Fuzzy norms, probabilistic norms and fuzzy metrics

Fuzzy Sets and Systems 36 (1990) 137-144 North-Holland

137

FUZZY NORMS, PROBABILISTIC NORMS AND FUZZY METRICS* WU Cong-Xin and MA Ming Department of Mathematics, Harbin Institute of Technology, Harbin, China Received May 1988 Revised November 1988

Abstract: In this paper, we discuss the relations among Katsaras fuzzy norms, H6hle probabilistic norms and Kaleva-Seikkala fuzzy metrics.

Keywords: Fuzzy norms; probabilistic norms; fuzzy metrics.

1. Introduction In 1984, A.K. Katsaras and Wu Congxin and Fang Ginxuan separately introduced the definitions of fuzzy norms (see [3] and [7]). Although they have different forms, they are almost the same in the sense that one determines the other. The relations between the two definitions of fuzzy norms have been discussed in [5]. In this paper, we first give several necessary and sufficient conditions of fuzzy norms in the sense of Katsaras and fuzzy metrics in the sense of Kaleva and Seikkala and discuss the relations between them. Then we consider the relations between fuzzy metrics in the sense of Kaleva and Seikkala and probabilistic norms in the sense of H/Shle. In what follows, fuzzy metric is always in the sense of Kaleva and Seikkala and probabilistic norm is always in the sense of H6hle.

2. Preliminaries Let X be a vector space over a real or complex number field K. Recall that if )., #, /9 are fuzzy set in X, then ~ #(x) = sup min(Z(s), #(t)), X=S+t

kl~(x) = ~(x/k)

whenever k 4: O,

sup ~(x)

O~(x) =

whenever x = O,

otherwise.

* This work has been partly supported by the National Natural Science Foundation of China. 0165-0114/90/$3.50 ~) 1990, Elsevier Science Publishers B.V. (North-Holland)

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In what follows, 0 denotes the zero element of vector space X. For x ~ X, x ~ p is defined by x ~ p ( y ) =/~(y - x). A fuzzy set p in X is called (1) convex if r# + (1 - r)p <~ # for all r ~ [0, 1]: (2) balanced if kp ~

0} = 1. An absolutely convex absorbing fuzzy set p in X is called a fuzzy seminorm (see [3]). If, in addition, inf{p(kx): k > 0} = 0 for x #=0, then p is called a fuzzy norm. Next, the definition of probabilistic norm on X will be recalled (see [1, 3]). Let ~ + ( R +) denote the set of all fuzzy sets of R + = [0, ~) such that (a) a~ is increasing and left continuous; (b) a~(0) = 0 and limk_,~ o~(k) = 1. The element eo of ~ + ( R ) is defined by e0(0)= 0 and eo(k)= 1 if k > 0 . For a~, j6 ~ ~+(R), the set a~ ~ / ~ (x) = sups+t=x min(o~(s),/~(t)). A probabilistic seminorm, on a vector space X, is a mapping P:X-->~+(R) with the following properties: (i) P(O) = e0; (ii) P(kx)(t) = P(x)/(t/llcl), for all k e K, k ~ 0; (iii) p ( x ) ~ P ( y ) ~ P ( x +y) for all x, y e X . If, in addition, P(x)(0 ÷) = 0 for each x 4: 0, then P is called a probabilistic norm. Finally, we recall that a fuzzy number is a fuzzy set on the real axis, i.e., a mapping tr: R ~ [0, 1], and a fuzzy number a~ is called nonnegative if a~(t) = 0 for all t > 0. Denote the set of all upper semicontinuous normal convex nonnegative fuzzy numbers by G. For a nonempty set E, d is a mapping from E x E to G and let mappings L, R:[0, 1] x [0, 1]---~ [0, 1] by symmetric, nondecreasing in both arguments and satisfy L(0, 0) = 0 and R(1, 1) = 1. Denote

[d(x,y)]r=[)~r(X,y),pr(X,y)] forx, y ~ E , 0~i L(d(x, z)(s), d(z, y)(t)) when s ~<).l(x, z), t ~<).l(z, y) and s + t ~<).l(X, y); (2) d(x, y)(s + t) <- R(d(x, z)(s), d(z, y)(t)) when s/>),1(x, z), t I> ~.l(z, y) and s + t >1 ).l(X, y). (See [2].) In what follows X* denotes the set of all fuzzy points in X.

