On some properties of distance measures

Fuzzy Sets and Systems 117 (2001) 355–361

www.elsevier.com/locate/fss

On some properties of distance measures Jiu-Lun Fan a ; ∗ , Yuan-Liang Ma a , Wei-Xin Xie b a

College of Marine Engineering, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China b Shenzhen University, Shenzhen 518060, People’s Republic of China Received May 1997; received in revised form August 1998

Abstract Based on the axiom deÿnition of distance measure, some new formulas of fuzzy entropy induced by distance measure and some new properties of distance measure are given. A characterization of a -distance measure is a metric deÿned on all c 2001 Elsevier Science B.V. All rights reserved. fuzzy sets is obtained. Keywords: Fuzzy entropy; Distance measure; -distance measure; Metric

1. Introduction Distance measure is a term that describes the difference between fuzzy sets. Liu [5] gave the axiom deÿnition of distance measure and discussed the relationships between distance measure and fuzzy entropy, similarity measure. Distance measure can be considered as a dual concept of similarity measure. Many researchers, such as Yager [10], Kosko [8] and Kaufmann [7] had used distance measure to deÿne fuzzy entropy. Liu [5] extended Yager’s formula to give a general relationship between distance (or similarity measure) and fuzzy entropy, obtained some important conclusions. Using the concept of distance measure, as deÿned by Liu, we have studied some properties of -distance measure and obtained a result about fuzzy entropy induced by -distance measure [9]. This result is a generalization of Kosko’s formula [8]. In this paper, some new properties of ∗ Correspondence address: Department of Computer Science, Xi’an Institute of Posts and Telecommunications, Xi’an, 710061, People’s Republic of China.

distance measure are given, and some new formulas of fuzzy entropy induced by distance measure are shown, among which Kaufmann’s formula is extended to some extent. In the study of distance measure, one interesting problem is the relation between distance measure and metric deÿned on all fuzzy sets. We shed some light on this problem in this paper, to obtain a characterization of a -distance measure as a metric deÿned on all fuzzy sets. 2. Fuzzy entropy and distance measure Throughout this paper, R+ = [0; +∞); F(X ) expresses the set of all fuzzy sets on universal set X , P(X ) expresses the set of all crisp sets on universal set X . A (x) is the membership function of A ∈ F(X ); [a] is the fuzzy set of X for which  [a] (x) = a, ∀x ∈ X (a ∈ [0; 1]). For fuzzy set A, we use Ac to express the complement of A, i.e., Ac (x) = 1 − A (x); ∀x ∈ X . For two fuzzy sets A and B, A ∪ B the union of A and B is deÿned as A∪B (x) =

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 ( 9 8 ) 0 0 3 8 7 - X

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J.-L. Fan et al. / Fuzzy Sets and Systems 117 (2001) 355–361

max(A (x); B (x)); A ∩ B the intersection of A and B is deÿned as A∩B (x) = min(A (x); B (x)). The fuzzy set A∗ is called the sharpness of A, if A∗ (x)¿A (x) when A (x)¿ 12 and A∗ (x)6A (x) when A (x)6 12 . For fuzzy set A, the crisp sets Anear , Afar ∈ P(X ) are deÿned as ( 1; A (x)¿ 12 ; Anear (x) = 0; A (x)¡ 12 ; ( 0; A (x)¿ 12 ; Afar (x) = 1; A (x)¡ 12 :

of A; A ∩ D and A ∪ D (D; Dc ∈ P(X )), Xuechang [5] introduced the concept of -entropy. Similar method is also used for distance measure. Deÿnition 2.3 (Liu Xuechang [5]). A fuzzy entropy e is called a -entropy on F(X ), if e satisÿes e(A) = e(A ∩ D) + e(A ∩ Dc );

∀D ∈ P(X ):

(2.1)

Deÿnition 2.4 (Liu Xuechang [5]). A distance measure d is called a -distance measure on F(X ), if for any A; B ∈ F(X ) and D ∈ P(X ),

To make the statements more clear, the following deÿnitions will be based on F(X ), and not on a subclass of F(X ) as done by Xuechang [5].

d(A; B) = d(A ∩ D; B ∩ D) + d(A ∩ Dc ; B ∩ Dc ) (2.2)

Deÿnition 2.1 (Liu Xuechang [5]). A real function e : F(X ) → [0; 1] is called an entropy on F(X ), if e has the following properties: (EP1) e(D) = 0; ∀D ∈ P(X ); (EP2) e([ 12 ]) = maxA∈F(X ) e(A); (EP3) e(A∗ )6e(A); (EP4) e(A) = e(Ac ); ∀A ∈ F(X ).

