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Completeness of the standard fuzzy real line for FNS
Fuzzy Sets and Systems 51 (1992) 95-103 North-Holland
95
Completeness of the standard fuzzy real line for FNS Hassan M. EI-Hamouly and Abdelwahab M. E1-Abyad
Dept. of Mathematics', Military TechnicalCollege, Cairo, Egypt Received February 1990 Revised March 1991
Abstract: We consider a special class of fuzzy metric neighbourhood spaces, where the fuzzy distance is induced by a fuzzy pseudo-norm. This class is contained in Lowen's well known category FNS of Fuzzy Neighbourhood Spaces. We define the fuzzy limit of a sequence, a fuzzy Cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces. We prove that the set of all nonnegative fuzzy real numbers R*(I) under the N-Euclidean metric (the standard fuzzy real line for FNS) is complete, but that the smallest fuzzy neighbourhood real algebra M(1) including R*(1) is not complete. We also characterize the fuzzy Cauchy sequences in M(I) in terms of their corresponding (real) sequences of tr-cuts.
incomplete and R*(1) is complete as a subset of M(1) over the N-Euclidean metric, are developed in Section 4. Let us start with some definitions, in which R* will denote the set of nonnegative real numbers. Definition 1.1 [5, 13]. A nonnegative fuzzy real number ~ is a nonascending, left continuous function from R* into I = [0, 1] with ~ ( 0 ) = 1 and ~(+o0-) = 0. The set of all n0nnegative fuzzy real numbers is denoted by R*(I). The partial order >! on R*(I) is the natural ordering of real functions. R* is canonically embedded in R*(I) in the following fashion; for every r • R * we associate the fuzzy real number f defined by ~(t) = {~
if t • [0, r], if t • (r, o0).
Then 13 is the smallest element in R*(I).
Keywords: Fuzzy metric neighbourhood spaces; fuzzy convergence; completeness.
1. Introduction In many application problems in optimization and approximation theories, the completeness of the underlying space is crucial to guarantee the existence of a solution. For example, the minimum norm problem [8] and the fixed point theorerla [3] need the underlying space to be complete. In this section we give some premiminary definitions that we will need in our study. Definitions of fuzzy Cauchy sequence, fuzzy convergence and fuzzy limit of a sequence in fuzzy metric neighbourhood spaces are given in Section 2. We characterize the fuzzy Cauchy sequences in M(1) in terms of their corresponding (real) sequences of a~-cuts in Section 3. The main results of this research, that M(1) is
Correspondence to: Dr. A. M. E1-Abyad, Dept. of Mathematics, Military Technical College, Cairo, Egypt.
The three main operations on R * ( 1 ) addition, nonnegative scalar multiplication and multiplication - are defined through Zadeh's Extension Principle [18] as follows: Definition 1.2 [13, 14]. Let ~, ~ e R*(1), t • R* Then: (i) Addition: (~ ~ ~)(t) = sup{~(a) ^ ~(b): a + b = t}. (ii) Scalar multiplication by a nonnegative real number r t> 0: {()(
(r~)(t) =
t/r)
if r = 0 , if r > 0 .
(iii) Multiplication: (~¢)(t) = sup{~(a) ^ ¢(b): ab = t}. It is well known [14] that the above operations are well defined on R*(1), and preserve the order on R*(I), and R*(1) is closed under these operations. The algebraic properties of addition
0165-0114/92/$05.(10t~) 1992--Elsevier SciencePublishers B.V. All rights reserved
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H.M. EI-Hamouly, A.M. EI-Abyad / Completenessof the standardfuzzy real line
and scalar multiplication on R*(1) enable us to embed R*(I) in a smallest real vector space M(1) as follows: Definition 1.3 [9]. The set M(1) is the Cartesian product R * ( I ) x R * ( I ) modulo an equivalence relation - defined by (~,¢)-(~,#)
iff
iff
Definition 1.6 [9]. The N-Euclidean norm on M(1) is the fuzzy norm defined by" For (~, ~) ~ M(I), ~(~, ¢)] = inf{Z ¢ R*(I): Z/> (~, ¢) and 3,/> (~, ~)} = inf{A ~ R*(I): 3. ~ ~/> and ~ . ~ >
~#~¢~..