3. Some properties of fuzzy normed spaces Proposition 3.1. Let (X, It) be a fuzzy normed space and denote Ilxll,, ( r ) = inf{t > 0: xr ~ tp} (where ~ means quasi-coincidence, i.e., (tlz)(x) + r > 1). Then

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lI" II~ (') is called Minko wski functional of # and satisfies: (1) Ilxll,(r) -- Ofor some r e (0, 1) if and only if x = O; (2) Ilkxll. (r) = Ikl Ilxll. (r), k e K;

(3) IIx +yll,. (r) ~< Ilxll. (r) + IlYlI~, (r); (4) I1"11.(') is nonincreasing and left continuous

for r.

P r o o f . For any fixed xr ~ X * , sup,>o (t#)(x) = 1 since # is a absorbing fuzzy set. H e n c e there exists a t' > 0 with (t'#)(x) > 1 - r, i.e., x~ ~ t ' # and t h e r e f o r e we have shown that I1"11. (') is meaningful for every xr e x * . In what follows, to show the proposition, it suffices to verify that II'lt. (') satisfies (1)-(4). F o r the p r o o f of ( 2 ) - ( 4 ) , see [6]. (1) If x = 0, then II011. ( r ) = i n f { t > 0 : O ~ t # } . By the preceding proof, we know there exists a t' with 0r ~ t'#, and hence 0r g t# for all t > 0 by tot = Or. T h e r e f o r e I[ 011. (r) = 0 for all r e (0, 1]. Conversely, if we assume the conclusion is false, then there exists an x d: 0 with Ilxll. (r) = 0 for some r e (0, 1). Since # is balanced and 0 = Ilxll. (r) = inft>0 {t: xr d t#}, it follows that (tg)(x) > 1 - r for all t > O, and hence inf,>o (t#)(x) =/=O; this contradicts with # being a fuzzy norm.

Proposition 3.2. Let X be a vector space and II'll ('):X*~

[0, 09 satisfy ( 1 ) - ( 4 ) of Proposition 3.1. Then there exits a unique fuzzy norm # such that I['l[ (') is a Minkowski functional of #. P r o o f . L e t II'll(') by a mapping from X * to [0, oo). D e n o t e # = U { x l - , : [[x[[ (r) < 1}. W e will verify # is a fuzzy n o r m and [1"[[ (') is a Minkowski functional of #. First, we prove an equivalent relation: xr ~ # if and only if [Ix 1[ (r) < 1.

(*)

I f x r d #, then # ( x ) > 1 - r. Notice that # = U {xl-r : [[x[[ (r) < 1}, and there is an r' with [[x[[ (r') < 1 such that 1 - r < 1 - r', i.e., r > r' and [[x[[ (r) < 1. H e n c e [Ix[[ ( r ' ) t > [[x[[ (r) since [1.[[ (.) is nondecreasing for r. Conversely, if [[x[[ ( r ) < 1, then there is a 6 > 0 with [[x[[ ( r ) < 1 - 6. Since H'I[ (') is left continuous for r, therefore for rn increasing to r, limn__,~ [[xll (rn) = Ilxl[ (r). H e n c e there is an r,, with [[x[[ (rm) < 1 - 6 < 1. This implies # ( x ) I> 1 - rm > 1 - r, i.e., xr~#. In o r d e r to show that [[.[[ (.) is a fuzzy n o r m , it suffices to show that # is an absolutely convex absorbing fuzzy set and inft>0 (t#)(x) = 0 for any x q: 0. # is balanced: F o r any 0 < t ~ < 1 , if xr g t # , then (1/t)Xr ~ # , and hence [[(1/t)x[[ (r) < 1. By (*), it follows that [[x][ (r) < t ~< 1, and this implies xr ~ # by (*). H e n c e t# c / z . # is absorbing: For any x e X , [[(1/(llx[[ ( r ) + 1))x[[ ( r ) < l , i.e., by (*) (1/([Ix[[ (r) + 1))xr g #. Hence ([[x[[ (r) + 1)#(x) + r > 1. This implies (supt>0 t#)(x) > 1 - r for all r e (0, 1], and hence supt>0 t#(x) = 1. # is convex: Consider x~ g t# + (1 - t)#, i.e. supy+z=x m i n ( t # ( y ) , (1 - t)#(z)) > l - r . This implies that there exists y ' , z' with y ' + z ' = x , t # ( y ' ) > l - r and (1-t)#(z')> 1-r. With regard to (*), II(1/t')y')[[ ( r ) < 1 and [ [ 1 / ( l + t ) z ' [ [ ( r ) < l . F r o m (3) of the conditions, we infer that [ I x [ [ ( r ) = [[y' + z'[[ (r) ~< J[y'[[ (r) + [[z'[[ (r) < t + (1 - t) = 1, and this implies xr d # by (*). H e n c e t# + (1 - t)# c #.

