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Decomposable measures and information measures for intuitionistic fuzzy sets
Fuzzy Sets and Systems 123 (2001) 103–117
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Decomposable measures and information measures for intuitionistic fuzzy sets A.I. Ban ∗ ; 1 , S.G. Gal Department of Mathematics, University of Oradea, str. Armatei Romˆane 5, 3700 Oradea, Romania Received 27 October 1998; received in revised form 29 March 2000; accepted 5 June 2000
Abstract The aim of this paper is to construct a decomposable measure on the class of measurable intuitionistic fuzzy sets, extending thus the corresponding result for fuzzy sets in a very recent paper. The main tool in this construction is a non-trivial approximation (with restrictions) result concerning the intuitionistic fuzzy sets. As an application we construct c 2001 Elsevier Science B.V. All rights a compositive information measure on the class of intuitionistic fuzzy sets. reserved. Keywords: Intuitionistic fuzzy sets; Non-additive measures; Possibility measures; Measures of information
1. Introduction In a very recent paper [2], a decomposable measure of measurable fuzzy sets based on continuous tconorms is constructed. The main purpose of this paper is to generalize the result for the class of measurable intuitionistic fuzzy sets. Section 2 contains some de:nitions and notations, introduced by Atanassov (see [1]). Moreover, the new concept of decomposable intuitionistic fuzzy measure is introduced. In Section 3 we prove an interesting result concerning the approximation of measurable intuitionistic fuzzy sets by discrete intuitionistic fuzzy sets, which is essential in the proofs of the main results. In comparison with the corresponding approximation result for usual fuzzy sets, when it is direct consequence of a classical theorem in real functions theory, the result for intuitionistic fuzzy sets is one of approximations with restrictions which cannot be directly derived from a classical result and it is solved by using [6]. ∗
Corresponding author. E-mail address: [email protected] (A.I. Ban). 1 While working on this paper the author has been partially supported by a grant of Romanian Agency for Science, Technology and Inovation ANSTI 410=2000. c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 1 0 6 - 8
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Section 4 contains the construction of the decomposable measure for intuitionistic fuzzy sets. Also, we point out some inadvertences in the de:nition of the decomposable measure in [2], which have been eliminated in our more general de:nitions. In Section 5 we indicate a bijection between the class of compositive information measures and the class of decomposable fuzzy measures on intuitionistic fuzzy sets, which allows that starting from the decomposable measure constructed in Section 4, we obtain a compositive information measure. 2. Preliminaries Firstly, we give some de:nitions and notations regarding the intuitionistic fuzzy sets. Denition 1 (Atanassov [1]). Let X = ∅ be a given set. An intuitionistic fuzzy set in X is an expression A˜ given by A˜ = {x; A˜ (x); A˜ (x): x ∈ X } where A˜ : X → [0; 1]; A˜ : X → [0; 1] satisfy the condition 06 A˜ (x) + A˜ (x)61, for all x ∈ X . We will denote by IFS(X ) the set of all the intuitionistic fuzzy sets in X . Finite union and intersection, inclusion and complementation are de:ned for every A;˜ B˜ ∈ IFS(X ) (see [1]): A˜ ∪ B˜ = {x; max{A˜ (x); B˜ (x)}; min{A˜ (x); B˜ (x)}: x ∈ X }; A˜ ∩ B˜ = {x; min{A˜ (x); B˜ (x)}; max{A˜ (x); B˜ (x)}: x ∈ X }; A˜ ⊆ B˜ if and only if A˜ (x)6B˜ (x) and A˜ (x)¿B˜ (x) for all x ∈ X; A˜ c = {x; A˜ (x); A˜ (x): x ∈ X }: Countable union and intersection are de:ned as above, by replacing max with sup and min with inf. For ; ∈ [0; 1], + 61 and the intuitionistic fuzzy set A˜ = {x; A˜(x); A˜ (x): x ∈ X } we de:ne the crisp set A ; by A ; = {x ∈ X : A˜ (x) = ; A˜ (x) = }: Let A ⊆ X ,
A˜ (x) = A˜ (x) =
0
if x ∈ A; if x ∈ A;
1
if x ∈ A; if x ∈ A;
where 06 + 61. The intuitionistic fuzzy set A˜ = {x; A˜ (x); A˜ (x): x ∈ X } will be denoted by x; ; A . For example, ∅˜ = x; 0; 1X and X˜ = x; 1; 0X . Denition 2. An intutionistic fuzzy set A˜ = {x; A˜ (x); A˜ (x): x ∈ X } is called discrete if the functions A˜ and A˜ take :nite number of values. Note that, if A˜ = {x; A˜ (x); A˜(x): x ∈ X } is a discrete intuitionistic fuzzy set then we can consider A˜ (x) ∈ { 1 ; : : : ; n } = M1 ; A˜ (x) ∈ {1 ; : : : ; n } = M2 , that is A˜ and A˜ take the same number of values if we naturally accept the fact that the elements of each between M1 and M2 , are not necessarily distinct. Moreover, we can consider −1 ( k ) = −1 (k ), ∀k ∈ {1; : : : ; n}. A˜ A˜
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Indeed, to prove this, let us suppose that A˜ (x) ∈ { 1 ; : : : ; p }, i = j if i = j; A˜ (x) ∈ {1 ; : : : ; q }, i = j if ( i ) = Yi , i ∈ {1; : : : ; p} and −1 (i ) = Zi ; i ∈ {1; : : : ; q}. We have Yi ∩ Yj = ∅ if i = j, i = j. Let us denote −1 A˜ A˜ p q p q Zi ∩ Zj = ∅ if i = j and X = i=1 Yi = j=1 Zj . We obviously have X = i=1 j=1 Mij , where Mij = Yi ∩ Zj n and the sets Mij are mutually disjoint. If we omit the sets Mij that are empty, we can rewrite X = k=1 Xk , where Xk are mutually disjoint, each Xk is non-empty of the form Xk = Mij and 16n6pq. We rede:ne A˜ and A˜ in the following way: A∗˜ (x) = k∗ = i
if x ∈ Xk = Mij ;
A∗˜ (x) = k∗ = j
if x ∈ Xk = Mij :
Obviously, both ∗A˜ and ∗A˜ have n-values and −1 ( k∗ ) = −1 (k∗ ) = Xk ; ∀k ∈ {1; : : : ; n}. A˜ A˜ As a conclusion we can always write A˜ by means of the decomposition A˜ =
n
x; k ; k A k ; k ;
k=1
˜ if i = j. where x; i ; i A i ; i ∩ x; j ; j A j ; j = ∅, For example, the intuitionistic fuzzy sets A˜ = {x; A˜ (x); A˜ (x): x ∈ R} where 0:1 if x ∈ (−∞; −1]; A˜ (x) = 0:3 if x ∈ (−1; 1); 0:6 if x ∈ [−1; +∞) and
A˜ (x) =
0:4 0:2
if x ∈ (−∞; 0); if x ∈ [0; +∞)
which can be rewritten as A˜ = {x; ∗A˜(x); ∗A˜(x): x ∈ R} where 0:1 0:3 A∗˜ (x) = 0:3 0:6
and
0:4 0:4 ∗A˜ (x) = 0:2 0:2
if x ∈ (−∞; −1]; if x ∈ (−1; 0); if x ∈ [0; 1); if x ∈ [1; +∞)
if if if if
x x x x
∈ (−∞; −1]; ∈ (−1; 0); ∈ [0; 1); ∈ [1; +∞):
The decomposition corresponding to A˜ as above is A˜ = x; 0:1; 0:4(−∞;−1] ∪ x; 0:3; 0:4(−1;0) ∪ x; 0:3; 0:2[0;1) ∪ x; 0:6; 0:2[1;+∞) : Now, we recall two well known de:nitions.
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Denition 3 (See e.g. Dubois and Prade [4; p: 126] or Grabisch [7]). A fuzzy measure over a measurable space (X; A) is a map m : A → [0; 1] with the following properties: (i) m(∅) = 0, (ii) m(X ) = 1, (iii) A ⊆ B implies m(A)6m(B). Denition 4 (See Dubois and Prade [5] and Pap [9]). A fuzzy measure is said to be ∗-decomposable if there exists the composition law ∗ : [0; 1] × [0; 1] → [0; 1] such that m(A ∪ B) = m(A) ∗ m(B);
∀A; B ∈ A; A ∩ B = ∅:
The structure (X; A; m; ∗) will be called decomposable space. Remark 1. It is obvious that ∗ should behave as a triangular conorm on the range of m (that is increasing, associative, commutative and x ∗ 0 = x). With respect to the operations of union, intersection and complementation introduced after De:nition 1 we can adapt the de:nitions of -algebra, fuzzy measure and decomposable fuzzy measure to intuitionistic fuzzy sets in the following way. Denition 5. Let X = ∅. A family A˜ of intuitionistic fuzzy sets in X is called intuitionistic fuzzy -algebra ˜ will be called if ∅˜ ∈ A,˜ A˜ ∈ A˜ implies A˜c ∈ A˜ and A˜i ∈ A;˜ i ∈ N implies i∈N A˜i ∈ A.˜ The pair (X; A) intuitionistic fuzzy measurable space. ˜ is a map m˜ : A˜→ [0; 1] An intuitionistic fuzzy measure over an intuitionistic fuzzy measurable space (X; A) ˜ ˜ ˜ ˜ ˜ ˜ with the properties: m( ˜ ∅) = 0; m( ˜ X ) = 1 and A ⊆ B implies m( ˜ A)6m( ˜ B). The intuitionistic fuzzy measure m˜ is said to be ∗˜ -decomposable if there exists the composition law ∗˜ : [0; 1] × [0; 1] → [0; 1] such that ˜ ˜ ∗˜ m( ˜ B˜ ∈ A;˜ A˜ ∩ B˜ = ∅: ˜ = m( ˜ ∀A; m( ˜ A˜ ∪ B) ˜ A) ˜ B); Also, we can consider Denition 6. Let (X; A) be a measurable space. We say that A˜ = {x; A˜(x); A˜(x): x ∈ X } ∈ IFS(X ) is Ameasurable if and only if A˜ and A˜ are A-measurable functions. Remark 2. Let (X; A) be a measurable space. The family A˜ = {A˜ ∈ IFS(X ): A˜ is A-measurable} is an intuitionistic fuzzy -algebra.
