∈ F µ∅ (ω) = 0 and ν∅ (ω) = 1 for each ω ∈ Ω. µ (A ∪ B) + µ (A ∩ B) ≥ µ (A) + µ (B) . For arbitrary f,g : Ω → [0, 1] we denote by f ∧g and f ∨g the functions defined by formulas It is called submodular, if for every A,B ∈ F µ (A ∪ B) + µ (A ∩ B) ≤ µ (A) + µ (B) . f ∧ g (ω) = min(f (ω) ,g (ω)) ; f ∨ g (ω) = max (f (ω) ,g (ω)) , ω ∈ Ω. If the above inequalities hold for each pair of disjoint events A and B , then µ is called suWe repeat Atanassov’s definition of the basic peradditive and subadditive respectively. µ is
operations on intuitionistic fuzzy sets.
called additive if it is both sub- and superadditive.
Definition 3 Let A,B ∈ IFS (Ω) with A = {ω,µA (ω) ,ν A (ω) : ω ∈ Ω} and B = For an arbitrary set I we denote with #I the {ω, µB (ω) ,ν B (ω) : ω ∈ Ω}. Then cardinality of I . Ac = {ω, νA (ω) ,µA (ω) : ω ∈ Ω} ; Definition 6 A monotone measure A∪B = µ : F → R+ is called k-monotone, k ≥ 2, if {ω, µA ∨ µB (ω) ,ν A ∧ ν B (ω) : ω ∈ Ω} ; for every A1, A2 ,...,Ak ∈ F A∩B = {ω, µA ∧ µB (ω) ,ν A ∨ ν B (ω) : ω ∈ Ω} . #I µ (∩ Ai ) ≥ 0 (−1) Two intuitionistic fuzzy sets are disjoint if and µ ∪ki=1Ai + i∈I I ⊂{1 , 2 ...,k } , only if A ∩ B = ∅. I =∅
k-alternating, k ≥ 2, A1,A2,...,Ak ∈ F while it is
Let µ be a monotone measure on a σ -algebra F of subsets of Ω. For each A ∈ IFS (F) the extension µ˜ of the measure µ is the interval µ˜ (A) = [µmin (A) , µmax (A)], where µmin (A) and µmax (A) are the Choquet
if for every Definition 9
#I µ (∪i∈I Ai ) ≤ 0. (−1) I ⊂{1,2...,k}, I =∅
µ ∩ki=1Ai +
µ is called totally monotone if it is k-monotone for every k ≥ 2.
integrals of the form
µmin (A) = (C) µA dµ and Ω
µmax (A) = (C) (1 − νA ) dµ. With any function f : Ω → R+ we assoΩ ciate the family Cf = {Cf (x) : x ∈ R+ }, with To show some basic properties of the Cf (x) = {ω ∈ Ω : f (ω) > x}. Each function f : Ω → R introduces on Ω a semi-order rela- extended measure, we introduce the following notations: 0 = [0, 0], µ = [µ (Ω) ,µ (Ω)] and tion: µ (A) = [µ (A) ,µ (A)] for A ∈ F . We also introduce the following order on the family of ω1 x) dx = 1 − FµA (x) dx. (3) 0
0
[3] K. Atanassov (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Physica-
Verlag.
[4] G. Choquet (1953-1954). Theory of capacities. In Ann. Inst. Fourier, volume 5, pages 131-295. [5] D. Denneberg ( 2000). Non-additive Measure and Integral, Basic Concepts and Their Role for Applications. In Fuzzy
Measures and Integrals. Theory and Applications. Physica-Verlag Heilderberg, pages 42-69.
[6] T. Gerstenkorn, J. Ma´nko (1991). Probability of Fuzzy Intuitionistic Sets. In BUSEFAL 45, pages 128-136. [7] P. Grzegorzewski , E. Mrówka (2002). Probability of Intuitionistic Fuzzy Events. In Soft Methods in Probability. Statistics and data Analysis. Springer - Physica Verlag, Heidelberg, 2002, pages 105-115. [8] E. Szmidt E., J. Kacprzyk (1999a). A Concept of a Probability of an Intuitionistic Fuzzy Event. In Proceedings of the
1999 IEEE International Fuzzy Systems Conference, pages 1346-1349, Seoul,1999.
[9] E. Szmidt, J. Kacprzyk (1999b). Probability of Intuitionistic Fuzzy Events and Their Applications in Decision Making. In Proceedings of the 9th Congreso EUSFLAT’99, pages 457-460. [10] G. Vitali (1925). Sulla definizione di integrale delle funzioni di una variabile. In Ann. Mat. Pura ed Appl. IV, 2, pages 111121.