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Finally, inft>o (tl~)(x) = 0 for any x ~ O. Otherwise, there exists an x ~: 0 with inft>o (ttQ(x) = r > O. Take 0 < 6 < r; then from inft>o (t#)(x) = r we infer Xl_r+ 6 g t~t for any t > 0, i.e., II1/tll (1 - r + 6) < 1. H e n c e Ilxll (1 - r + 6) < t for any t > 0 , and this implies Ilxll (l-r+ 6)=0. By (1) of Proposition 3.1, we conclude the proof. Note 1. The consequences of Propositions 3.1 and 3.2 are also true for fuzzy seminorms only when (1') replaces (1). (1') II011 (r) = 0 for any r • (0, 1]. Note 2. The fuzzy norm defined by Wu Congxin and Fang Ginxuan [7] is a mapping from X* to [0, ~) and satisfies (2)-(4) and (1"): (1") Ilxll (r) = 0 for some r • (0, 1] if and only if x = 0. Example. Let I1"11 be a norm on a vector space X. Then I1"11(') defined by I1"II (r) = I1"II for all r • (0, 1] is a fuzzy norm in the sense of W u Congxin and Fan Ginxuan I1"II ('), called an induced fuzzy norm. For a fuzzy norm on a vector space in the sense of [7] denote # = U{xl-r:llxll ( r ) < l } . Then { x + t l ~ A r * : t > O , r > l - ) . } is a q-neighborhood basis of xx for a unique fuzzy topology T. W e say that the fuzzy topology T is determined by I1"II ('). Proposition 3.3. Let I1"111('), 11"112(') be f u z z y norms in the sense of [7]. Then II'lh ('), II'lh(') determine the same f u z z y topology T if and only if there exist functions a(r), b(r), r • (0, 1], with infr~tr,,11 a(r) =p~, > 0, sup,~l~, 11b(r) = q,, < ~, r • (0, 1], such that a(r) Ilxlh (r) ~< Ilxlh (r) ~< b(r) Ilxlll (r)

for any x, • X*.

Proof. Sufficiency. From preceding theorems of [6], it suffices to show that for any r there exist t', t" such that t'~q f3 r* c t~2 f3 r* c t"~tl f3 r*, where gi = U (Xl-r" Ilxlli (r) < 1} (i = 1, 2). For a fixed x, g t'#l As*, then IIx/t'lll (r) < 1, and this implies Ilxlh (r) < t'. H e n c e Ilxlh (r) < b(r) Ilxlll (r) < t'q~/2. Take t' < 1/q~n; we infer x~ g/z2. This implies t'/A 1 c ~2" The other inclusion is similar. Necessity. Since I1"11~('), 11"112(') determine the same fuzzy topology T, there exist t', t" such that t'~t~ f3 r* c/~2 f3 r* c t"#l f3 r*. D e n o t e n(),) = inf{t' : t'/z~ t3 ),* c / z 2 N ,~*}. Then n().) is a nondecreasing function and (n(),) + 1)/.t I f"l ~,* c /~2 N r*. This implies II(1/(n(1) + 1))xlll (r) < Ilxlh (r). Proposition 3.4. Let (X, I1"II (')) be a f u z z y normed space in the sense of [7]. Then I1"11(') is equivalent to an induced f u z z y norm if and only if I1"11"is equivalent to I1.111, where I1.11~ is defined by Ilxll r = Ilxll (r). Proof. From Proposition 3.3, there exist a(r), b(r) such that a(r) Ilxlll (r) ~< Ilxlh (r) ~< b(r)llxlll (r), since I1"111('), II'lh (') are equivalent. On the other hand, 11"112(') is induced by I1"II, which implies II" I1~ is equivalent to I1"II. So we infer I1"11~is equivalent to I1"I1~-