3. Approximation of intuitionistic fuzzy sets by discrete intuitionistic fuzzy sets A classical approximation result in the real functions theory (see e.g. [10, pp. 124 –126]), assures that if A : X → [0; 1] is a measurable fuzzy set, then there exists a sequence An : X → [0; 1]; n ∈ N, of discrete fuzzy sets (i.e. An are step functions), such that An → A, uniformly on X and An ⊆ An+1 ⊆ A; n ∈ N, where ⊆ represents the usual inclusion between two fuzzy sets. More exactly, we can write ∀ ¿0; ∃n0 ∈ N; such that 06A(x) − An (x)¡ ; n¿n0 ; x ∈ X
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and An (x)6An+1 (x)6A(x); n ∈ N; x ∈ X: At this point, it is natural to ask the following question: how the above result to the intuitionistic case can be extended? Let (X; A) be a measurable space and A˜ ∈ IFS(X ); A˜ = {x; A˜(x); A˜(x): x ∈ X }; supposed to be Ameasurable. A direct application of the above result to each A˜ and A˜ yields an approximation sequence (n) (n) (n) of discrete intuitionistic fuzzy sets A˜n = {x; A(n) ˜ (x); A˜ (x): x ∈ X }, n ∈ N; such that A˜ A˜; A˜ A˜ uniformly on X , but for which the monotony property (i.e. A˜n ⊆ A˜n+1 ; n ∈ N) fails to exist. But as we see in the next section, this monotony property is essential in the de:nition of the decomposable measure. In order to answer the above question in analytical terms, we have to solve the following non-trivial problem of approximation with restrictions: If f; g : X → [0; 1] are measurable and satisfy f(x) + g(x)61; then we construct two sequences of step functions fn ; gn : X → [0; 1], such that fn (x) + gn (x)61; n ∈ N; x ∈ X and moreover fn (x)6fn+1 (x)6f(x); gn (x)¿gn+1 (x)¿g(x); for all n ∈ N; x ∈ X . The following result solves the question. Theorem 1. If A˜ ∈ IFS(X ); A˜ = {x; A˜(x); A˜(x): x ∈ X } is A-measurable; then there exists a sequence of (n) (n) (n) discrete intuitionistic fuzzy sets A˜n = {x; A(n) ˜ (x); A˜ (x): x ∈ X }; n ∈ N; such that A˜ A˜; A A˜ uniformly on X; (i.e. A˜n ⊆ A˜n+1 ⊆ A;˜ for all n ∈ N). Proof. By the classical approximation result in [10, pp. 124 –126], there exist the sequences of step functions Fn : X → [0; 1], hn : X → [0; 1]; n ∈ N, such that Fn A˜ + A˜; hn A˜; uniformly on X . More exactly, by the above result we have 06A˜(x) − hn (x)6
1 2n
for all n ∈ N; x ∈ X:
Now, by using the idea in [6], we obtain a monotonous decreasing sequence of step functions, h∗n (x) = hn (x) +
2 ; 2n−1
x ∈ X; n ∈ N;
i.e. h∗n → A˜ uniformly on X and h∗n (x) − h∗n+1 (x) = hn (x) − hn+1 (x) +
2 ¿0; 2n
x ∈ X; n ∈ N;
because hn (x) − hn+1 (x) 6 |hn (x) − hn+1 (x)|6|hn (x) − A˜(x)| + |A˜(x) − hn+1 (x)| 1 2 1 6 n + n+1 6 n ; x∈X; n∈N: 2 2 2 Because h∗n (x) can take values greater than 1, let us de:ne h∗n (x) if h∗n (x)61; gn (x) = 1 if h∗n (x)¿1: Obviously, gn (x) ∈ [0; 1] and gn is step function, n ∈ N.
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By A˜ (x)6gn (x)6h∗n (x); x ∈ X; n ∈ N; it immediately follows that gn → A˜, uniformly on X . Also, by simple reasonings we get gn+1 (x)6gn (x) for all x ∈ X; n ∈ N. As a conclusion, gn : X → [0; 1]; n ∈ N; is a sequence of step functions satisfying gn A˜;
uniformly on X:
Now, let us de:ne + A(n) ˜ (x) = (Fn − gn ) (x) (i:e: the positive part of Fn (x) − gn (x))
and A(n) ˜ (x) = gn (x); x ∈ X; n ∈ N. (n) (n) Obviously, A(n) ˜ is step function, n ∈ N. Then, 06A˜ (x) and if 0¡A˜ (x) then A(n) ˜ (x) = Fn (x) − gn (x)6A˜(x) + A˜(x) − gn (x)6A˜(x)61; i.e. A(n) ˜ ∈ [0; 1], for all x ∈ X; n ∈ N. Also, + A(n) ˜ (x) = (Fn − gn ) (x) = max{(Fn − gn ) (x); 0}
Fn (x) − gn (x) + |Fn (x) − gn (x)| n→∞ A˜(x) + A˜(x) − A˜(x) + |A˜(x) + A˜(x) − A˜(x)| → 2 2 A˜(x) + |A˜(x)| = A+˜ (x) = A˜(x) = 2
=
uniformly on X . Moreover, by simple reasoning we get (n+1) (x) A(n) ˜ (x)6A˜
for all x ∈ X; n ∈ N:
Finally, (n) + 0 6 A(n) ˜ (x) + A˜ (x) = (Fn − gn ) (x) + gn (x) Fn (x)61 if Fn (x) − gn (x)¿0; = gn (x)61 if Fn (x) − gn (x)¡0
which proves the theorem. Remark 3. If A˜ ∈ IFS(X ) is of the form A˜ = 1 − A˜ (i.e. if A˜ one reduces to a classical fuzzy set), then Theorem 1 is an obvious consequence of the classical result in the real functions theory. Indeed, by A(n) ˜ A˜ (n) it follows that A(n) ˜ = 1 − A˜ 1 − A˜ = A˜. But as we have seen in the proof of Theorem 1, in the general
case A˜ (x) + A˜ (x)61; this is not an obvious one. In the next section we also need the following:
Denition 7. Let A˜1 = {x; 1 (x); 1 (x): x ∈ X }, A˜2 = {x; 2 (x); 2 (x): x ∈ X } ∈ IFS(X ) and ¿0. We say that A˜1 is an -approximation of A˜2 if |2 (x) − 1 (x)|¡ and |2 (x) − 1 (x)|¡ ; ∀x ∈ X .
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Remark 4. If A˜ ∈ IFS(X ) is A-measurable then for any ¿0 there exist the discrete intuitionistic fuzzy sets ˜ The :rst inclusion is an E˜ and F˜ which are -approximations of A˜ and they moreover satisfy E˜ ⊆ A˜ ⊆ F. ˜ immediate consequence of Theorem 1. Because the A-measurability of A implies A˜c is A-measurable and the complementation of a discrete intuitionistic fuzzy set is a discrete intuitionistic fuzzy set from which we obtain the second inclusion.
4. Construction of the decomposable intuitionistic fuzzy measure In this section, we construct a decomposable intuitionistic fuzzy measure for a suitable composition law ∗˜ , starting from a decomposable space (X; A; m; ∗). We mention that in [2] a similar result is obtained for fuzzy sets. First, we recall a well known result. Theorem 2. (See e.g. Bertoluzza and Cariolaro [2] or Klement and Mesiar [8]). Let D = {]ai ; bi [: i ∈ N ⊆ N} be a countable family of open disjoint subintervals of [0; 1] and let F = {fi : [ai ; bi ] → R | i ∈ N ⊆ N} be a family of continuous and strictly increasing functions with fi (ai ) = 0: The map ∗ de8ned on [0; 1] × [0; 1] by x∗y =
fi(−1) (fi (x) + fi (y)) if (x; y) ∈ [ai ; bi ] × [ai ; bi ]; max(x; y) otherwise
(where fi(−1) (x) = fi−1 (min{x; fi (bi )}) is the pseudo-inverse of fi ) is a triangular conorm; and conversely; any continuous triangular conorm has this form. Owing to the previous theorem, in the sequel we will consider only continuous triangular conorms. The following two lemmas will be used in the proof of the main result of this section Lemma 1 (Bertoluzza and Cariolaro [2]). Let ∗ be a continuous triangular conorm and (]ai ; bi [)i∈N as in the previous theorem. The following properties hold: (i) x ∗ y¿max(x; y); (ii) x ∈ if and only if x ∗ x = x; (iii) x ∈ implies x ∗ y = max(x; y); (iv) (x; y) ∈ ]ai ; bi [ × ]ai ; bi [ if and only if x ∗ y¿max(x; y); (v) x ∈ ]ai ; bi [; y ∈ ]aj ; bj [; j = i implies x ∗ y = max(x; y); where = [0; 1]\ i ∈ N ]ai ; bi [. ˜ x; ; A = ∅˜ and x; +; ,B = ∅˜ then A ∩ B = ∅. Lemma 2. If x; ; A ∩ x; +; ,B = ∅; Proof. Let us assume that A ∩ B = ∅, therefore, ∃x0 ∈ X such that x0 ∈ A and x0 ∈ B. The hypothesis x; ; A = ∅˜ implies = 0 or = 1. ˜ This implies , = 1 and = 1. Then + ¿1, If = 0 then + = 0 because x; ; A ∩ x; +; ,B = ∅. contradiction. Analogously, if = 1 then , = 1; + = 0 and we get the contradiction 1¡+ + ,, which proves the lemma. We also introduce
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Denition 8. Let D = {( ; ) ∈ [0; 1] × [0; 1]: +61}. A function G : D × [0; 1] → [0; 1] is called intuitionistic structure function if and only if the following conditions hold: (i) G(0; 1; x) = 0; (ii) G(1; 0; x) = x; (iii) 6 and ¿ implies G( ; ; x)6G( ; ; x), (iv) x 6x implies G( ; ; x )6G( ; ; x ). Example 1. The function G : D × [0; 1] → [0; 1] de:ned by G( ; ; x) = n (1 − n )x; n ∈ {1; 2; : : :}, :xed, is a simple example of intuitionistic structure function. The :rst main result is Theorem 3. Let G be an intuitionistic structure function and ∗; ∗˜ be continuous triangular conorms represented by the countable families D = {]ai ; bi [: i ∈ I }; D˜ = {]a˜j ; b˜j [: j ∈ J }; respectively; and the families of additive generators F = {fi : [ai ; bi ] → R: i ∈ I }; F˜ = {fj˜ : [a˜j ; b˜j ] → R: j ∈ J }; respectively. We assume that ∀j ∈ J; ∀( ; ) ∈ D\{(0; 1)}; ∃i ∈ I : ]a˜j ; b˜j [ ⊆ G( ; ; ]ai ; bi [) and (X; A; m; ∗) is a decomposable space. We den note by M A˜ = max{G( i ; i ; m(A i ; i )): i ∈ {1; : : : ; n}}; where A˜ = i=1 x; i ; i A i ; i is a A-measurable and discrete intuitionistic fuzzy set. Let Id be the class of all A-measurable and discrete intuitionistic fuzzy sets in X . Let us de8ne m˜ : Id → [0; 1] in the following way: ˜ = 0; – m( ˜ ∅) ˜ then we take m( ˜ = M ˜; – if M A˜ ∈ ; ˜ A) A ˜ ⊆ G( ; ; ]a; b[) with ( ; ) ∈ {( 1 ; 1 ); : : : ; ( n ; n )}; ]a; b[ ∈ D; ˜ ˜ b[ – if M A˜ ∈= then M A˜ belongs to some ]a; and we take