Systems Research Institute Polish Academy of Sciences ul. Newelska 6, 01-447 Warszawa, Poland [email protected]
Abstract Our main goal is to extend the domain of a monotone measure to a class of intuitionistic fuzzy sets. The extension is made by using of the Choquet integral. Keywords: Monotone Measure, Fuzzy Measure, Intuitionistic Fuzzy Set, Choquet Integral, Probability of an Intuitionistic Fuzzy Event
1 Introduction
Real- or complex-valued integrals can be treated as functionals defined on a certain space of functions. One of the fundamental properties of this functionals is that the integral of the characteristic function of a set is equal to its measure. Therefore it is obvious that an extension of a monotone measure to intuitionistic fuzzy sets is the first step to define an integral of imprecisely defined functions or random variables. The extension of the domain of a monotone measure to a class of intuitionistic fuzzy sets is the main goal of this paper. We also prove some basic properties of such extended measures. The proposed definition is in agreement with the notion of probability of intuitionistic fuzzy events, which was introduced and discussed by Gerstenkorn and Ma´nko (see [6]) and Szmidt and Kacprzyk ([8], [9]) as well as Grzegorzewski and Mrówka in [7]. Section 2 of the paper includes basic definitions and notations. Section 3 is devoted to the definition and basic properties of the proposed extension of a monotone measure to a class of intuitionistic fuzzy sets. In its third subsection we show that the probability of intuitionistic fuzzy events is an extended monotone measure.
In the real world we often meet imprecisely defined notions. Fuzzy sets were introduced to deal with them. The intuitionistic fuzzy sets, defined by Atanassov in [1], can be regarded as generalizations of fuzzy sets, because they are described by two functions: one expressing the degree of membership and the other one expressing the degree of nonmembership, which do not need to sum up to one. Classical measure theory and classical theory of integration do not describe all real world situations too. Therefore monotone measures, called fuzzy measures are considered. It is not possible to integrate with respect to them in the classical way, since their additivity is not assumed. Choquet integral, proposed by Vitali 2 Intuitionistic Fuzzy Sets ([10]) and Choquet ([4]), is one of the most important integrals defined without the assump- We first recall the definition of intuitionistic tion of the σ-additivity of the measure. fuzzy sets and introduce some basic notations.
Definition 1 Let Ω denote a universe of dis-
course. Then an intuitionistic fuzzy set Ω is a set of ordered triples
A in
3
Extension Measures
of
Monotone
3.1 Monotone Measures
A = {ω,µA (ω) ,νA (ω) : ω ∈ Ω} , Definition 4 Let Ω be an arbitrary set and let where µA ,ν A : Ω → [0, 1] are such that 0 ≤ F be a σ-algebra of its subsets. Then every µA (ω)+ν A (ω) ≤ 1 for each ω ∈ Ω. The family function µ : F → R+ = [0, ∞ ) satisfying the
of all intuitionistic fuzzy sets on Ω is denoted by IFS (Ω).
For each ω the numbers µA (ω) and ν A (ω) represent the degree of membership and degree of nonmembership of an element ω ∈ Ω to A, respectively. The function πA (ω) = 1 − µA (ω) − ν A (ω) is interpreted as hesitation margin.
properties: 1. , 2. for every ⊂ ∈ F ≤ is called a monotone measure or a fuzzy measure. We say that is continuous from below if for every sequence { n }n=1 ⊂ F and ∈ F
µ (∅) = 0
A B
µ (A) µ (B)
µ
C ∞ C Cn C ⇒ µ (Cn) µ (C ) . In general we do not assume that µ is con-
Definition 2 Let A,B ∈ IFS (Ω) with tinuous from below. If Ω is finite, it is often A = {ω,µA (ω) ,ν A (ω) : ω ∈ Ω} and B = assumed that F = 2Ω . {ω, µB (ω) ,ν B (ω) : ω ∈ Ω}. A ⊆ B ⇔ ∀ω ∈ Ω µA (ω) ≤ µB (ω) and ν A (ω) ≥ ν B (ω). Definition 5 A monotone measure Consequently, A = B ⇔ A ⊆ B and B ⊆ A. µ : F → R+ is supermodular, if for every
∅ denotes the intuitionistic fuzzy set with A,B ∈ F µ∅ (ω) = 0 and ν∅ (ω) = 1 for each ω ∈ Ω. µ (A ∪ B) + µ (A ∩ B) ≥ µ (A) + µ (B) . For arbitrary f,g : Ω → [0, 1] we denote by f ∧g and f ∨g the functions defined by formulas It is called submodular, if for every A,B ∈ F µ (A ∪ B) + µ (A ∩ B) ≤ µ (A) + µ (B) . f ∧ g (ω) = min(f (ω) ,g (ω)) ; f ∨ g (ω) = max (f (ω) ,g (ω)) , ω ∈ Ω. If the above inequalities hold for each pair of disjoint events A and B , then µ is called suWe repeat Atanassov’s definition of the basic peradditive and subadditive respectively. µ is
operations on intuitionistic fuzzy sets.