Theorem 2.1. If d is a -distance measure on F(X ); then ehdi is a -entropy on F(X ).

Deÿnition 2.2 (Liu Xuechang [5]). A real function d : F(X ) × F(X ) → [0; 1] is called a distance measure, if d has the following properties: (DP1) d(A; B) = d(B; A); ∀A; B ∈ F(X ); (DP2) d(A; A) = 0; ∀A ∈ F(X ); (DP3) d(D; Dc ) = maxA; B∈F(X ) d(A; B); ∀D ∈P(X ); (DP4) if A ⊂ B ⊂ C, then d(A; C)¿d(A; B) and d(A; C)¿d(B; C). If we normalize e and d, we can make 06e(A)61 for A ∈ F(X ), 06d(A; B)61 for A; B ∈ F(X ). In this paper, we only consider the entropy and distance measure that are normalizations. Proposition 2.1 (Liu Xuechang [5]). If d is a distance measure on F(X ); deÿne c

e(A) = 1 − d(A; A ); ∀A ∈ F(X ): Then e is an entropy on F(X ). We call e mentioned in Proposition 2.1 the entropy generated by distance measure d and denote it by ehdi. To describe the relation between the fuzziness

holds.

Proof. The theorem can be proved by Theorems 8.1 and 7.2 in [5]. In [5], it was proved that e is a -entropy i e(A) + e(B) = e(A ∩ B) + e(A ∪ B) for any A; B ∈ F(X ). Next we give another expression of -entropy. Theorem 2.2. Let e be an entropy on F(X ); then e is a -entropy i for any A ∈ F(X ) and D ∈ P(X ); e(A) = e(A ∪ D) + e(A ∪ Dc );

∀D ∈ P(X )

(2.3)

holds. Proof. We only prove that (2:1) implies (2:3). The proof of the conclusion of (2:3) implies that (2:1) is similar. From (2:1), we have e(A ∪ D) = e((A ∪ D) ∩ D) + e((A ∪ D) ∩ Dc ) = e(D) + e(A ∩ Dc ) = e(A ∩ Dc ): Similarly, e(A ∪ Dc ) = e(A ∩ D): Thus e(A ∪ D) + e(A ∪ Dc ) = e(A ∩ D) + e(A∩Dc ) = e(A) and the conclusion follows.

J.-L. Fan et al. / Fuzzy Sets and Systems 117 (2001) 355–361

Theorem 2.3. If e is an entropy on F(X ); then e0 = e=(2 − e) is also an entropy on F(X ). Proof. It is clear that 06e0 (A)61 for any A ∈ F(X ) and e0 satisÿes (EP1), (EP2) and (EP4). For (EP3), from e(A∗ )6e(A), we have 2−e(A)62−e(A∗ ). Thus e0 (A∗ ) =

e(A∗ ) e(A) 6 = e0 (A): 2 − e(A∗ ) 2 − e(A)

3. Fuzzy entropy induced by distance measure One commonly used -distance measure between fuzzy sets is the Hamming distance dH [5]. Let X = {x1 ; x2 ; : : : ; xn } be a ÿnite set, then dH is deÿned as n 1X |A (xi ) − B (xi )|: dH (A; B) = n i=1

Kaufman [7] deÿned an entropy generated by distance measure which is dierent from ehdi: e Ka (A) =

n 2 X

n1=p

[|A (xi ) − Anear (xi )|p ]1=p ;

i=1

p ∈ [1; +∞)

Pn For p = 1; e Ka (A) = 2=n i=1 |A (xi ) − Anear (xi )| = 2dH (A; Anear ). Using dH , Kosko [8] also deÿned an entropy which is dierent from the entropy ehdi : e K (A) = dH (A; Anear )=dH (A; Afar ). Using Kosko’s idea, we have given a general conclusion about the relation between entropy and distance measure in [9] (see Theorem 3.1). In this section, we will give another relation between entropy and distance measure which extends Kaufmann’s formula to some extent. First we state some properties about -distance measure; the proofs are contained in [9]. Proposition 3.1 (Fan and Xie [9]). Let d be a -distance measure on F(X ); then (1) d(A; Anear )¿d(A∗ ; Anear ); (2) d(A; Afar )6d(A∗ ; Afar ). For crisp set D ⊂ X , use 12 D to express that the fuzzy set satisÿes ( 1 ; x ∈ D;  (1=2)D (x) = 2 0; x ∈= D:

357

Proposition 3.2 (Fan and Xie [9]). Let d be a -distance measure on F(X ); if d satisÿes d( 12 D; [0]) = d( 12 D; D) for any crisp set D, then d has the following properties: (1) d(A; Anear )6d(A; Afar ); (2) d(A; Anear ) = d(A; Acfar ); d(A; Afar ) = d(A; Acnear ). Theorem 3.1 (Fan and Xie [9]). Let d be a distance measure on F(X ); if d satisÿes the following properties, (1) d( 12 D; [0]) = d( 12 D; D); ∀D ∈ P(X ); (2) d(Ac ; Bc ) = d(A; B); ∀A; B ∈ F(X ), then eK (A) = d(A; Anear )=d(A; Afar ) is a fuzzy entropy. Theorem 3.2. Let d be a -distance measure on F(X ); if d satisÿes the following properties, (1) d( 12 D; [0]) = d( 12 D; D); ∀D ∈ P(X ); (2) d(Ac ; Bc ) = d(A; B); ∀A; B ∈ F(X ); then eF (A) = d(A; Anear ) + 1 − d(A; Afar ) is a fuzzy entropy. Proof. From Proposition 3.2, we have 06eF (A)61. (i) eF satisÿes (EP1): ∀D ⊂ P(X ), Dnear = D; Dfar = Dc , so eF (D) = 0. (ii) eF satisÿes (EP2): because [ 12 ]near = [1]; [ 12 ]far = [0], thus eF ([ 12 ]) = d([ 12 ]; [1]) + 1 − d([ 12 ]; [0]) = 1: (iii) eF satisÿes (EP3): Let A∗ is the sharpened version of A; then A∗near = Anear , A∗far = Afar . From Proposition 3.1, we have eF (A∗ ) = d(A∗ ; A∗near ) + 1 − d(A∗ ; A∗far ) = d(A∗ ; Anear ) + 1 − d(A∗ ; Afar ) 6 d(A; Anear ) + 1 − d(A; Afar ) = eF (A): (iv) eF satisÿes (EP4): From d(A; B) = d(Ac ; Bc ), obtain d(A; Anear ) = d(Ac ; (Anear )c ) = d(Ac ; Afar ); d(A; Afar ) = d(Ac ; (Afar )c ) = d(Ac ; Anear ): From Proposition 3.2, we have eF (A) = d(A; Anear ) + 1 − d(A; Afar ) = d(Ac ; Afar ) + 1 − d(Ac ; Anear ) = d(Ac ; Acnear ) + 1 − d(Ac ; Acfar ) = eF (Ac ):

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Note: If d also satisÿes d(A; Anear ) = 1 − d(A; Afar ), then eF (A) = 2d(A; Anear ) is an entropy. For Hamming distance dH , we have dH (A; Anear ) = 1 − dH (A; Afar ). Therefore, eF (A) = 2dH (A; Anear ) is an entropy. So Theorem 3.2 can be considered as a generalization of Kaufmann’s formula to some extent.

Proposition 3.4. If d is a distance measure and satisÿes d(A; B) = d(Ac ; Bc ), then e1 = e2 . We will now state two induced relations; the proofs are similar to the proof of Theorem 4.1 and are omitted.

Proposition 3.3. For Hamming distance dH and entropy eF (A) = 2dH (A; Anear ), we have eF0 = e K , where eF0 is deÿned in Theorem 2.3.

Theorem 3.4. If d is a distance measure on F(X ) and satisÿes d([ 12 ]; X ) = d([ 12 ]; 0), deÿne

Proof. For any A ∈ F(X ), we have

e3 (A) =

2dH (A; Anear ) eF (A) = 2 − eF (A) 2 − 2dH (A; Anear ) dH (A; Anear ) dH (A; Anear ) = = e K (A): = 1 − dH (A; Anear ) dH (A; Afar )

eF0 (A) =

d(A ∪ Ac ; X ) : d(A ∪ Ac ; 0)

Then e3 is a fuzzy entropy. Theorem 3.5. If d is a distance measure on F(X ) and satisÿes d([ 12 ]; X ) = d([ 12 ]; 0), deÿne d(A ∩ Ac ; 0) : d(A ∩ Ac ; X )

In the following, we will give some formulas of fuzzy entropy induced by distance measure which are dierent from the above.