˜ = f˜(−1) m( ˜ A) f˜ ◦ G( r ; r ; m(A r ; r )) ; ˜ and f˜ is the function where the sum is extended to all the values of r for which G( r ; r ; m(A r ; r )) ∈ ]a; ˜ b[ ˜ Then m˜ is a ∗˜ -decomposable intuitionistic fuzzy measure. which determines ∗˜ in ]a; ˜ b[. ˜ = G( ; ; m(A)). Indeed, if Proof. First, we prove that for A˜ = x; ; A ∈ Id ; m˜ (A) ˜ then m( ˜ =M˜ MA˜ = max{G( ; ; m(A))} = G( ; ; m(A)) ∈ ˜ A) A by construction. ˜ ⊆ G( ; ; ] a; b[) with ]a; b[ ∈ D and we have ˜ then by hypothesis M ˜ ∈ ]a; If M A˜ ∈= ˜ b[ A ˜ ˜ = f˜(−1) (f(G( ; ; m(A)))) m( ˜ A) −1 ˜ ˜ ˜ b))) ˜ = f (min(f(G( ; ; m(A))); f( ˜ ; m(A)))) = G( ; ; m(A)); = f˜−1 (f(G( ; ˜ (see Theorem 2). because f˜ is strictly increasing on ]a; ˜ b[ Because X˜ = x; 1; 0X ; m( ˜ X˜) = G(1; 0; m(X )) = 1: Now, we will prove that ˜ ˜ ∗˜ m( ˜ B˜ ∈ Id ; A˜ ∩ B˜ = ∅: ˜ = m( ˜ ∀A; m( ˜ A˜ ∪ B) ˜ A) ˜ B); ˜ First, let the discrete intuitionistic fuzzy sets be of the form A˜ = x; ; A ; B˜ = x; +; ,B such that A˜ ∩ B˜ = ∅. ˜ ˜ ˜ ˜ ˜ ˜ Then A = ∅ or B˜ = ∅ or A ∩ B = ∅ (see Lemma 2). If A = ∅ or B˜ = ∅ then the equality is evident. If A ∩ B = ∅
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then A˜ ∪ B˜ = x; ; A ; ∪ x; +; ,A+; , ; M A∪ ˜ B˜ = max{G( ; ; m(A)); G(+; ,; m(B))}; M A˜ = G ( ; ; m(A)) and MB˜ =G(+; ,; m(B)). The following situations are possible: ˜ and M ˜ ∈ . ˜ Then M ˜ ˜ ∈ ˜ and by Lemma 1(iii), we have (i) M A˜ ∈ B A∪B ˜ ∗˜ m( ˜ = MA˜ ∗˜ MB˜ = max{G( ; ; m(A)); G(+; ,; m(B))} m( ˜ A) ˜ B) ˜ ˜ A˜ ∪ B): = MA∪ ˜ B˜ = m( ˜ and M ˜ ∈= ˜ or M ˜ ∈= ˜ and M ˜ ∈ . ˜ If M ˜ ˜ ∈ ˜ the proof is analogous with (i). Let M ˜ ∈ ˜ (ii) M A˜ ∈ B B A A∪B A ˜ therefore ˜ If M ˜ ˜ ∈= ˜ then M ˜ ˜ = G(+; ,; m(B)) ∈ ]a; ˜ b[, and MB˜ ∈= . A∪B A∪B ˜ ˜ = f˜(−1) (f(G(+; m( ˜ A˜ ∪ B) ,; m(B)))) = G(+; ,; m(B)): On the other hand, ˜ ∗˜ m( ˜ = G( ; ; m(A)) ∗˜ G(+; ,; m(B)) m( ˜ A) ˜ B) = max{G( ; ; m(A)); G(+; ,; m(B))} = MA∪ ˜ B˜ = G(+; ,; m(B)); by using again Lemma 1(iii). ˜ and M ˜ ∈ ˜ is completely analogous. The subcase M A˜ ∈= B ˜ ˜ ˜ ˜ (iii) M A˜ ∈= and MB˜ ∈= . Then M A˜ ∈ ]a˜i ; b˜ i [; MB˜ ∈ ]a˜j ; b˜j [ and M A∪ ˜ B˜ ∈ ]a˜i ; bi [ or M A∪ ˜ B˜ ∈ ]a˜j ; bj [. If i = j then ˜ ˜ ˜ = f˜(−1) (f(G( ; m( ˜ A˜ ∪ B) ; m(A))) + f(G(+; ,; m(B)))) ˜ ∗˜ m( ˜ = G( ; ; m(A)) ∗˜ G(+; ,; m(B)) = m( ˜ A) ˜ B) (see Theorem 2, with fi denoted by f). If i = j and M A∪ ˜ B˜ = M A˜ then ˜ ˜ = f˜(−1) (f(G( ; ; m(A)))) = MA˜ m( ˜ A˜ ∪ B) and by Lemma 1(v), we get ˜ ∗˜ m( ˜ = MA˜ ∗˜ MB˜ = max(MA˜; MB˜ ) = MA∪ ˜ m( ˜ A) ˜ B) ˜ A˜ ∪ B): ˜ B˜ = MA˜ = m( The proof is analogous if i = j and M A∪ ˜ B˜ = MB˜ . n ˜ i = j, then it is easy to prove that ˜ Now, if A = i=1 x; i ; i A i ; i ∈ Id ; x; i ; i A i ; i ∩ x; j ; j A j ; j = ∅; n n ˜ = m˜ m( ˜ A) x; i ; i A i ; i = ∗˜ m(x; ˜
i ; i A i ;i ); i=1
i=1
which immediately implies ˜ ∗˜ m( ˜ = m( ˜ m( ˜ A˜ ∪ B) ˜ A) ˜ B);
˜ ˜ B˜ ∈ Id ; A˜ ∩ B˜ = ∅: ∀A;
Finally, we observe that it is suKcient to prove that m˜ is monotone for two intuitionistic fuzzy sets A˜1 = x; ; A and A˜2 = x; +; ,A ; A˜1 ; A˜2 ∈ Id . If A˜1 ⊆ A˜2 then 6+ and ¿, and m( ˜ A˜1 ) = G( ; ; m(A)6G(+; ,; m(A)) = m( ˜ A˜2 ); which completes the proof.
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The second main result is Theorem 4. In the conditions of Theorem 3; the function m˜ : A˜→ [0; 1] de8ned by ˜ = sup sup{m( ˜ E˜ ⊆ A} ˜ ˜ E˜ ∈ Id is an -approximation of A; m( ˜ A) ˜ E): ¿0
is a ∗˜ -decomposable fuzzy measure on the class of all A-measurable intuitionistic fuzzy sets. Proof. First, note that from Remark 4 in Section 3, m˜ is well de:ned. Obviously, m˜ is an intuitionistic fuzzy ˜ = 0 and m( measure, because ∅˜ and X˜ are discrete intuitionistic fuzzy sets with m( ˜ ∅) ˜ X˜ ) = 1 (see the proof of Theorem 3) and m˜ is monotone (which easily follows from de:nition and from the fact proved in Theorem 3 that the restriction of m˜ to Id is monotone). ˜ ¿0, and P˜ ∈ Id , an The only problem is to show that m˜ is ∗˜ -decomposable. Let A;˜ B˜ ∈ A,˜ A˜∩ B˜ = ∅, ˜ ˜ ˜ ˜ ˜ ˜ ˜ -approximation of A ∪ B, P ⊆ A ∪ B, A = {x; A˜(x); A˜(x): x ∈X }, B = {x; B˜ (x); B˜ (x): x ∈ X } and P˜ = {x; P˜(x); P˜(x): x ∈ X }. We de:ne P˜A ; P˜B ∈ Id by P˜ (x) if A˜(x)¿0; P˜A (x) = 0 otherwise; P˜ (x) if A˜(x)¡1; P˜A (x) = 1 otherwise; P˜ (x) if B˜ (x)¿0; P˜B (x) = 0 otherwise; P˜ (x) if B˜ (x)¡1; P˜B (x) = 1 otherwise: ˜ P˜A ⊆ A,˜ P˜B ⊆ B. ˜ Also, if P˜ is an -approximation of A˜∪ B˜ then It is easy to see that P˜ = P˜A ∪ P˜B , P˜A ∩ P˜B = ∅, ˜ ˜ ˜ ˜ because PA is an -approximation of A and PB is an -approximation of B, A˜(x) − P˜A (x) = 0 if A˜(x) = 0 and A˜(x) − P˜A (x) = A˜(x) − P˜ (x) = A∪ ˜ B˜ (x) − P˜ (x) if A˜(x)¿0; P˜A (x) − A˜(x) = P˜ (x) − A˜(x) = P˜ (x) − A∪ ˜ B˜ (x)
if A˜(x)¡1;
P˜A (x) − A˜(x) = 0 if A˜(x) = 1: The proof in the case of P˜B and B˜ is similar. Therefore by Theorem 3 we get ˜ m( ˜ P˜ A ) ∗˜ m( ˜ P˜ B ) = m( ˜ P):
(1)
˜ we easily obtain Let ¿0 be :xed. Since P˜ generates P˜A and P˜B (as above), passing to supremum over all P, ˜ P˜ ∈ Id ; P˜ is an -approximation of A˜ ∪ B; ˜ P˜ ⊆ A˜ ∪ B} ˜ sup{m( ˜ P): ˜ P˜ A ⊆ A} ˜ 6 sup{m( ˜ P˜ A ): P˜ A ∈ Id ; P˜ A is an -approximation of A; ˜ P˜ B ⊆ B}: ˜ ∗˜ sup{m( ˜ P˜ B ): P˜ B ∈ Id ; P˜ B is an -approximation of B;
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113
Passing now to supremum with ¿0 and taking into account the continuity of ∗˜ , we get ˜ ∗˜ m( ˜ m( ˜ m( ˜ A˜ ∪ B)6 ˜ A) ˜ B):
(2)
˜ 6 m( ˜ :rstly :xing P˜B and passing to supremum with P˜A , then passing to Conversely, by (1), by m( ˜ P) ˜ A˜∪ B), supremum with P˜B and :nally, passing to supremum with ¿0 (here we take account that ∗˜ is continuous), we obtain ˜ ∗˜ m( ˜ m( ˜ m( ˜ A) ˜ B)6 ˜ A˜ ∪ B); which together with (2) proves the theorem. Remark 5. It is evident that Theorem 4 is a generalization of Proposition 3:1 in [2]. But looking to De:nition 3:2 in [2], we have discovered some inadvertences, which fortunately do not ˜ the so-called standard approximation of the usual inLuence the proofs in the paper. Thus, it follows that P, ˜ in the sense of the inclusion in fuzzy set theory. But in this case, in De:nition 3:3 fuzzy set A,˜ satis:es A˜⊆ P, we would obtain ˜ = sup{m( ˜ A˜ ⊆ P}; ˜ P˜ ∈ P(A); ˜ m( ˜ A) ˜ P): which obviously is not correct. As a consequence, we think that in De:nition 3:2 we have to replace i by i−1 in the formula of P˜ and of course to consider 0 as the :rst point of the division of [0; 1]. In the sequel, we study the ∨-decomposable measures (see De:nitions 4 and 5 with ∗ = ∨), where ∨ is the triangular conorm de:ned by x ∨ y = max(x; y), ∀x; y ∈ [0; 1]. Theorem 5. Let (X; A; m) be a fuzzy measure space. If m is a ∨-decomposable fuzzy measure then the intuitionistic fuzzy measure m˜ constructed by Theorem 4 is a ∨-decomposable intuitionistic fuzzy measure. ˜ = [0; 1] and m˜ : Id → [0; 1] is deProof. With the notations in Theorem 3, ∗ = ∗˜ = ∨, D = D˜ = ∅, = :ned by ˜ = max{G( i ; i ; m(A i ; i )): i ∈ {1; : : : ; n}} m( ˜ A) n if A˜= i=1 x; i ; i A i ; i and Theorem 4 assures that m˜ : A˜→ [0; 1] is a ∨-decomposable intuitionistic fuzzy measure. ˜ (as in Theorem 4) is reduced to If m is a ∨-decomposable fuzzy measure then the computation of m˜ (A) the maximization of a real-valued positive function. Theorem 6. Let (X; A; m) be a fuzzy measure space. If m is a ∨-decomposable fuzzy measure then ˜ = sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61}; m( ˜ A) where A ; is the crisp set de8ned by A ; = {x ∈ X : A˜(x)¿ and A˜(x)6}: Proof. Let ; ∈ [0; 1]; + 6 1. Because x; ; A ; ⊆ A,˜ we have ˜ m(x; m( ˜ A)¿ ˜