The set M*(I) is defined by
M*(I) = {(~, ¢) e M(I): (~, ¢) ~ (0, 0)} = {(~, ¢) e M(I): ~ I> ¢}. R*(1) is canonically embedded in M*(I) by representing each ~ ~ R*(I) as (~, 0) e g ( l ) . Also, R is embedded in M(1) as follows: If r ~>0 it is identified with (f, 0), and with (0, ( - f ) ) if r < 0. Definition 1.4 [9]. Addition and scalar multiplication are defined on M(I) as follows: For (~, ~), (3., #) ~ M(1) and t e R: (i) Addition: (~, ~) ¢ (Z, ~,) = (~ • ~, ~ ¢ #). (ii) Scalar multiplication:
t(~,
(iii) Ilxll ~:O for all x ~e0 in X.
¢~#=¢~.
The partial order I> on M(I) is defined by (~,~)~>(X,#)
further satisfies:
t >I 0, ~) j"[ (t~, t~), (It[ ~, It[ ~5), t < O.
¢}.
It is proved in [9] that this function [] is indeed a fuzzy norm on M(1), Definition 1.7 [12]. Multiplication on M(1) is defined by: For (~, ~), (~., #) ~ M(I),
(~, ¢)(L ~,) = (~x • ~ , ~, ¢ ~z). Theorem 1.2 [12]. (i) Multiplication on M(I) is
well defined. (ii) The canonical embedding of R*(I) into M (1) preserves multiplication. (iii) Under addition, scalar multiplication and multiplication, M(1) is a real commutative and associative algebra with unit element i = (i, 0). (iv) M(I) is not an integral domain. (v) (M(I), ~ ~) is a fuzzy normed vector space and is a fuzzy normed algebra under multiplication i.e. for every x, y ~ M(I) we have
[xy~<~x]~y]. (vi) M*(I) is closed under multiplication.
[]
Theorem 1.1 [9]. The above addition and scalar
multiplication are well defined on M(1). Under these two operations, M(I) is the smallest real vector space containing R*(I). In particular, the canonical embedding of R *(1) into M (1) preserves addition and scalar multiplication, while the canonical embedding of R into M(1) is a vector space embedding. [] Definition 1.5 [9]. A fuzzy pseudo-norm on a vector space X is a mapping l[ II :X-->R*(1) which satisfies: For x, y e X and r e R: (i) I}rx[} = Ir[ [Ix}l, (ii) IIx + y II ~< IIx II ~ IIY II (triangle inequality). (X, 11II) is called a fuzzy pseudo-normed space, and it is called a fuzzy normed space if it
Definition 1.8 [10]. A fuzzy pseudo-metric on a nonempty set X is a mapping d : X × X-->R*(1) which satisfies the following: For x, y e X: (i) d(x, x) = O, (ii) d(x, y) = d(y, x) (symmetry), (iii) d(x, z) <~d(x, y) ~) d(y, z) (triangle inequality). (X, d) is called fuzzy pseudo-metric space, and it is called a fuzzy metric space if it further satisfies: (iv) d(x, y) ~ 0 for all x :/:y in X. The fuzzy pseudo-metric associated with a fuzzy pseudo-normed space (X, II II) is given by: For x, y e X,
d(x, y ) = IlY -xll.
H.M. EI-Hamouly,A.M. EI-Abyad / Completenessof the standardfuzzy realline
97
In particular, the fuzzy metric on M(1), induced in this manner from the N-Euclidean norm on M(I), is called the N-Euclidean metric. Same terminology applies to the restriction of this fuzzy metric to R*(1)~ M(I).
The ring M(I) is clearly a fuzzy neighbourhood ring, as defined in [1].