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Conversely, since I1.11x is equivalent to I1"11", there exists M(r) such that Ilxll 1 ~< Ilxllr <~M(r) IIxlIL Denote g(r) = inf{M: Ilxll~ ~
4. The relations between fuzzy metric and fuzzy norms Proposition 4.1. (X, d, max, max) is a fuzzy metric if and only if there exist mappings p :X* x X* --->R +, )~:X* x X* -->R + satisfying (1))~(Xr, yr)=O for all xr, Yr, EX*; (2) p(x,, Yr) = 0 for all r • (0, 1] if and only if x = y; (3) p(xr, Yr) = D(Yr, Xr)'~

(4) o(xr, Yr) p(Xr, Zr) + O(Zr, Yr); (5) p is nondecreasing and left continuous for r.

Proof. Necessity. From Lemma 3.1 of [2], we obtain ).l(X, y) = 0 for all x, y • X whenever L/> max. Hence [d(x, y)]r = [~,r(X ' y ) , pr(X, y)] ---- [0, pr(X, y)]. Define p(Xr, Yr)= pr(X, y), p(Xr, Yr,)= Pmin(r,,,)(x, y). Then p : X * xX*--->R + satisfies

(2)-(5): (2) If p(Xr, Yr)----0 for all r • (0, 1], then pr(X, y ) = 0 for all r • (0, 1], i.e.,

d(x, y) = 0. This implies x = y. Conversely, we infer [d(x, x)] r = 0 for any x e X, r • (0, 1] from d(x, x) = ~). Hence p(Xr, Xr) = Dr(X, X) = O. (3) From d(x, y) = d(y, x) it follows [d(x, y)]r = [d(y, x)] ~. This implies p(x~, Yr) = p~(x, y) = Pr(Y, x) = P(Yr, Xr)" (4) By Lemma 3.2 of [2], it is trivial. (5) Since d(x, y) is a semi-continuous, normal convex fuzzy number, we infer p(Xr, y~) = pr(X, y) is nondecreasing and left continuous for r. Sufficiency. Denote pr(X, y) = p(Xr, Yr) and [d(x, y)]r = [0, p,(X, Y)I" Then d(x, y) • G and it satisfies (i)-(iii) of a fuzzy metric. In order to show d ( x , y ) • G for any x , y • X , it suffices to show that [0, pr(X, y)] = [d(x, y)]r satisfies: (a) [d(x, y)]r is a closed interval; (b) [d(x, y)]rl = [d(x, y)lr2 whenever 0 < rl ~< rE ~< 1; (C) For any nondecreasing sequence rn converging to r, N~=I [d(x, y)]rn= [d(x, y)]r. From the definition of [d(x, y)]r, (a), (b) are trivial since p~(x, y) is nondecreasing for r. Now we verify (c). For any nondecreasing sequence rn converging to r, [d(x, y)]~n = [0, pr~(X, y)]- This implies I

n=l

[d(x, y)] rn ~ ~-~ [ 0 , [~rn(X, y ) ] n~l

:

[0, lim prn(x, Y)I" L tl-----~~

On the other hand, pr(x,y) is left continuous for r. Therefore [0, limn--,®prn(X, y)] = [0, p,(X, y)]. Finally we verify that d : X x X---> G satisfies (i)-(iii) of a fuzzy metric. (iii) is the consequence of Lemma 3.2 in [2]. For (i), if x = y, then pr(X, y) = 0 for all r • (0, 1]. This implies d(x, y ) = 0 . Conversely, we infer pr(X, y ) = 0 for any r from d(x,y), therefore x = y . For (ii), from [d(x,y)]~=[O, p~(x,y)]= [0, Pr(Y, X)] = d(y, x) r, we infer the conclusion.