; A ; ) = G( ; ; m(A ; )):
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Therefore, ˜ m( ˜ A)¿ sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61}:
n In order to show the converse inequality, let ¿0 and E˜ = i=1 x; i ; i E i ;i with x; i ; i E i ;i ∩ ˜ if i = j, be an -approximation of A;˜ E˜ ⊆ A˜ (see Remark 4). Because m˜ is a ∨-decomx; j ; j E j ;j = ∅, posable intuitionistic fuzzy measure, we have ˜ = max m( ˜ E)
i∈{1;:::; n}
m(x; ˜
i ; i E i ; i ) = max
i∈{1;:::; n}
G( i ; i ; m(E i ; i )):
Also, E i ; i ⊆ A i ; i , ∀i ∈ {1; : : : ; n}, therefore m(E i ; i ) 6 m(A i ; i ), ∀i ∈ {1; : : : ; n}. We obtain ˜ m( ˜ E)6 sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61}: The inequality ˜ m( ˜ A)6 sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61} follows when E˜ varies among all -approximations of A;˜ E˜ ⊆ A,˜ with ¿0. Example 2. Let X = ∅ and x0 ∈ X . The function mx0 : P(X ) → [0; 1] de:ned by 0 if x0 ∈= A; mx0 (A) = 1 if x0 ∈ A is a ∨-decomposable fuzzy measure. By using Theorem 6 we obtain a ∨-decomposable intuitionistic fuzzy measure m˜x0 : IFS(X ) → [0; 1] corresponding to intuitionistic structure function in Example 1. This is de:ned by ˜ = sup{ n (1 − n )mx0 (A ; ): ; ∈ [0; 1]; + 61} m˜ x0 (A) = ˜nA˜(x0 )(1 − ˜nA˜(x0 )); because
n (1 − n )mx0 (A ; ) =
n (1 − n ) if ˜A˜(x0 )¿ and ˜A˜(x)6 0 otherwise:
Example 3. Let X = {x1 ; : : : ; xp } and f : X → [0; 1] be such that maxx∈X f(x) = 1. The function m : P(X ) → [0; 1] de:ned by max f(x) if A = ∅; x∈A m(A) = 0 if A = ∅ is a ∨-decomposable fuzzy measure. By using Theorem 6, with the help of the intuitionistic structure function in Example 1, n = 1, we obtain a ∨-decomposable intuitionistic fuzzy measure m˜ : IFS(X ) → [0; 1], de:ned by ˜ = sup{ (1 − )m(A ; ): ; ∈ [0; 1]; + 61} m( ˜ A)
f(xk ): ; ∈ [0; 1]; + 61 : = sup (1 − ) max k: A˜(xk )¿ ; A˜(xk )6
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115
In the particular case p = 2 we consider A˜ = {x1 ; A˜(x1 ); A˜(x1 ); x2 ; A˜(x2 ); A˜(x2 )} ∈ IFS(X ): Because
A ;
{x1 ; x2 } {x } 1 = {x2 } ∅
if A˜(x1 )¿ ; A˜(x2 )¿ ; A˜(x1 )6; A˜(x2 )6; if A˜(x1 )¿ ; A˜(x1 )6 and (A˜(x2 )¡ or A˜(x2 )¿); if A˜(x2 )¿ ; A˜(x2 )6 and (A˜(x1 )¡ or A˜(x1 )¿); otherwise;
we obtain ˜ = sup{ (1 − )m(A ; ): ; ∈ [0; 1]; + 61} m( ˜ A) = max{min{A˜(x1 ); A˜(x2 )}(1 − max{A˜(x1 ); A˜(x2 )})m({x1 ; x2 }); A˜(x1 )(1 − A˜(x1 ))m({x1 }); A˜(x2 )(1 − A˜(x2 ))m({x2 })} = max{A˜(x1 )(1 − A˜(x1 ))f(x1 ); A˜(x2 )(1 − A˜(x2 ))f(x2 )}: 5. Compositive intuitionistic information measure In this section :rst we introduce the concept of intuitionistic compositive information measure. Then we study its connection with the concept of decomposable intuitionistic fuzzy measure. The following axiomatic de:nition of information measure is given in [3]. Denition 9. Let (T; 4) be a lattice with minimum m and maximum M. An information measure over M + , where R M + = R+ ∪ {+∞}, with the properties (T; 4) is a map J : T → R (i) J (M) = 0; (ii) J (m) = + ∞; (iii) x4y implies J (x)¿J (y); ∀(x; y) ∈ T × T. If T is a family of intuitionistic fuzzy sets then we obtain the de:nition of intuitionistic information measure. We will denote by J˜ such an information measure. Remark 6. If A˜ is an intuitionistic -algebra and ⊆ is the inclusion between intuitionistic fuzzy sets, then (A;˜ ⊆ ) is a lattice with minimum ∅˜ and maximum X˜ . Inspired by [3], we can give the following: ˜ Denition 10. We say that an intuitionistic information measure J˜ is F-compositive if there exists a function M M M ˜ F : R+ × R+ → R+ such that ˜ J˜ (B)); ˜ = F( ˜ J˜ (A); ˜ A˜ ∩ B˜ = ∅˜ implies J˜ (A˜ ∪ B) where ∩ and ∪ represent the usual operations between the intuitionistic fuzzy sets. Remark 7. It is easy to see that the function F˜ satis:es the properties ˜ y) = F(y; ˜ x), (i) F(x;
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˜ F(x; ˜ y); z) = F(x; ˜ F(y; ˜ z)), (ii) F( ˜ (iii) F(x; ∞) = x, ˜ y)6F(x ˜ ; y), (iv) x6x implies F(x; for every x; x ; y in the range of J˜. Let M(X ) be the class of all decomposable intuitionistic fuzzy measures (on a set X ) and C(X ) be the class of all compositive intuitionistic information measures (on X ). Theorem 7. The function V : M(X ) → C(X ) de8ned by V (m) ˜ = J˜; where 1 − 1 if m( ˜ ˜ A)¿0; ˜ ˜ ˜ ˜ A) J (A) = m( ˜ =0 +∞ if m( ˜ A) is a bijection. Proof. We assume that m˜ is a ∗˜ -decomposable intuitionistic fuzzy measure and J˜ = V (m). ˜ ˜ = + ∞; J˜(X˜ ) = 0 and A˜ ⊆ B˜ implies J˜(A)¿ ˜ J˜(B), ˜ therefore J˜ is an intuitionistic It is easy to see that J˜(∅) ˜ information measure. We show that J˜ is F-compositive, where 1 − 1 if x; y ∈ [0; +∞); 1 1 ∗˜ x+1 y+1 ˜ y) = F(x; if x = +∞; y x if y = +∞: ˜ Let A;˜ B˜ be such that A˜ ∩ B˜ = ∅. ˜ ˜ ¡+∞ and J˜(B)¡+∞. ˜ ˜ ˜ If m( ˜ A)¿0 and m( ˜ B)¿0 then m( ˜ A˜ ∪ B)¿0; J˜(A) Therefore, ˜ J˜ (B)) ˜ J˜ (A); ˜ = F(
1 1 1 ˜ −1= − 1 = J˜ (A˜ ∪ B): −1= 1 1 ˜ ∗˜ m( ˜ ∪ B) ˜ ˜ m( ˜ A) ˜ B) m( ˜ A ∗˜ ˜ + 1 J˜ (B) ˜ +1 J˜ (A)
˜ ˜ ˜ = 0 then m( ˜ ˜ = +∞, If m( ˜ A)¿0 and m( ˜ B) ˜ A˜ ∪ B)¿0; J˜(A)¡ + ∞ and J˜(B) ˜ J˜ (B)) ˜ = ˜ J˜ (A); ˜ = J˜ (A) F( =
1 1 −1= −1 ˜ ˜ ∗˜ m( ˜ m( ˜ A) m( ˜ A) ˜ B)
1 ˜ − 1 = J˜ (A˜ ∪ B): ˜ m( ˜ A˜ ∪ B)
˜ = 0 and m( ˜ ∗˜ m˜ (B) ˜ = J˜(B) ˜ J˜(B)) ˜ = 0 then m( ˜ = m( ˜ = 0, J˜(A) ˜ = + ∞, therefore F( ˜ J˜(A); ˜ = If m( ˜ A) ˜ B) ˜ A˜ ∪ B) ˜ A) ˜ + ∞ = J˜(A˜ ∪ B). We observe that because of the properties of ∗˜ , the function F˜ has the properties in Remark 7. ˜ The function V is obviously injective. Also, it is easy to prove that if J˜ is a F-compositive intuitionistic information measure, then the function m˜ de:ned by 1 ˜ is :nite; if J˜ (A) ˜ ˜ ˜ m( ˜ A) = J (A) + 1 ˜ = +∞ 0 if J˜ (A)
A.I. Ban, S.G. Gal / Fuzzy Sets and Systems 123 (2001) 103–117
is a ∗˜ -decomposable intuitionistic fuzzy 1 1−y 1 − x F˜ ; +1 x y x ∗˜ y = y x
117
measure, where if x; y ∈ (0; 1]; if x = 0; if y = 0
and in addition V (m) ˜ = J˜. Remark 8. Let m˜ : A˜→ [0; 1] M + given Theorem 7, J˜ : A˜→ R 1 − 1 if ˜ ˜ ˜ ˜ A) J (A) = m( +∞ if
be the ∗˜ -decomposable intuitionistic fuzzy measure in Theorem 4. Then by by ˜ m( ˜ A)¿0; ˜ =0 m( ˜ A)
˜ is a F-compositive intuitionistic information measure, where F˜ is that one in the proof of Theorem 7. 6. Future research and possible applications The results in the paper might be used in several directions. Thus, the approximation result in Section 3 could be applied to approximation of continuous intuitionistic fuzzy logics by discrete intuitionistic fuzzy logics. On the other hand, sometimes the usage of intuitionistic fuzzy sets can be :ner than that obtained with standard fuzzy sets. Thus, the usage of decomposable measures on classical sets and on fuzzy sets, in decision making theory is well known so using the decomposable intuitionistic fuzzy measures might obtain better results. Acknowledgements The authors would like to express their thanks to the referees and editors for their useful comments. References [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. [2] C. Bertoluzza, D. Cariolaro, On the measure of a fuzzy set based on continuous t-conorms, Fuzzy Sets and Systems 88 (1997) 355–362. [3] C. Bertoluzza, T. Brezmes, G. Naval, in: B. Bouchon-Meunier, R.R. Yager, L.A. Zadeh (Eds.), Compositive Information Measure of a Fuzzy Set, World Scienti:c, Singapore, 1995, pp. 353–359. [4] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [5] D. Dubois, H. Prade, A class of fuzzy measures based on triangular norms, Internat. J. Gen. Systems 8 (1982) 43–61. [6] S.G. Gal, Sur les thQeorRemes d’approximation de Weierstrass, Mathematica (Cluj) 23 (46) (1) (1981) 25–30. [7] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69 (1995) 279–298. [8] E.P. Klement, R. Mesiar, Triangular norms, Tatra Mountains Math. Publ. 13 (1997) 169–193. [9] E. Pap, Decomposable measures and nonlinear equations, Fuzzy Sets and Systems 92 (1997) 205–221. [10] A. Precupanu, Mathematical Analysis, Real Functions, Editura DidacticTa sUi PedagogicTa, Bucharest, 1976 (in Romanian).