Definition 1.9 [5]. For every r • R, the fuzzy subset Lr of R*(I) is defined by: For every
U (~) = {x ~ X : U ( x ) > o~}.
Definition I . U [17]. Let U be a fuzzy subset of a universe X and let tr • (0, 1). The it-cut of U is the crisp subset of X
Therefore, for each ~ • R * ( I ) its a~-cut is a bounded interval of reals
• R*(1), Lr(~) = 1 - ~(r).
~(~) = [0, t) or [0, t] It is obvious that Lr(~) is a real left continuous nondescending function in r and is nonascending in ~.
Definition 1.10 [10]. Let (X, d) be a fuzzy pseudo-metric space. The fuzzy open ball with center x • X and radius r > 0 is the fuzzy subset B(x, r) of X defined by: For every y • X
B(x, r)(y) = Lr(d(x, y)). In [9], it is proved prefilterbases
fl(x)={B(x,r):r>O},
that
the
family
of
x•X,
is a fuzzy neighbourhood basis, in the sense of Lowen [6], for a fuzzy neighbourhood topology t(d) on X. The fuzzy neighbourhood space (fns) (X, t(d)) is called fuzzy metric meighbourhood space. The fns (M(l),t(~])) is called the
N-Euclidean space. The N-Euclidean space is the fuzzy analogue, for FNS, of the real line and its usual topology for the following reasons: 1. The N-Euclidean space is used [9] to introduce a property on fuzzy topological spaces (fts's) called N-complete regularity. This property characterizes the well known Lowen fuzzy uniformizable fts's. 2. It is also used [11] in formulating a fuzzy Urysohn lemma for the characterization of the axiom of fuzzy normality on fns's. 3. It is shown in [12] that the three operations, addition, scalar multiplication and multiplication, are continuous in the NEuclidean space. 4. The N-Euclidean space plays an essential role in our theory of fuzzy inner product spaces [4].
where t -- sup{x • R*: ~(x) > oc}. We identify ~(") with this real number t. It is easily proved that taking a~-cuts preserves the three operations on R*(I), and its order, in the following sense: For ~, ~ • R*(1), tr • (0, 1) and r/> 0 we have (i) (~ ~ ~)(~) -- ~(~) + ~("), (ii) (r~) (") = r~ (~), (iii) ( ~ ) ( ~ ) = ~(")~("), (iv) ~ ~ ~ iff ~(~) ~ ~(") for every tre (0, 1). Definition 1.12 [4]. Let (~, ~ ) • M(I) and ce • (0, 1). We define the ol-cut of (~, ~) to be the real number
We also define the 1-cut of (~, ~ ) • M(1) to be the real number (~, ~ ) l = lim (~, ~)("). o~+1
This 1-cut exists, since clearly we have (~, ~)1= lim (~(~)-~(~)) ac~
1-
= inf ~(oo_ inf ~("). a'>O
~>0
Also, (~, ~)l is unambiguously defined because so are all a-cuts of (~, ~).
Proposition 1.1 [4]. (i) The o:-cut (~, ~)(~) is well defined on M(I). (ii) (~, ~)--()., ~u) iff they have the same indexed family of re-cuts. (iii) (~, ~) • M*(1) iff (~, ~)(~) >10. (iv) Taking re-cuts is an order preserving real algebra homomorphism from M(I) onto R, for every fixed (r e (0, 1). []
H.M. El-Hamouly, A.M. EI-Abyad / Completeness of the standard fuzzy real line
98
2. Fuzzy convergence in fuzzy metric spaces
Grabiec in [3] gave a definition for a Cauchy sequence and a convergent sequence in a fuzzy metric space. Also, he defined a complete fuzzy metric space. In this section, we will modify these definitions in terms of fuzzy pseudometrics. Grabiec used the Schweizer and Sklar distance [16] which is the probabilistic metric (a probability nondescending distribution function). In contrast, we will use as a distance the fuzzy pseudo-metric as used in [4, 9, 10, 12], which is a nonnegative fuzzy real number [Definition 1.1]. The advantage of using the latter metric is that the triangle inequality takes the usual sense, while it is reversed in the former one. Definition 2.1.