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Proposition 4.2. Let (X, Iz) be a f u z z y normed space in the sense of Katsaras. Then f u z z y norm I~ can induce a f u z z y metric d such that (X, d, max, max) is a f u z z y metric space and satisfies: (a) d(x + z, y + z) = d(x, y); (b) d(Ax, Ay) = I;q d(x, y). Proof. Define ~.,(x, y) = 0 and p r ( X , y) = IIx - y II. (r) = inf{t > 0: (x - Y)r d t/t} for any x, y e X. It is easy to verify that Ar, Pr satisfy (1)-(5) of Proposition 4.1. Therefore, a unique fuzzy metric d is determined by [d(x,y)]r= [Zr(X, y), pr(X, y)]. Now we show that d satisfies (a) and (b). (a) For any z e X,

[d(x + z, y + z)] r -~" [~,r(X "~-Z, y + Z), pr(X -t- Z, y + Z)] = [0, IIx +z

- (y + z)ll~ (r)] = [0, IIx - YlI,, (r)]

= [Zr(X, y), pr(X, y)] = [d(x, y)]~. This implies d(x + z, y + z) = d(x, y). The proof of (b) is similar. Proposition 4.3. Let X be a vector space, (X, d, max, max) be a f u z z y metric space and satisfy (a), (b) and (c) If [d(x, y)]r = {0} for some r, then [d(x, y)]r = {0} for all r ~ (0, 1). Then Ilxll (r) -- re(x, O) is a Minkowski functional of a f u z z y norm in the sense of Katsaras.

Proof. From Proposition 4.1, we infer that [d(x,y)]r=[Ar(X,y), p r ( x , y ) ] = [0, pr(x, y)]. Denote itxll (r) -- re(x, 0). According to Proposition 3.2, in order to show the proposition it suffices to show H"If (') satisfies (1)-(4) of Proposition 3.1. (1) If x = 0, then d(x, 0) = 0 and this implies Ilxll (r) = re(x, 0) -- 0 for all r e (0, 1]. Conversely, if IIxll (r) = pr(X, 0) = 0 for some r • (0, 1), then from (c), we infer d(x, 0) = 0. This implies x = 0. (2) From (b) we infer I1~11 (r) = pr(kx, 0) = Ikl pr(x, 0) = Ikl Ilxll (r). (3) According to Lemma 3.2 of [2], Pr(', ") satifies the triangle inequality. This implies (3) by the definition of I1"11(')(4) Since d(x, y) is a semicontinuous normal convex fuzzy number we infer that I1"11(') satisfies (4) from Ilxll (r) = pr(X, 0).

5. The relations between fuzzy metric and probabilistic norm Proposition 5.1. Let (X, P) be a probabilistic normed space. Denote d(x, y)(t) = 1-P(x-y)(t) whenever t>~O, and d ( x , y ) ( t ) = O whenever t
(b) d ( ~ , Xy) = IXl d(x, y); (c) d(x, y)(O +) =O for any x =/=y.

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143

Proof. F r o m the definition of probabilistic n o r m we infer d(x, y ) ~ G for any x, y ~ X. N o w we verify d(x, y ) satisfies ( 1 ) - ( 3 ) of a fuzzy metric. (1) If x = y , then P ( x - y ) = e0, and this implies d(x, y ) = 0. C o n v e r s e l y , if d(x, y ) = 0, then P ( x - y ) ( t ) = 1 - d(x, y ) ( t ) w h e n e v e r t i> 0. Since P is a probabilistic n o r m , P ( x ) ( 0 +) = 0 for any x 4: 0. This implies x = y. (2) W e only consider t i> 0. T h e n d(x, y ) ( t ) = 1 - P ( x - y ) ( t ) = 1 - P ( y - x ) ( t ) = d ( y , x). (3) Part (1) of (iii) is meaningless by L e m m a 3.1 of [2]. N o w we show the second part of (iii). W e have d(x, y ) ( s + t) = 1 - P ( x - y ) ( s + t) = 1 - e ( x - z + z - y ) ( s + t).