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Decomposable measures and information measures for intuitionistic fuzzy sets A.I. Ban ∗ ; 1 , S.G. Gal Department of Mathematics, University of Oradea, str. Armatei Romˆane 5, 3700 Oradea, Romania Received 27 October 1998; received in revised form 29 March 2000; accepted 5 June 2000
Abstract The aim of this paper is to construct a decomposable measure on the class of measurable intuitionistic fuzzy sets, extending thus the corresponding result for fuzzy sets in a very recent paper. The main tool in this construction is a non-trivial approximation (with restrictions) result concerning the intuitionistic fuzzy sets. As an application we construct c 2001 Elsevier Science B.V. All rights a compositive information measure on the class of intuitionistic fuzzy sets. reserved. Keywords: Intuitionistic fuzzy sets; Non-additive measures; Possibility measures; Measures of information
1. Introduction In a very recent paper [2], a decomposable measure of measurable fuzzy sets based on continuous tconorms is constructed. The main purpose of this paper is to generalize the result for the class of measurable intuitionistic fuzzy sets. Section 2 contains some de:nitions and notations, introduced by Atanassov (see [1]). Moreover, the new concept of decomposable intuitionistic fuzzy measure is introduced. In Section 3 we prove an interesting result concerning the approximation of measurable intuitionistic fuzzy sets by discrete intuitionistic fuzzy sets, which is essential in the proofs of the main results. In comparison with the corresponding approximation result for usual fuzzy sets, when it is direct consequence of a classical theorem in real functions theory, the result for intuitionistic fuzzy sets is one of approximations with restrictions which cannot be directly derived from a classical result and it is solved by using [6]. ∗
Corresponding author. E-mail address: [email protected] (A.I. Ban). 1 While working on this paper the author has been partially supported by a grant of Romanian Agency for Science, Technology and Inovation ANSTI 410=2000. c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 1 0 6 - 8
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Section 4 contains the construction of the decomposable measure for intuitionistic fuzzy sets. Also, we point out some inadvertences in the de:nition of the decomposable measure in [2], which have been eliminated in our more general de:nitions. In Section 5 we indicate a bijection between the class of compositive information measures and the class of decomposable fuzzy measures on intuitionistic fuzzy sets, which allows that starting from the decomposable measure constructed in Section 4, we obtain a compositive information measure. 2. Preliminaries Firstly, we give some de:nitions and notations regarding the intuitionistic fuzzy sets. Denition 1 (Atanassov [1]). Let X = ∅ be a given set. An intuitionistic fuzzy set in X is an expression A˜ given by A˜ = {x; A˜ (x); A˜ (x): x ∈ X } where A˜ : X → [0; 1]; A˜ : X → [0; 1] satisfy the condition 06 A˜ (x) + A˜ (x)61, for all x ∈ X . We will denote by IFS(X ) the set of all the intuitionistic fuzzy sets in X . Finite union and intersection, inclusion and complementation are de:ned for every A;˜ B˜ ∈ IFS(X ) (see [1]): A˜ ∪ B˜ = {x; max{A˜ (x); B˜ (x)}; min{A˜ (x); B˜ (x)}: x ∈ X }; A˜ ∩ B˜ = {x; min{A˜ (x); B˜ (x)}; max{A˜ (x); B˜ (x)}: x ∈ X }; A˜ ⊆ B˜ if and only if A˜ (x)6B˜ (x) and A˜ (x)¿B˜ (x) for all x ∈ X; A˜ c = {x; A˜ (x); A˜ (x): x ∈ X }: Countable union and intersection are de:ned as above, by replacing max with sup and min with inf. For ; ∈ [0; 1], + 61 and the intuitionistic fuzzy set A˜ = {x; A˜(x); A˜ (x): x ∈ X } we de:ne the crisp set A ; by A ; = {x ∈ X : A˜ (x) = ; A˜ (x) = }: Let A ⊆ X ,
A˜ (x) = A˜ (x) =
0
if x ∈ A; if x ∈ A;
1
if x ∈ A; if x ∈ A;
where 06 + 61. The intuitionistic fuzzy set A˜ = {x; A˜ (x); A˜ (x): x ∈ X } will be denoted by x; ; A . For example, ∅˜ = x; 0; 1X and X˜ = x; 1; 0X . Denition 2. An intutionistic fuzzy set A˜ = {x; A˜ (x); A˜ (x): x ∈ X } is called discrete if the functions A˜ and A˜ take :nite number of values. Note that, if A˜ = {x; A˜ (x); A˜(x): x ∈ X } is a discrete intuitionistic fuzzy set then we can consider A˜ (x) ∈ { 1 ; : : : ; n } = M1 ; A˜ (x) ∈ {1 ; : : : ; n } = M2 , that is A˜ and A˜ take the same number of values if we naturally accept the fact that the elements of each between M1 and M2 , are not necessarily distinct. Moreover, we can consider −1 ( k ) = −1 (k ), ∀k ∈ {1; : : : ; n}. A˜ A˜
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Indeed, to prove this, let us suppose that A˜ (x) ∈ { 1 ; : : : ; p }, i = j if i = j; A˜ (x) ∈ {1 ; : : : ; q }, i = j if ( i ) = Yi , i ∈ {1; : : : ; p} and −1 (i ) = Zi ; i ∈ {1; : : : ; q}. We have Yi ∩ Yj = ∅ if i = j, i = j. Let us denote −1 A˜ A˜ p q p q Zi ∩ Zj = ∅ if i = j and X = i=1 Yi = j=1 Zj . We obviously have X = i=1 j=1 Mij , where Mij = Yi ∩ Zj n and the sets Mij are mutually disjoint. If we omit the sets Mij that are empty, we can rewrite X = k=1 Xk , where Xk are mutually disjoint, each Xk is non-empty of the form Xk = Mij and 16n6pq. We rede:ne A˜ and A˜ in the following way: A∗˜ (x) = k∗ = i
if x ∈ Xk = Mij ;
A∗˜ (x) = k∗ = j
if x ∈ Xk = Mij :
Obviously, both ∗A˜ and ∗A˜ have n-values and −1 ( k∗ ) = −1 (k∗ ) = Xk ; ∀k ∈ {1; : : : ; n}. A˜ A˜ As a conclusion we can always write A˜ by means of the decomposition A˜ =
n
x; k ; k A k ; k ;
k=1
˜ if i = j. where x; i ; i A i ; i ∩ x; j ; j A j ; j = ∅, For example, the intuitionistic fuzzy sets A˜ = {x; A˜ (x); A˜ (x): x ∈ R} where 0:1 if x ∈ (−∞; −1]; A˜ (x) = 0:3 if x ∈ (−1; 1); 0:6 if x ∈ [−1; +∞) and
A˜ (x) =
0:4 0:2
if x ∈ (−∞; 0); if x ∈ [0; +∞)
which can be rewritten as A˜ = {x; ∗A˜(x); ∗A˜(x): x ∈ R} where 0:1 0:3 A∗˜ (x) = 0:3 0:6
and
0:4 0:4 ∗A˜ (x) = 0:2 0:2
if x ∈ (−∞; −1]; if x ∈ (−1; 0); if x ∈ [0; 1); if x ∈ [1; +∞)
if if if if
x x x x
∈ (−∞; −1]; ∈ (−1; 0); ∈ [0; 1); ∈ [1; +∞):
The decomposition corresponding to A˜ as above is A˜ = x; 0:1; 0:4(−∞;−1] ∪ x; 0:3; 0:4(−1;0) ∪ x; 0:3; 0:2[0;1) ∪ x; 0:6; 0:2[1;+∞) : Now, we recall two well known de:nitions.
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Denition 3 (See e.g. Dubois and Prade [4; p: 126] or Grabisch [7]). A fuzzy measure over a measurable space (X; A) is a map m : A → [0; 1] with the following properties: (i) m(∅) = 0, (ii) m(X ) = 1, (iii) A ⊆ B implies m(A)6m(B). Denition 4 (See Dubois and Prade [5] and Pap [9]). A fuzzy measure is said to be ∗-decomposable if there exists the composition law ∗ : [0; 1] × [0; 1] → [0; 1] such that m(A ∪ B) = m(A) ∗ m(B);
∀A; B ∈ A; A ∩ B = ∅:
The structure (X; A; m; ∗) will be called decomposable space. Remark 1. It is obvious that ∗ should behave as a triangular conorm on the range of m (that is increasing, associative, commutative and x ∗ 0 = x). With respect to the operations of union, intersection and complementation introduced after De:nition 1 we can adapt the de:nitions of -algebra, fuzzy measure and decomposable fuzzy measure to intuitionistic fuzzy sets in the following way. Denition 5. Let X = ∅. A family A˜ of intuitionistic fuzzy sets in X is called intuitionistic fuzzy -algebra ˜ will be called if ∅˜ ∈ A,˜ A˜ ∈ A˜ implies A˜c ∈ A˜ and A˜i ∈ A;˜ i ∈ N implies i∈N A˜i ∈ A.˜ The pair (X; A) intuitionistic fuzzy measurable space. ˜ is a map m˜ : A˜→ [0; 1] An intuitionistic fuzzy measure over an intuitionistic fuzzy measurable space (X; A) ˜ ˜ ˜ ˜ ˜ ˜ with the properties: m( ˜ ∅) = 0; m( ˜ X ) = 1 and A ⊆ B implies m( ˜ A)6m( ˜ B). The intuitionistic fuzzy measure m˜ is said to be ∗˜ -decomposable if there exists the composition law ∗˜ : [0; 1] × [0; 1] → [0; 1] such that ˜ ˜ ∗˜ m( ˜ B˜ ∈ A;˜ A˜ ∩ B˜ = ∅: ˜ = m( ˜ ∀A; m( ˜ A˜ ∪ B) ˜ A) ˜ B); Also, we can consider Denition 6. Let (X; A) be a measurable space. We say that A˜ = {x; A˜(x); A˜(x): x ∈ X } ∈ IFS(X ) is Ameasurable if and only if A˜ and A˜ are A-measurable functions. Remark 2. Let (X; A) be a measurable space. The family A˜ = {A˜ ∈ IFS(X ): A˜ is A-measurable} is an intuitionistic fuzzy -algebra.