Let (X, d) be a fuzzy pseudometric space. Let (x,)~=l be a sequence in X. Then the f u z z y limit of (x,)~=1 is a fuzzy set in I x denoted by L. For all x • X, the fuzzy limit L<~.) is defined by L{x,)(x) = inf liminf B(x, e)(x,) = inf liminf Le [d(x, x.)] = inf liminf[1 -
d(x, xn)(e)]
[7, Lemma 2.2.1]. Hence, taking limsup of both sides we get
d(x, y)(2e) ~
d(x, y)(e) = 0
Ve > O.
So, from Definition 1.8 we have x =y, hence the limit is unique. []
and
Definition 2.3. A sequence (xn) in a fuzzy pseudo-metric space (X, d) is called a f u z z y Cauchy sequence if 1 = inf liminf Le[d(xm, x,)] = 1 - sup limsup d(xm, xn)(e), where liminf, limsup are taken as m, n tend to infinity, i.e. (x,)~=~ is fuzzy Cauchy sequence iff lim d(Xm, Xn)(e)
= 0
for all e > 0.
Theorem 2.2. Let (x~)~=l be a sequence in a fuzzy metric space (X, d) which converges to x e X . Then (xn)~=l is a fuzzy Cauchy sequence in (X, d)
= 1 - sup limsup d(x, xn)(e). Inf and sup will always be taken over all positive e and liminf and limsup will be taken as n tends to infinity. This convention will be followed throughout. We should keep in mind that 0_(~.) measures the degree of convergence of ( x , ) to x. For instance, in the classical limit, d(x, x,) tends to zero and 0_<~,>goes to 1 as x, tends to x. Definition 2.2. A sequence in a fuzzy pseudo-
metric space (X, d) is said to be convergent to x • X if 0_(y) = 1 for x, y • X.
Proof. Since (x~)~=l converges to x e X ,
we
have Q-(x.)(x) = 1 - sup limsup d(x, x,)(e) = 1. Therefore limsup d(x, x,)(e) = 0
Ve > O.
But, from the triangle inequality,
d(xm, x.)(2e) -< (d(Xm, x) ~ d(x, x.))(2e) <<-d(xm, x)(e) v d(x, x.)(e). Taking limsup as m, n tend to infinity, we get limsup d(xm, x~)(e) = 0
Ve > O,
i.e. {x~)~=l is a fuzzy Cauchy sequence in X. []
This yields, for all e > O,
From the triangle inequality, we have
Definition 2.4. A fuzzy metric space (X, d) is complete iff for every fuzzy Cauchy sequence (xn)~=l in X there exists x • X such that
d(x, y)(Ze) ~< (d(x, x,,) ~ d(x,,, y))(Ze)
0_(x.>(x) = 1.
limsup d(x, x,,)(e) = limsup d(y, x,,)(e) = O.
d(x, x,)(e) v d(x,, y)(e),
If (X, d) is a fuzzy normed space, then it is
99
H.M. EI-Hamouly, A.M. El-Abyad / Completenessof the standard fuzzy real line
called fuzzy Banach space, and if it is a fuzzy inner product space [4] it is called f u z z y Hilbert space.
Proposition 1.1. Since z(a:) is right continuous, and nonascending on (0, 1) and using the above definition of ~, we get ~<°°=z(a:) Va:e (0, 1). []
3. a-cuts in M(1)
In the next proposition, we characterize the set of nonnegative fuzzy real numbers R*(1) as a subset of M(1), by means of a:-cuts.