e4 (A) =

Theorem 3.3. If d is a distance measure on F(X ), deÿne

Proposition 3.5. If d is a distance measure and satisÿes d(A; B) = d(Ac ; Bc ), then e3 = e4 .

e1 (A) =

d(A ∪ Ac ; X ) : d(A ∩ Ac ; X )

4. Some properties of -distance measure

Then e1 is a fuzzy entropy. Proof. e1 obviously satisÿes 06e1 (A)61, (EP1), (EP2) and (EP4). For (EP3), from c

c

0 ⊂ A∗ ∩ A∗ ⊂ A ∩ Ac ⊂ A ∪ Ac ⊂ A∗ ∪ A∗ ⊂ X; ∗

∗c

c

∗

∗c

we have d(A ∪A ; X )6d(A ∪A ; X ); d(A ∩A ; X ) ¿d(A ∩ Ac ; X ). Thus c

e1 (A∗ ) =

d(A∗ ∪ A∗ ; X ) d(A ∪ Ac ; X ) 6 = e1 (A): d(A∗ ∩ A∗c ; X ) d(A ∩ Ac ; X )

Similarly, we have the following theorem. Theorem 3.4. If d is a distance measure on F(X ), deÿne d(A ∩ Ac ; 0) : e2 (A) = d(A ∪ Ac ; 0) Then e2 is a fuzzy entropy.

Then e4 is a fuzzy entropy.

Theorem 4.1. Let d be a distance measure on F(X ); then d is a -distance measure i for any A; B ∈ F(X ) and D ∈ P(X ), d(A; B) = d(A ∪ D; B ∪ D) + d(A ∪ Dc ; B ∪ Dc ) (4.1) holds. Proof. We only prove that (2:2) implies (4:1). The proof of the conclusion of (4:1) implies (2:2) is similar. From (2:2), we have d(A ∪ D; B ∪ D) = d((A ∪ D) ∩ D; (B ∪ D) ∩ D) + d((A ∪ D) ∩ Dc ; (B ∪ D) ∩ Dc ) = d(D; D) + d(A ∩ Dc ; B ∩ Dc ) = d(A ∩ Dc ; B ∩ Dc ): Similarly d(A ∪ Dc ; B ∪ Dc ) = d(A ∩ D; B ∩ D):

J.-L. Fan et al. / Fuzzy Sets and Systems 117 (2001) 355–361

359

= d(A ∩ D; B ∩ D) + d(A ∩ Dc ; B ∩ Dc )

Proof. Let D = {x ∈ X | A (x)6B (x)}; we have proved in Proposition 4.1 that d(A; A ∪ B) = d(B; A ∩ B) = d(A ∩ D; B ∩ D). Similarly, we can prove that d(A; A ∩ B) = d(B; A ∪ B) = d(A ∩ Dc ; B ∩ Dc ). From (2:2), we have

= d(A; B)

d(A; B) = d(A ∩ D; B ∩ D) + d(A ∩ Dc ; B ∩ Dc )

Thus d(A ∪ D; B ∪ D) + d(A ∪ Dc ; B ∪ Dc )

= d(A; A ∪ B) + d(A; A ∩ B)

and the conclusion follows. Proposition 4.1. If d is a -distance measure on F(X ); then we have d(A; A ∪ B) = d(B; A ∩ B);

(4.2)

d(A; A ∩ B) = d(B; A ∪ B):

(4.3)

= d(B; A ∩ B) + d(B; A ∪ B): In the proof of Theorem 8.2 in [5] (step (6)), it was proved that s(A; B) = s(A ∩ B; A ∪ B) for a normal -similarity measure s. Because d = 1 − s we have d(A; B) = d(A ∩ B; A ∪ B). Theorem 4.2. If d is a -distance measure on F(X ); A; B; C ∈ F(X ), then we have

Proof. We only prove (4:2). The proof of the conclusion of (4:3) is similar. Let D = {x ∈ X | A (x)6B (x)}; from (2:2) we have

d(A; B) + d(A; C) = d(A; B ∩ C) + d(A; B ∪ C):

d(A; A ∪ B) = d(A ∩ D; (A ∪ B) ∩ D)

Proof. Let D = {x ∈ X | B (x)6C (x)}; from (2:2) we have

+ d(A ∩ Dc ; (A ∪ B) ∩ Dc )

(4.4)

d(A; B ∩ C) c

c

= d(A ∩ D; B ∩ D) + d(A ∩ D ; A ∩ D )