3. Approximation of intuitionistic fuzzy sets by discrete intuitionistic fuzzy sets A classical approximation result in the real functions theory (see e.g. [10, pp. 124 –126]), assures that if A : X → [0; 1] is a measurable fuzzy set, then there exists a sequence An : X → [0; 1]; n ∈ N, of discrete fuzzy sets (i.e. An are step functions), such that An → A, uniformly on X and An ⊆ An+1 ⊆ A; n ∈ N, where ⊆ represents the usual inclusion between two fuzzy sets. More exactly, we can write ∀ ¿0; ∃n0 ∈ N; such that 06A(x) − An (x)¡ ; n¿n0 ; x ∈ X
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and An (x)6An+1 (x)6A(x); n ∈ N; x ∈ X: At this point, it is natural to ask the following question: how the above result to the intuitionistic case can be extended? Let (X; A) be a measurable space and A˜ ∈ IFS(X ); A˜ = {x; A˜(x); A˜(x): x ∈ X }; supposed to be Ameasurable. A direct application of the above result to each A˜ and A˜ yields an approximation sequence (n) (n) (n) of discrete intuitionistic fuzzy sets A˜n = {x; A(n) ˜ (x); A˜ (x): x ∈ X }, n ∈ N; such that A˜ A˜; A˜ A˜ uniformly on X , but for which the monotony property (i.e. A˜n ⊆ A˜n+1 ; n ∈ N) fails to exist. But as we see in the next section, this monotony property is essential in the de:nition of the decomposable measure. In order to answer the above question in analytical terms, we have to solve the following non-trivial problem of approximation with restrictions: If f; g : X → [0; 1] are measurable and satisfy f(x) + g(x)61; then we construct two sequences of step functions fn ; gn : X → [0; 1], such that fn (x) + gn (x)61; n ∈ N; x ∈ X and moreover fn (x)6fn+1 (x)6f(x); gn (x)¿gn+1 (x)¿g(x); for all n ∈ N; x ∈ X . The following result solves the question. Theorem 1. If A˜ ∈ IFS(X ); A˜ = {x; A˜(x); A˜(x): x ∈ X } is A-measurable; then there exists a sequence of (n) (n) (n) discrete intuitionistic fuzzy sets A˜n = {x; A(n) ˜ (x); A˜ (x): x ∈ X }; n ∈ N; such that A˜ A˜; A A˜ uniformly on X; (i.e. A˜n ⊆ A˜n+1 ⊆ A;˜ for all n ∈ N). Proof. By the classical approximation result in [10, pp. 124 –126], there exist the sequences of step functions Fn : X → [0; 1], hn : X → [0; 1]; n ∈ N, such that Fn A˜ + A˜; hn A˜; uniformly on X . More exactly, by the above result we have 06A˜(x) − hn (x)6
1 2n
for all n ∈ N; x ∈ X:
Now, by using the idea in [6], we obtain a monotonous decreasing sequence of step functions, h∗n (x) = hn (x) +
2 ; 2n−1
x ∈ X; n ∈ N;
i.e. h∗n → A˜ uniformly on X and h∗n (x) − h∗n+1 (x) = hn (x) − hn+1 (x) +
2 ¿0; 2n
x ∈ X; n ∈ N;
because hn (x) − hn+1 (x) 6 |hn (x) − hn+1 (x)|6|hn (x) − A˜(x)| + |A˜(x) − hn+1 (x)| 1 2 1 6 n + n+1 6 n ; x∈X; n∈N: 2 2 2 Because h∗n (x) can take values greater than 1, let us de:ne h∗n (x) if h∗n (x)61; gn (x) = 1 if h∗n (x)¿1: Obviously, gn (x) ∈ [0; 1] and gn is step function, n ∈ N.
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By A˜ (x)6gn (x)6h∗n (x); x ∈ X; n ∈ N; it immediately follows that gn → A˜, uniformly on X . Also, by simple reasonings we get gn+1 (x)6gn (x) for all x ∈ X; n ∈ N. As a conclusion, gn : X → [0; 1]; n ∈ N; is a sequence of step functions satisfying gn A˜;
uniformly on X:
Now, let us de:ne + A(n) ˜ (x) = (Fn − gn ) (x) (i:e: the positive part of Fn (x) − gn (x))
and A(n) ˜ (x) = gn (x); x ∈ X; n ∈ N. (n) (n) Obviously, A(n) ˜ is step function, n ∈ N. Then, 06A˜ (x) and if 0¡A˜ (x) then A(n) ˜ (x) = Fn (x) − gn (x)6A˜(x) + A˜(x) − gn (x)6A˜(x)61; i.e. A(n) ˜ ∈ [0; 1], for all x ∈ X; n ∈ N. Also, + A(n) ˜ (x) = (Fn − gn ) (x) = max{(Fn − gn ) (x); 0}
Fn (x) − gn (x) + |Fn (x) − gn (x)| n→∞ A˜(x) + A˜(x) − A˜(x) + |A˜(x) + A˜(x) − A˜(x)| → 2 2 A˜(x) + |A˜(x)| = A+˜ (x) = A˜(x) = 2
=
uniformly on X . Moreover, by simple reasoning we get (n+1) (x) A(n) ˜ (x)6A˜
for all x ∈ X; n ∈ N:
Finally, (n) + 0 6 A(n) ˜ (x) + A˜ (x) = (Fn − gn ) (x) + gn (x) Fn (x)61 if Fn (x) − gn (x)¿0; = gn (x)61 if Fn (x) − gn (x)¡0
which proves the theorem. Remark 3. If A˜ ∈ IFS(X ) is of the form A˜ = 1 − A˜ (i.e. if A˜ one reduces to a classical fuzzy set), then Theorem 1 is an obvious consequence of the classical result in the real functions theory. Indeed, by A(n) ˜ A˜ (n) it follows that A(n) ˜ = 1 − A˜ 1 − A˜ = A˜. But as we have seen in the proof of Theorem 1, in the general
case A˜ (x) + A˜ (x)61; this is not an obvious one. In the next section we also need the following:
Denition 7. Let A˜1 = {x; 1 (x); 1 (x): x ∈ X }, A˜2 = {x; 2 (x); 2 (x): x ∈ X } ∈ IFS(X ) and ¿0. We say that A˜1 is an -approximation of A˜2 if |2 (x) − 1 (x)|¡ and |2 (x) − 1 (x)|¡ ; ∀x ∈ X .
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Remark 4. If A˜ ∈ IFS(X ) is A-measurable then for any ¿0 there exist the discrete intuitionistic fuzzy sets ˜ The :rst inclusion is an E˜ and F˜ which are -approximations of A˜ and they moreover satisfy E˜ ⊆ A˜ ⊆ F. ˜ immediate consequence of Theorem 1. Because the A-measurability of A implies A˜c is A-measurable and the complementation of a discrete intuitionistic fuzzy set is a discrete intuitionistic fuzzy set from which we obtain the second inclusion.
4. Construction of the decomposable intuitionistic fuzzy measure In this section, we construct a decomposable intuitionistic fuzzy measure for a suitable composition law ∗˜ , starting from a decomposable space (X; A; m; ∗). We mention that in [2] a similar result is obtained for fuzzy sets. First, we recall a well known result. Theorem 2. (See e.g. Bertoluzza and Cariolaro [2] or Klement and Mesiar [8]). Let D = {]ai ; bi [: i ∈ N ⊆ N} be a countable family of open disjoint subintervals of [0; 1] and let F = {fi : [ai ; bi ] → R | i ∈ N ⊆ N} be a family of continuous and strictly increasing functions with fi (ai ) = 0: The map ∗ de8ned on [0; 1] × [0; 1] by x∗y =
fi(−1) (fi (x) + fi (y)) if (x; y) ∈ [ai ; bi ] × [ai ; bi ]; max(x; y) otherwise
(where fi(−1) (x) = fi−1 (min{x; fi (bi )}) is the pseudo-inverse of fi ) is a triangular conorm; and conversely; any continuous triangular conorm has this form. Owing to the previous theorem, in the sequel we will consider only continuous triangular conorms. The following two lemmas will be used in the proof of the main result of this section Lemma 1 (Bertoluzza and Cariolaro [2]). Let ∗ be a continuous triangular conorm and (]ai ; bi [)i∈N as in the previous theorem. The following properties hold: (i) x ∗ y¿max(x; y); (ii) x ∈ if and only if x ∗ x = x; (iii) x ∈ implies x ∗ y = max(x; y); (iv) (x; y) ∈ ]ai ; bi [ × ]ai ; bi [ if and only if x ∗ y¿max(x; y); (v) x ∈ ]ai ; bi [; y ∈ ]aj ; bj [; j = i implies x ∗ y = max(x; y); where = [0; 1]\ i ∈ N ]ai ; bi [. ˜ x; ; A = ∅˜ and x; +; ,B = ∅˜ then A ∩ B = ∅. Lemma 2. If x; ; A ∩ x; +; ,B = ∅; Proof. Let us assume that A ∩ B = ∅, therefore, ∃x0 ∈ X such that x0 ∈ A and x0 ∈ B. The hypothesis x; ; A = ∅˜ implies = 0 or = 1. ˜ This implies , = 1 and = 1. Then + ¿1, If = 0 then + = 0 because x; ; A ∩ x; +; ,B = ∅. contradiction. Analogously, if = 1 then , = 1; + = 0 and we get the contradiction 1¡+ + ,, which proves the lemma. We also introduce
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Denition 8. Let D = {( ; ) ∈ [0; 1] × [0; 1]: +61}. A function G : D × [0; 1] → [0; 1] is called intuitionistic structure function if and only if the following conditions hold: (i) G(0; 1; x) = 0; (ii) G(1; 0; x) = x; (iii) 6 and ¿ implies G( ; ; x)6G( ; ; x), (iv) x 6x implies G( ; ; x )6G( ; ; x ). Example 1. The function G : D × [0; 1] → [0; 1] de:ned by G( ; ; x) = n (1 − n )x; n ∈ {1; 2; : : :}, :xed, is a simple example of intuitionistic structure function. The :rst main result is Theorem 3. Let G be an intuitionistic structure function and ∗; ∗˜ be continuous triangular conorms represented by the countable families D = {]ai ; bi [: i ∈ I }; D˜ = {]a˜j ; b˜j [: j ∈ J }; respectively; and the families of additive generators F = {fi : [ai ; bi ] → R: i ∈ I }; F˜ = {fj˜ : [a˜j ; b˜j ] → R: j ∈ J }; respectively. We assume that ∀j ∈ J; ∀( ; ) ∈ D\{(0; 1)}; ∃i ∈ I : ]a˜j ; b˜j [ ⊆ G( ; ; ]ai ; bi [) and (X; A; m; ∗) is a decomposable space. We den note by M A˜ = max{G( i ; i ; m(A i ; i )): i ∈ {1; : : : ; n}}; where A˜ = i=1 x; i ; i A i ; i is a A-measurable and discrete intuitionistic fuzzy set. Let Id be the class of all A-measurable and discrete intuitionistic fuzzy sets in X . Let us de8ne m˜ : Id → [0; 1] in the following way: ˜ = 0; – m( ˜ ∅) ˜ then we take m( ˜ = M ˜; – if M A˜ ∈ ; ˜ A) A ˜ ⊆ G( ; ; ]a; b[) with ( ; ) ∈ {( 1 ; 1 ); : : : ; ( n ; n )}; ]a; b[ ∈ D; ˜ ˜ b[ – if M A˜ ∈= then M A˜ belongs to some ]a; and we take