In this section, we characterize the fuzzy Cauchy sequences in M ( I ) in terms of their corresponding (real) sequences of a:-cuts. If we consider R*(1) on its own, it is just an Abelian cancellation monoid under fuzzy addition. We do not have subtraction, and so we cannot speak of a fuzzy norm. However, if we consider R*(1) as a subset canonically embedded in M(I), we can define the above three concepts as follows: For ~, ~ ~ R*(1) ~ M(I): (i) ~ - ¢ = (~, 13) - (¢, 13) = (~, 13) ~ (13, ¢) = (~, ¢)
in M(I),
(ii) d(~, ~) = [[(~, ¢)] ~ R*(1),
(iii) ~ ] = [I(~, 13)]] = inf{¢ e R*(I): ~ ~) ~/> 13 and ~ 1 3 ~ > ~} --~. The main objective of this research is to show that the N-Euclidean space (M(I), ~ ) is an incomplete fuzzy metric space, while R*(1), as a subset of M ( I ) and under the same N-Euclidean metric, is complete. To prove this, we now build the needed mathematical machinery. L e m m a 3.1. Let z be a real nonnegative function
on the interval (0, 1]. Then the following two statements are equivalent: (i) z is right continuous and nonascending on (0, 1). (ii) There is ~ ~ R*(1) such that ~(~) = z(a:)
for all a: ~ (0, 1).
Proof. ( i i ) ~ (i): Is obvious from the definition of a:-cuts. (i) ~ (ii): Let ~ : R* --~ I be defined as follows:
~(t) = sup{a:: z(a:) I> t}
for all t/> 0.
Since z(a:) is nonascending on (0, 1), it follows that ~ is nonascending and left continuous on R*. Hence ~ e R * ( l ) . This ~ is unique by
P r o p o s i t i o n 3.1. Let x e M(1). Then x e R*(I) iff x (~), as a function of a:, is nonnegative, nonascending and right continuous on (0, 1). Proof. This follows from the
Lemma 3.1 and Proposition 1.1.
conjunction []
of
3.1. Let x e M ( I ) , e > O and a:o (0, 1), and let Q be a dense subset of [0, 1]. Then:
Theorem
[x(~)[ ~< e for all a: ~ [a:o, 1] fl Q
if and only if IM(~ +) ~< a:o. We break the proof of this theorem into the following two propositions, in which the above notations are maintained: 3.2. If x • M(1), (0, 1) are such that Proposition
e > 0 and a:o
M ( ~ + ) <~ a:o,
then
Ix<~)l ~< e for all a: 6 [a:0, 1]. Proof. By the right continuity of x (~) at a:o, and
by its left continuity at 1, it suffices to prove that Ix(~')l<~e for all a:e(a:o, 1).
(3.1)
Let x -- ()., ~) and suppose that (3.1) does not hold. Then, Ix<~) I = IZ <~) - u<~)l > e,. for some a: z (O~o, 1) and some el > e. Hence, for that pair (a:, el), we have either Z(~) -/~(~) > el,
(3.2)
or / ~ ) _ 3(,0 > el.
(3.3)
H.M. EI-Hamouly, A.M. EI-Abyad / Completeness of the standard fuzzy real line
100
If (3.2) holds, then
4, # e R * ( I ) . Then our assumption becomes
4 (~) > #(~) + el. for all a~ e (Cro, 1) f3 Q.
By definition of a~-cut, 4 ~), we have 4(# (~) + el) > re.
(3.4)
Now, for all r / • R * ( I ) such that
Define a function ¢ : R* --~ I by ¢(t)
T/~#~>4
(3.5)
(3.9)
1 [ a~o A (A ~ #)(t)
if t • [0, el,
if t • (e, ~).
Then it is easily seen ¢(e +) ~ ~o. Now, for all s • (e, oo),
we get = sup{q(b) A #(c): b + c = #(~) + e~}.
that
~•R*(1)
and
(3.6)
But, whenever c > #(oo we get by definition of
#(a),
= sup{ ¢(t) A 4(S -- t): t • [0, S]} = sup {1 ^ 4(s - t): t • [0, e]}
u(c) ~< a.
v sup{o~o ^ (4 ~ #)(t) ^ 4(s - t): t • (e, S]}
Therefore, (3.6) yields
= 4 ( s - e)
a~
v sup{o~o A 4(b) A #(C) A 4(S -- b - c): b + c • (e, s]}
Hence 3b/> e~ such that
(because 4 is nonascending). ~(b) >
In particular, by taking b = 0 and c - - s above, we get for all s • (e, ~),
Consequently, (3.7)
~(E1) > ¢:lf.