= d(A ∩ D; B ∩ C ∩ D) + d(A ∩ Dc ; B ∩ C ∩ Dc )

= d(A ∩ D; B ∩ D);

= d(A ∩ D; B ∩ D) + d(A ∩ Dc ; C ∩ Dc );

d(B; A ∩ B) = d(B ∩ D; A ∩ B ∩ D)

d(A; B ∪ C) = d(A ∩ D; (B ∪ C) ∩ D) + d(A ∩ Dc ; (B ∪ C) ∩ Dc )

+ d(B ∩ Dc ; A ∩ B ∩ Dc ) = d(B ∩ D; A ∩ D) + d(B ∩ Dc ; B ∩ Dc ) = d(B ∩ D; A ∩ D): Therefore d(A; A ∪ B) = d(B; A ∩ B). Proposition 4.2. If d is a -distance measure on F(X ), then for A; B ∈ F(X ) we have d(A; B) = d(A ∩ B; A ∪ B) = d(A; A ∩ B) + d(A; A ∪ B) = d(B; A ∩ B) + d(B; A ∪ B):

= d(A ∩ D; C ∩ D) + d(A ∩ Dc ; B ∩ Dc ): Thus, d(A; B ∩ C) + d(A; B ∪ C) = d(A ∩ D; B ∩ D) + d(A ∩ Dc ; C ∩ Dc ) + d(A ∩ D; C ∩ D) + d(A ∩ Dc ; B ∩ Dc ) = d(A; B) + d(A; C): Corollary. If d is a -distance measure on F(X ), then for any A; B; C ∈ F(X ) we have (1) d(A; B ∪ C)6d(A; B) + d(A; C); (2) d(A; B ∩ C)6d(A; B) + d(A; C).

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J.-L. Fan et al. / Fuzzy Sets and Systems 117 (2001) 355–361

5. A relation between distance measure and metric Metric is an important concept in mathematics; we call  a metric on set U , if  satisÿes [4] (1) (x; y)¿0 and (x; y) = 0 i x = y; (2) (x; y) = (y; x) for all x; y ∈ U ; (3) (x; z)6(x; y) + (y; z) for all x; y; z ∈ U . Now we give a condition for which a distance measure is a metric on F(X ). First we give a lemma. LemmaS5.1 (Liu Xuechang [5]). If D1 ; D2 ; : : : ; Dn ∈ n 6= j and d is a P(X ); i=1 Di = X; Di ∩ Dj = 0 for iP n -distance measure, then d(A; B) = i=1 d(A ∩ Di ; Pn B ∩ Di ). Especially d(A; B) = i=1 d(A(xi ); B(xi )) when X = {x1 ; x2 ; : : : ; xn } is a ÿnite set. Theorem 5.1. Let d be a -distance measure on F(X ). If d satisÿes (1) d(A; B) = 0 i A = B; (2) d(A; B) + d(B; C) = d(A; C) for A ⊂ B ⊂ C. Then d is a metric on F(X ). Proof. It is clear that d satisÿes (1) and (2) of the deÿnition of metric. Now we prove d satisÿes (3) of the deÿnition of metric. For fuzzy sets A; B; C ∈ F(X ), from (4:4), we have d(A; B) + d(B; C) = d(B; A ∩ C) + d(B; A ∪ C): Thus we can suppose that A ⊂ C. Let D1 = {x ∈ X | B (x)6A (x)6C (x)}; D2 = {x ∈ X | A (x)¡B (x)6C (x)}; D3 = {x ∈ X | A (x)¡C (x)¡B (x)}: From Lemma 5.1, we have d(A; B) = d(A ∩ D1 ; B ∩ D1 ) + d(A ∩ D2 ; B ∩ D2 ) + d(A ∩ D3 ; B ∩ D3 ); d(B; C) = d(B ∩ D1 ; C ∩ D1 ) + d(B ∩ D2 ; C ∩ D2 ) + d(B ∩ D3 ; C ∩ D3 ); d(A; C) = d(A ∩ D1 ; C ∩ D1 ) + d(A ∩ D2 ; C ∩ D2 ) + d(A ∩ D3 ; C ∩ D3 ): From B ∩ D1 ⊂ A ∩ D1 ⊂ C ∩ D1 , we have d(A ∩ D1 ; C ∩ D1 )