˜ = f˜(−1) m( ˜ A) f˜ ◦ G( r ; r ; m(A r ; r )) ; ˜ and f˜ is the function where the sum is extended to all the values of r for which G( r ; r ; m(A r ; r )) ∈ ]a; ˜ b[ ˜ Then m˜ is a ∗˜ -decomposable intuitionistic fuzzy measure. which determines ∗˜ in ]a; ˜ b[. ˜ = G( ; ; m(A)). Indeed, if Proof. First, we prove that for A˜ = x; ; A ∈ Id ; m˜ (A) ˜ then m( ˜ =M˜ MA˜ = max{G( ; ; m(A))} = G( ; ; m(A)) ∈ ˜ A) A by construction. ˜ ⊆ G( ; ; ] a; b[) with ]a; b[ ∈ D and we have ˜ then by hypothesis M ˜ ∈ ]a; If M A˜ ∈= ˜ b[ A ˜ ˜ = f˜(−1) (f(G( ; ; m(A)))) m( ˜ A) −1 ˜ ˜ ˜ b))) ˜ = f (min(f(G( ; ; m(A))); f( ˜ ; m(A)))) = G( ; ; m(A)); = f˜−1 (f(G( ; ˜ (see Theorem 2). because f˜ is strictly increasing on ]a; ˜ b[ Because X˜ = x; 1; 0X ; m( ˜ X˜) = G(1; 0; m(X )) = 1: Now, we will prove that ˜ ˜ ∗˜ m( ˜ B˜ ∈ Id ; A˜ ∩ B˜ = ∅: ˜ = m( ˜ ∀A; m( ˜ A˜ ∪ B) ˜ A) ˜ B); ˜ First, let the discrete intuitionistic fuzzy sets be of the form A˜ = x; ; A ; B˜ = x; +; ,B such that A˜ ∩ B˜ = ∅. ˜ ˜ ˜ ˜ ˜ ˜ Then A = ∅ or B˜ = ∅ or A ∩ B = ∅ (see Lemma 2). If A = ∅ or B˜ = ∅ then the equality is evident. If A ∩ B = ∅
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111
then A˜ ∪ B˜ = x; ; A ; ∪ x; +; ,A+; , ; M A∪ ˜ B˜ = max{G( ; ; m(A)); G(+; ,; m(B))}; M A˜ = G ( ; ; m(A)) and MB˜ =G(+; ,; m(B)). The following situations are possible: ˜ and M ˜ ∈ . ˜ Then M ˜ ˜ ∈ ˜ and by Lemma 1(iii), we have (i) M A˜ ∈ B A∪B ˜ ∗˜ m( ˜ = MA˜ ∗˜ MB˜ = max{G( ; ; m(A)); G(+; ,; m(B))} m( ˜ A) ˜ B) ˜ ˜ A˜ ∪ B): = MA∪ ˜ B˜ = m( ˜ and M ˜ ∈= ˜ or M ˜ ∈= ˜ and M ˜ ∈ . ˜ If M ˜ ˜ ∈ ˜ the proof is analogous with (i). Let M ˜ ∈ ˜ (ii) M A˜ ∈ B B A A∪B A ˜ therefore ˜ If M ˜ ˜ ∈= ˜ then M ˜ ˜ = G(+; ,; m(B)) ∈ ]a; ˜ b[, and MB˜ ∈= . A∪B A∪B ˜ ˜ = f˜(−1) (f(G(+; m( ˜ A˜ ∪ B) ,; m(B)))) = G(+; ,; m(B)): On the other hand, ˜ ∗˜ m( ˜ = G( ; ; m(A)) ∗˜ G(+; ,; m(B)) m( ˜ A) ˜ B) = max{G( ; ; m(A)); G(+; ,; m(B))} = MA∪ ˜ B˜ = G(+; ,; m(B)); by using again Lemma 1(iii). ˜ and M ˜ ∈ ˜ is completely analogous. The subcase M A˜ ∈= B ˜ ˜ ˜ ˜ (iii) M A˜ ∈= and MB˜ ∈= . Then M A˜ ∈ ]a˜i ; b˜ i [; MB˜ ∈ ]a˜j ; b˜j [ and M A∪ ˜ B˜ ∈ ]a˜i ; bi [ or M A∪ ˜ B˜ ∈ ]a˜j ; bj [. If i = j then ˜ ˜ ˜ = f˜(−1) (f(G( ; m( ˜ A˜ ∪ B) ; m(A))) + f(G(+; ,; m(B)))) ˜ ∗˜ m( ˜ = G( ; ; m(A)) ∗˜ G(+; ,; m(B)) = m( ˜ A) ˜ B) (see Theorem 2, with fi denoted by f). If i = j and M A∪ ˜ B˜ = M A˜ then ˜ ˜ = f˜(−1) (f(G( ; ; m(A)))) = MA˜ m( ˜ A˜ ∪ B) and by Lemma 1(v), we get ˜ ∗˜ m( ˜ = MA˜ ∗˜ MB˜ = max(MA˜; MB˜ ) = MA∪ ˜ m( ˜ A) ˜ B) ˜ A˜ ∪ B): ˜ B˜ = MA˜ = m( The proof is analogous if i = j and M A∪ ˜ B˜ = MB˜ . n ˜ i = j, then it is easy to prove that ˜ Now, if A = i=1 x; i ; i A i ; i ∈ Id ; x; i ; i A i ; i ∩ x; j ; j A j ; j = ∅; n n ˜ = m˜ m( ˜ A) x; i ; i A i ; i = ∗˜ m(x; ˜
i ; i A i ;i ); i=1
i=1
which immediately implies ˜ ∗˜ m( ˜ = m( ˜ m( ˜ A˜ ∪ B) ˜ A) ˜ B);
˜ ˜ B˜ ∈ Id ; A˜ ∩ B˜ = ∅: ∀A;
Finally, we observe that it is suKcient to prove that m˜ is monotone for two intuitionistic fuzzy sets A˜1 = x; ; A and A˜2 = x; +; ,A ; A˜1 ; A˜2 ∈ Id . If A˜1 ⊆ A˜2 then 6+ and ¿, and m( ˜ A˜1 ) = G( ; ; m(A)6G(+; ,; m(A)) = m( ˜ A˜2 ); which completes the proof.
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The second main result is Theorem 4. In the conditions of Theorem 3; the function m˜ : A˜→ [0; 1] de8ned by ˜ = sup sup{m( ˜ E˜ ⊆ A} ˜ ˜ E˜ ∈ Id is an -approximation of A; m( ˜ A) ˜ E): ¿0
is a ∗˜ -decomposable fuzzy measure on the class of all A-measurable intuitionistic fuzzy sets. Proof. First, note that from Remark 4 in Section 3, m˜ is well de:ned. Obviously, m˜ is an intuitionistic fuzzy ˜ = 0 and m( measure, because ∅˜ and X˜ are discrete intuitionistic fuzzy sets with m( ˜ ∅) ˜ X˜ ) = 1 (see the proof of Theorem 3) and m˜ is monotone (which easily follows from de:nition and from the fact proved in Theorem 3 that the restriction of m˜ to Id is monotone). ˜ ¿0, and P˜ ∈ Id , an The only problem is to show that m˜ is ∗˜ -decomposable. Let A;˜ B˜ ∈ A,˜ A˜∩ B˜ = ∅, ˜ ˜ ˜ ˜ ˜ ˜ ˜ -approximation of A ∪ B, P ⊆ A ∪ B, A = {x; A˜(x); A˜(x): x ∈X }, B = {x; B˜ (x); B˜ (x): x ∈ X } and P˜ = {x; P˜(x); P˜(x): x ∈ X }. We de:ne P˜A ; P˜B ∈ Id by P˜ (x) if A˜(x)¿0; P˜A (x) = 0 otherwise; P˜ (x) if A˜(x)¡1; P˜A (x) = 1 otherwise; P˜ (x) if B˜ (x)¿0; P˜B (x) = 0 otherwise; P˜ (x) if B˜ (x)¡1; P˜B (x) = 1 otherwise: ˜ P˜A ⊆ A,˜ P˜B ⊆ B. ˜ Also, if P˜ is an -approximation of A˜∪ B˜ then It is easy to see that P˜ = P˜A ∪ P˜B , P˜A ∩ P˜B = ∅, ˜ ˜ ˜ ˜ because PA is an -approximation of A and PB is an -approximation of B, A˜(x) − P˜A (x) = 0 if A˜(x) = 0 and A˜(x) − P˜A (x) = A˜(x) − P˜ (x) = A∪ ˜ B˜ (x) − P˜ (x) if A˜(x)¿0; P˜A (x) − A˜(x) = P˜ (x) − A˜(x) = P˜ (x) − A∪ ˜ B˜ (x)
if A˜(x)¡1;
P˜A (x) − A˜(x) = 0 if A˜(x) = 1: The proof in the case of P˜B and B˜ is similar. Therefore by Theorem 3 we get ˜ m( ˜ P˜ A ) ∗˜ m( ˜ P˜ B ) = m( ˜ P):
(1)
˜ we easily obtain Let ¿0 be :xed. Since P˜ generates P˜A and P˜B (as above), passing to supremum over all P, ˜ P˜ ∈ Id ; P˜ is an -approximation of A˜ ∪ B; ˜ P˜ ⊆ A˜ ∪ B} ˜ sup{m( ˜ P): ˜ P˜ A ⊆ A} ˜ 6 sup{m( ˜ P˜ A ): P˜ A ∈ Id ; P˜ A is an -approximation of A; ˜ P˜ B ⊆ B}: ˜ ∗˜ sup{m( ˜ P˜ B ): P˜ B ∈ Id ; P˜ B is an -approximation of B;
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Passing now to supremum with ¿0 and taking into account the continuity of ∗˜ , we get ˜ ∗˜ m( ˜ m( ˜ m( ˜ A˜ ∪ B)6 ˜ A) ˜ B):
(2)
˜ 6 m( ˜ :rstly :xing P˜B and passing to supremum with P˜A , then passing to Conversely, by (1), by m( ˜ P) ˜ A˜∪ B), supremum with P˜B and :nally, passing to supremum with ¿0 (here we take account that ∗˜ is continuous), we obtain ˜ ∗˜ m( ˜ m( ˜ m( ˜ A) ˜ B)6 ˜ A˜ ∪ B); which together with (2) proves the theorem. Remark 5. It is evident that Theorem 4 is a generalization of Proposition 3:1 in [2]. But looking to De:nition 3:2 in [2], we have discovered some inadvertences, which fortunately do not ˜ the so-called standard approximation of the usual inLuence the proofs in the paper. Thus, it follows that P, ˜ in the sense of the inclusion in fuzzy set theory. But in this case, in De:nition 3:3 fuzzy set A,˜ satis:es A˜⊆ P, we would obtain ˜ = sup{m( ˜ A˜ ⊆ P}; ˜ P˜ ∈ P(A); ˜ m( ˜ A) ˜ P): which obviously is not correct. As a consequence, we think that in De:nition 3:2 we have to replace i by i−1 in the formula of P˜ and of course to consider 0 as the :rst point of the division of [0; 1]. In the sequel, we study the ∨-decomposable measures (see De:nitions 4 and 5 with ∗ = ∨), where ∨ is the triangular conorm de:ned by x ∨ y = max(x; y), ∀x; y ∈ [0; 1]. Theorem 5. Let (X; A; m) be a fuzzy measure space. If m is a ∨-decomposable fuzzy measure then the intuitionistic fuzzy measure m˜ constructed by Theorem 4 is a ∨-decomposable intuitionistic fuzzy measure. ˜ = [0; 1] and m˜ : Id → [0; 1] is deProof. With the notations in Theorem 3, ∗ = ∗˜ = ∨, D = D˜ = ∅, = :ned by ˜ = max{G( i ; i ; m(A i ; i )): i ∈ {1; : : : ; n}} m( ˜ A) n if A˜= i=1 x; i ; i A i ; i and Theorem 4 assures that m˜ : A˜→ [0; 1] is a ∨-decomposable intuitionistic fuzzy measure. ˜ (as in Theorem 4) is reduced to If m is a ∨-decomposable fuzzy measure then the computation of m˜ (A) the maximization of a real-valued positive function. Theorem 6. Let (X; A; m) be a fuzzy measure space. If m is a ∨-decomposable fuzzy measure then ˜ = sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61}; m( ˜ A) where A ; is the crisp set de8ned by A ; = {x ∈ X : A˜(x)¿ and A˜(x)6}: Proof. Let ; ∈ [0; 1]; + 6 1. Because x; ; A ; ⊆ A,˜ we have ˜ m(x; m( ˜ A)¿ ˜