called additive if it is both sub- and superadditive.
Definition 3 Let A,B ∈ IFS (Ω) with A = {ω,µA (ω) ,ν A (ω) : ω ∈ Ω} and B = For an arbitrary set I we denote with #I the {ω, µB (ω) ,ν B (ω) : ω ∈ Ω}. Then cardinality of I . Ac = {ω, νA (ω) ,µA (ω) : ω ∈ Ω} ; Definition 6 A monotone measure A∪B = µ : F → R+ is called k-monotone, k ≥ 2, if {ω, µA ∨ µB (ω) ,ν A ∧ ν B (ω) : ω ∈ Ω} ; for every A1, A2 ,...,Ak ∈ F A∩B = {ω, µA ∧ µB (ω) ,ν A ∨ ν B (ω) : ω ∈ Ω} . #I µ (∩ Ai ) ≥ 0 (−1) Two intuitionistic fuzzy sets are disjoint if and µ ∪ki=1Ai + i∈I I ⊂{1 , 2 ...,k } , only if A ∩ B = ∅. I =∅
k-alternating, k ≥ 2, A1,A2,...,Ak ∈ F while it is
Let µ be a monotone measure on a σ -algebra F of subsets of Ω. For each A ∈ IFS (F) the extension µ˜ of the measure µ is the interval µ˜ (A) = [µmin (A) , µmax (A)], where µmin (A) and µmax (A) are the Choquet
if for every Definition 9
#I µ (∪i∈I Ai ) ≤ 0. (−1) I ⊂{1,2...,k}, I =∅
µ ∩ki=1Ai +
µ is called totally monotone if it is k-monotone for every k ≥ 2.
integrals of the form
µmin (A) = (C) µA dµ and Ω
µmax (A) = (C) (1 − νA ) dµ. With any function f : Ω → R+ we assoΩ ciate the family Cf = {Cf (x) : x ∈ R+ }, with To show some basic properties of the Cf (x) = {ω ∈ Ω : f (ω) > x}. Each function f : Ω → R introduces on Ω a semi-order rela- extended measure, we introduce the following notations: 0 = [0, 0], µ = [µ (Ω) ,µ (Ω)] and tion: µ (A) = [µ (A) ,µ (A)] for A ∈ F . We also introduce the following order on the family of ω1 x) dx = 1 − FµA (x) dx. (3) 0
0
[3] K. Atanassov (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Physica-
Verlag.
[4] G. Choquet (1953-1954). Theory of capacities. In Ann. Inst. Fourier, volume 5, pages 131-295. [5] D. Denneberg ( 2000). Non-additive Measure and Integral, Basic Concepts and Their Role for Applications. In Fuzzy
Measures and Integrals. Theory and Applications. Physica-Verlag Heilderberg, pages 42-69.
[6] T. Gerstenkorn, J. Ma´nko (1991). Probability of Fuzzy Intuitionistic Sets. In BUSEFAL 45, pages 128-136. [7] P. Grzegorzewski , E. Mrówka (2002). Probability of Intuitionistic Fuzzy Events. In Soft Methods in Probability. Statistics and data Analysis. Springer - Physica Verlag, Heidelberg, 2002, pages 105-115. [8] E. Szmidt E., J. Kacprzyk (1999a). A Concept of a Probability of an Intuitionistic Fuzzy Event. In Proceedings of the
1999 IEEE International Fuzzy Systems Conference, pages 1346-1349, Seoul,1999.
[9] E. Szmidt, J. Kacprzyk (1999b). Probability of Intuitionistic Fuzzy Events and Their Applications in Decision Making. In Proceedings of the 9th Congreso EUSFLAT’99, pages 457-460. [10] G. Vitali (1925). Sulla definizione di integrale delle funzioni di una variabile. In Ann. Mat. Pura ed Appl. IV, 2, pages 111121.