F r o m the definition of probabilistic n o r m , P ( x + y ) ( t ) >! P ( x ) ~ P ( y ) ( t ) , and we infer that 1 - e ( x - z + z - y ) ( s + t) <~ 1 - e ( x - z ) ~ e ( z - y ) ( s + t)

=

inf

max(1-P(x-z)(s'),l-P(z-y)(t'))

$'+t'=$+t

<~m a x ( d ( x , z ) ( s ) , d ( z , y)(t)).

In o r d e r to show d(x, y ) satisfies ( a ) - ( c ) it suffices to consider the case of t/> O. (a) follows f r o m d ( x + z, y + z ) ( t ) = 1 - P ( x + z - (y + z ) ) ( t ) = 1 - P ( x - y ) ( t ) = d(x, y)(t).

(b) follows f r o m d(Xx, ~.y)(t) = 1 - P(Xx - )Ly)(t) = 1 - P ( x - y ) ( t ] l Z ] ) = d ( x , y ) ( t / i Z l ) = IZl d(x, y ) ( t ) .

(c) F r o m P ( x ) ( 0 ÷) = 0 for any x ~ 0, we infer d ( x , y ) ( 0 ÷) = 1 - e ( x - y ) ( 0 ÷) = 1 for any x 4: 0.

Proposition 5.2. L e t X be a vector space and ( X , d, m a x , max) be a f u z z y metric space and satisfy ( a ) - ( c ) o f Proposition 5.1. Then d ( x , y ) can induce a unique probabilistic norm. Proof. D e n o t e P ( x ) ( t ) = 1 - d(x, O)(t). T h e n P ( x ) ~ ~ + ( R ) for any x e X. In o r d e r to show P ( . ) is a probabilistic n o r m it suffices to verify p ( . ) satisfies (i)-(iii), according to (c). (i) If x = 0, then d ( x , 0) = 0, and this implies P ( x ) = eo. (ii) This follows f r o m P ( A x ) ( t ) = 1 - d(Ax, O)(t) = 1 - IZl d(x, O)(t) = 1 - d(x, O)(t/IZl) = P ( x ) ( t / I Z l ) .

(iii) W e have P ( x ) ~ P ( y ) ( s + t) =

sup S'+t'=$+t

min(P(x)(s'), P(y)(t')).

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According to (a), for any s' + t' = s + t,

d(x + y, O)(s' + t') >I max{d(x + y, y)(s'), d(y, O)(t')} = max(d(x, O)(s'), d(y, O)(t')}. This implies

1-d(x+y)(s+t)<~

min

(1-d(x,O)(s'),l-d(y,O)(t')}.

s'+t'=s+t

Hence P(x) ~) P(y)(s + t) <~P(x + y)(s + t).

References [1] U. H6hle, Minkowski functionals of L-fuzy sets, in: P.P. Wang and S.K. Chang, Eds., Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems (Plenum Press, New York, 1980) 13-24. [2] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984) 215-229. [3] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984) 143-154. [4] A.K. Katsaras, Linear fuzzy neighborhood spaces, Fuzzy Sets and Systems 16 (1985) 25-40. [5] Ma Ming, A comparison between two definitions of fuzzy normed spaces, J. Harbin Inst. Technology, Suppl. Math. (1985) 47-49. [6] Pu Paoming and Liu Yingming, Fuzzy topology I, J. Math. Anal. Appl., 76 (1980) 571-599. [7] Wu Coagxin and Fang Ginxuan, Fuzzy generalization of Kolmogoroff's theorem, J. Harbin Inst. Technology (1984) (1) 1-7.