6 d(B ∩ D1 ; C ∩ D1 ) 6 d(B ∩ D1 ; C ∩ D1 ) + d(A ∩ D1 ; B ∩ D1 ): From A ∩ D2 ⊂ B ∩ D2 ⊂ C ∩ D2 , we have d(A ∩ D2 ; C ∩ D2 ) = d(A ∩ D2 ; B ∩ D2 ) + d(B ∩ D2 ; C ∩ D2 ): From A ∩ D3 ⊂ C ∩ D3 ⊂ B ∩ D3 ; we have d(A ∩ D3 ; C ∩ D3 ) 6 d(A ∩ D3 ; B ∩ D3 ) 6 d(A ∩ D3 ; B ∩ D3 ) + d(B ∩ D3 ; C ∩ D3 ): Summing up the above results, we have d(A; C)6 d(A; B) + d(B; C). Example 5.1. Let X = {x1 ; x2 ; : : : ; xn } be a ÿnite set, and ci be a family of constant Pn numbers which satisfy the following condition: i=1 ci = 1. Then the following distance measure dF (A; B) =

n X

ci |A(xi ) − B(xi )|

i=1

is a metric on F(X ). Especially when ci = 1=n, we obtain the Hamming distance. Lemma 5.2. Let  be a metric on [0; +∞); then (; ) = c| − | i  satisÿes the following conditions: If 66 ; then (; ) = (; ) + (; );

(5.1)

(;  + ) = (0; ):

(5.2)

Proof. If (; ) = c| − |, then obviously  satisÿes (5:1) and (5:2). In the following, we shall prove that if  satisÿes (5:1) and (5:2), then (; ) = c| − |. Let f() = (0; ); then f( + ) = (0;  + ) = (0; ) + (;  + ) = (0; ) + (0; ) = f() + f(): By [1, p. 34], we have f() = c.

J.-L. Fan et al. / Fuzzy Sets and Systems 117 (2001) 355–361

Because (; ) = (min(; ); min(; ) + | − |) = (0; | − |); thus (; ) = c| − |.

d(A; B) =

Note 1. If we only consider metric  deÿned on [0; 1] and valued in [0; 1], the above conclusion is also correct.

=

Note 2. The two conditions on Lemma 5.2 have no relation. (1) does not necessarily imply (2): deÿne 1 (; ) = |2 − 2 |, then 1 is a metric on [0; +∞) and satisÿes (; ) = (; ) + (; ) for 66 . From 1 (0; ) = 2 and 1 (;  + ) = |2 − ( + )2 | = ∗ (2 + ), we know that generally 1 (;  + ) 6= 1 (0; ): (2) does not necessarily imply (1): deÿne 2 (; ) = | − |=(1 + | − |), then 2 (;  + ) = 2 (0; ), but 2 (; ) 6= 2 (; ) + 2 (; ) for 66 . For example, 2 (0;

1 4)

+

2 ( 14 ; 12 ) =

1 4

1+

1 4

+

1 4

1+

1 4

= 25 = 6 2 (0; 12 ) =

1 4

=2 ∗ 1 2

1+

1 2

1+

x (y) =



0;

y=x y 6= x

i=1 n X i=1 n X

d(A(xi ); B(xi )) xi (A(xi ); B(xi )) cxi |A(xi ) − B(xi )|;

i=1

where cxi is a suitable constant positive number. We can select cxi to satisfy 06d(A; B)61, i.e. P n i=1 cxi = 1. 6. For further reading The following references are also of interest to the reader: [2,3,6,11]. Acknowledgements

1 4

= 13 :

We use x to express the fuzzy point. i.e.   ;

=

n X

361

(0661):

Theorem 5.2. Let X = {x1 ; x2 ; : : : ; xn } be a ÿnite set, and d be a -distance measure on F(X ); then d = dF i d satisÿes the following conditions: (1) d(x ; x ) = 0 i  = ; (2) d(x ; x ) + d(x ; x ) = d(x ; x ) for 0666

61 and x ∈ X ; (3) d(x ; x ) = d(x0 ; x|−| ), ∀x ∈ X and 0661, 0661. Proof. If d = dF , then d obviously satisÿes conditions (1) – (3) of Theorem 5.2. In the following, we will prove that if d satisÿes conditions (1)– (3) of Theorem 5.2, then d = dF . Deÿne x (; ) = d(x ; x ) ∀x ∈ X ; then x is a metric on [0; 1]. By Lemma 5.2, we know x (; ) = cx | − |. From Lemma 5.1, we know

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