; A ; ) = G( ; ; m(A ; )):
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Therefore, ˜ m( ˜ A)¿ sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61}:
n In order to show the converse inequality, let ¿0 and E˜ = i=1 x; i ; i E i ;i with x; i ; i E i ;i ∩ ˜ if i = j, be an -approximation of A;˜ E˜ ⊆ A˜ (see Remark 4). Because m˜ is a ∨-decomx; j ; j E j ;j = ∅, posable intuitionistic fuzzy measure, we have ˜ = max m( ˜ E)
i∈{1;:::; n}
m(x; ˜
i ; i E i ; i ) = max
i∈{1;:::; n}
G( i ; i ; m(E i ; i )):
Also, E i ; i ⊆ A i ; i , ∀i ∈ {1; : : : ; n}, therefore m(E i ; i ) 6 m(A i ; i ), ∀i ∈ {1; : : : ; n}. We obtain ˜ m( ˜ E)6 sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61}: The inequality ˜ m( ˜ A)6 sup{G( ; ; m(A ; )): ; ∈ [0; 1]; + 61} follows when E˜ varies among all -approximations of A;˜ E˜ ⊆ A,˜ with ¿0. Example 2. Let X = ∅ and x0 ∈ X . The function mx0 : P(X ) → [0; 1] de:ned by 0 if x0 ∈= A; mx0 (A) = 1 if x0 ∈ A is a ∨-decomposable fuzzy measure. By using Theorem 6 we obtain a ∨-decomposable intuitionistic fuzzy measure m˜x0 : IFS(X ) → [0; 1] corresponding to intuitionistic structure function in Example 1. This is de:ned by ˜ = sup{ n (1 − n )mx0 (A ; ): ; ∈ [0; 1]; + 61} m˜ x0 (A) = ˜nA˜(x0 )(1 − ˜nA˜(x0 )); because
n (1 − n )mx0 (A ; ) =
n (1 − n ) if ˜A˜(x0 )¿ and ˜A˜(x)6 0 otherwise:
Example 3. Let X = {x1 ; : : : ; xp } and f : X → [0; 1] be such that maxx∈X f(x) = 1. The function m : P(X ) → [0; 1] de:ned by max f(x) if A = ∅; x∈A m(A) = 0 if A = ∅ is a ∨-decomposable fuzzy measure. By using Theorem 6, with the help of the intuitionistic structure function in Example 1, n = 1, we obtain a ∨-decomposable intuitionistic fuzzy measure m˜ : IFS(X ) → [0; 1], de:ned by ˜ = sup{ (1 − )m(A ; ): ; ∈ [0; 1]; + 61} m( ˜ A)
f(xk ): ; ∈ [0; 1]; + 61 : = sup (1 − ) max k: A˜(xk )¿ ; A˜(xk )6
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In the particular case p = 2 we consider A˜ = {x1 ; A˜(x1 ); A˜(x1 ); x2 ; A˜(x2 ); A˜(x2 )} ∈ IFS(X ): Because
A ;
{x1 ; x2 } {x } 1 = {x2 } ∅
if A˜(x1 )¿ ; A˜(x2 )¿ ; A˜(x1 )6; A˜(x2 )6; if A˜(x1 )¿ ; A˜(x1 )6 and (A˜(x2 )¡ or A˜(x2 )¿); if A˜(x2 )¿ ; A˜(x2 )6 and (A˜(x1 )¡ or A˜(x1 )¿); otherwise;
we obtain ˜ = sup{ (1 − )m(A ; ): ; ∈ [0; 1]; + 61} m( ˜ A) = max{min{A˜(x1 ); A˜(x2 )}(1 − max{A˜(x1 ); A˜(x2 )})m({x1 ; x2 }); A˜(x1 )(1 − A˜(x1 ))m({x1 }); A˜(x2 )(1 − A˜(x2 ))m({x2 })} = max{A˜(x1 )(1 − A˜(x1 ))f(x1 ); A˜(x2 )(1 − A˜(x2 ))f(x2 )}: 5. Compositive intuitionistic information measure In this section :rst we introduce the concept of intuitionistic compositive information measure. Then we study its connection with the concept of decomposable intuitionistic fuzzy measure. The following axiomatic de:nition of information measure is given in [3]. Denition 9. Let (T; 4) be a lattice with minimum m and maximum M. An information measure over M + , where R M + = R+ ∪ {+∞}, with the properties (T; 4) is a map J : T → R (i) J (M) = 0; (ii) J (m) = + ∞; (iii) x4y implies J (x)¿J (y); ∀(x; y) ∈ T × T. If T is a family of intuitionistic fuzzy sets then we obtain the de:nition of intuitionistic information measure. We will denote by J˜ such an information measure. Remark 6. If A˜ is an intuitionistic -algebra and ⊆ is the inclusion between intuitionistic fuzzy sets, then (A;˜ ⊆ ) is a lattice with minimum ∅˜ and maximum X˜ . Inspired by [3], we can give the following: ˜ Denition 10. We say that an intuitionistic information measure J˜ is F-compositive if there exists a function M M M ˜ F : R+ × R+ → R+ such that ˜ J˜ (B)); ˜ = F( ˜ J˜ (A); ˜ A˜ ∩ B˜ = ∅˜ implies J˜ (A˜ ∪ B) where ∩ and ∪ represent the usual operations between the intuitionistic fuzzy sets. Remark 7. It is easy to see that the function F˜ satis:es the properties ˜ y) = F(y; ˜ x), (i) F(x;
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˜ F(x; ˜ y); z) = F(x; ˜ F(y; ˜ z)), (ii) F( ˜ (iii) F(x; ∞) = x, ˜ y)6F(x ˜ ; y), (iv) x6x implies F(x; for every x; x ; y in the range of J˜. Let M(X ) be the class of all decomposable intuitionistic fuzzy measures (on a set X ) and C(X ) be the class of all compositive intuitionistic information measures (on X ). Theorem 7. The function V : M(X ) → C(X ) de8ned by V (m) ˜ = J˜; where 1 − 1 if m( ˜ ˜ A)¿0; ˜ ˜ ˜ ˜ A) J (A) = m( ˜ =0 +∞ if m( ˜ A) is a bijection. Proof. We assume that m˜ is a ∗˜ -decomposable intuitionistic fuzzy measure and J˜ = V (m). ˜ ˜ = + ∞; J˜(X˜ ) = 0 and A˜ ⊆ B˜ implies J˜(A)¿ ˜ J˜(B), ˜ therefore J˜ is an intuitionistic It is easy to see that J˜(∅) ˜ information measure. We show that J˜ is F-compositive, where 1 − 1 if x; y ∈ [0; +∞); 1 1 ∗˜ x+1 y+1 ˜ y) = F(x; if x = +∞; y x if y = +∞: ˜ Let A;˜ B˜ be such that A˜ ∩ B˜ = ∅. ˜ ˜ ¡+∞ and J˜(B)¡+∞. ˜ ˜ ˜ If m( ˜ A)¿0 and m( ˜ B)¿0 then m( ˜ A˜ ∪ B)¿0; J˜(A) Therefore, ˜ J˜ (B)) ˜ J˜ (A); ˜ = F(
1 1 1 ˜ −1= − 1 = J˜ (A˜ ∪ B): −1= 1 1 ˜ ∗˜ m( ˜ ∪ B) ˜ ˜ m( ˜ A) ˜ B) m( ˜ A ∗˜ ˜ + 1 J˜ (B) ˜ +1 J˜ (A)
˜ ˜ ˜ = 0 then m( ˜ ˜ = +∞, If m( ˜ A)¿0 and m( ˜ B) ˜ A˜ ∪ B)¿0; J˜(A)¡ + ∞ and J˜(B) ˜ J˜ (B)) ˜ = ˜ J˜ (A); ˜ = J˜ (A) F( =
1 1 −1= −1 ˜ ˜ ∗˜ m( ˜ m( ˜ A) m( ˜ A) ˜ B)
1 ˜ − 1 = J˜ (A˜ ∪ B): ˜ m( ˜ A˜ ∪ B)
˜ = 0 and m( ˜ ∗˜ m˜ (B) ˜ = J˜(B) ˜ J˜(B)) ˜ = 0 then m( ˜ = m( ˜ = 0, J˜(A) ˜ = + ∞, therefore F( ˜ J˜(A); ˜ = If m( ˜ A) ˜ B) ˜ A˜ ∪ B) ˜ A) ˜ + ∞ = J˜(A˜ ∪ B). We observe that because of the properties of ∗˜ , the function F˜ has the properties in Remark 7. ˜ The function V is obviously injective. Also, it is easy to prove that if J˜ is a F-compositive intuitionistic information measure, then the function m˜ de:ned by 1 ˜ is :nite; if J˜ (A) ˜ ˜ ˜ m( ˜ A) = J (A) + 1 ˜ = +∞ 0 if J˜ (A)
A.I. Ban, S.G. Gal / Fuzzy Sets and Systems 123 (2001) 103–117
is a ∗˜ -decomposable intuitionistic fuzzy 1 1−y 1 − x F˜ ; +1 x y x ∗˜ y = y x
117
measure, where if x; y ∈ (0; 1]; if x = 0; if y = 0
and in addition V (m) ˜ = J˜. Remark 8. Let m˜ : A˜→ [0; 1] M + given Theorem 7, J˜ : A˜→ R 1 − 1 if ˜ ˜ ˜ ˜ A) J (A) = m( +∞ if
be the ∗˜ -decomposable intuitionistic fuzzy measure in Theorem 4. Then by by ˜ m( ˜ A)¿0; ˜ =0 m( ˜ A)
˜ is a F-compositive intuitionistic information measure, where F˜ is that one in the proof of Theorem 7. 6. Future research and possible applications The results in the paper might be used in several directions. Thus, the approximation result in Section 3 could be applied to approximation of continuous intuitionistic fuzzy logics by discrete intuitionistic fuzzy logics. On the other hand, sometimes the usage of intuitionistic fuzzy sets can be :ner than that obtained with standard fuzzy sets. Thus, the usage of decomposable measures on classical sets and on fuzzy sets, in decision making theory is well known so using the decomposable intuitionistic fuzzy measures might obtain better results. Acknowledgements The authors would like to express their thanks to the referees and editors for their useful comments. References [1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96. [2] C. Bertoluzza, D. Cariolaro, On the measure of a fuzzy set based on continuous t-conorms, Fuzzy Sets and Systems 88 (1997) 355–362. [3] C. Bertoluzza, T. Brezmes, G. Naval, in: B. Bouchon-Meunier, R.R. Yager, L.A. Zadeh (Eds.), Compositive Information Measure of a Fuzzy Set, World Scienti:c, Singapore, 1995, pp. 353–359. [4] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [5] D. Dubois, H. Prade, A class of fuzzy measures based on triangular norms, Internat. J. Gen. Systems 8 (1982) 43–61. [6] S.G. Gal, Sur les thQeorRemes d’approximation de Weierstrass, Mathematica (Cluj) 23 (46) (1) (1981) 25–30. [7] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems 69 (1995) 279–298. [8] E.P. Klement, R. Mesiar, Triangular norms, Tatra Mountains Math. Publ. 13 (1997) 169–193. [9] E. Pap, Decomposable measures and nonlinear equations, Fuzzy Sets and Systems 92 (1997) 205–221. [10] A. Precupanu, Mathematical Analysis, Real Functions, Editura DidacticTa sUi PedagogicTa, Bucharest, 1976 (in Romanian).