(¢ ~ 4)(s) 1> 4(s - e) v [fro A #(s)].
(3.10)
We have the following three cases: If s e [0, e], then since ~ @ 4 1> ~ we get
Thus, taking r / = Ux]]we get [x](e,) = [[4 - / z ] ( e , ) > a~.
(3.8)
If (3.3) holds, an exactly similar argument again yields (3.8). But, for all i r e (O~o, 1) and el • (e, o~), the assumption and the nonascendency of Ix] yield
~X]](EI)< OE,
(¢ ~) 4)(s)/> ¢(s) = 1 ~/~(s).
(3.11)
If s • (e, oo) and #(s) ~< tXo, then from (3.10), (3.12)
( ¢ ( ~ 4 ) ( s ) > ~ 4 ( s - e) v # ( s ) > ~ ( s ) .
If s e (e, ~) and #(s) > n~o, then the definition of a-cuts ensures that for all cr • (a~o, #(s)), /~('~) I> S.
which contradicts (3.8). So, (3.1) must hold. This completes the proof. []
This, coupled with (3.9), yields for all n~e (fro, #(s)) O Q, 4 (`0/>//(~) - e i> s -- e.
Proposition 3.3.
If x • M(I),
e > 0 and
O:o •
By the definition of or-cuts again, we get
(0, 1) are such that
4(s - e) = 4(s - e - ) I>
Ix(")l ~< ~ f o r all te • (teo, 1) tq Q,
then
for all a • (ao, #(s)) N Q, since Q is dense in L
~x](~ +) ~< ao-
4(s - e) I> u(s).
and
equivalently,
Using this, we deduce from (3.10) that Proof. We
have x = ( 4 ,
#)eM(1)
for some
(¢ ~) 4)(s) I> 4(s - e) 1> #(s).
(3.13)
H.M. EI-Hamouly, A.M. EI-Abyad / Completeness of the standard fuzzy real line
The above three inequalities (3.11), (3.12) and (3.13) complete the proof that
101
is a right continuous nonascending function of tr on (0, 1], and is left continuous at tr = 1.
Proposition 3.6. Let ( y , ) be a sequence of right But, the definition of ~ is symmetric in ~. and ~. Hence also, ~#~>~. Consequently, the definition of ~(3.,/z)~ ensures that
Ix] = ~(Z, U)B ~< .~Therefore, Ux](e +)
+)
continuous functions of tr which converges uniformly to a function y on [Cro, 1], for some o¢0e (0, 1). Then y is a right continuous function of tr on [Cro, 1].
Proof. This is an immediate consequence of Theorem 7.11, [15, p. 135]. [] The following proposition is a generalization of Lemma 3.1 to M(1).
C o,
which proves the proposition, and completes the proof of Theorem 3.1. []
Proposition 3.7. Let z be a real function on the
which converges to x • M ( l ) . Then (x~~)) converges to x (°0, uniformly on each interval [a~o, 1], a~o e (0, 1).
interval (0, 1]. Then the following two statements are equivalent: (i) z is right continuous and is of bounded variation on every interval [OCo, 1], Cro • (0, 1). (ii) There is x • M ( I ) such that x (~) = z(o 0 for all cr • (0, 1).
ProoL Since (xn) converges to x, we have
Proof. (ii)ff(i): Let x = (~, ~ ) • M(1) be such
lim Uxn - x](e) = 0
that
Proposition 3.4. Let (x.) be a sequence in M(1)
for all e > 0.
That is, for each or0 • (0, 1) and e > 0, there is a positive integer N~,~o such that -
<
6,
for all n>~N ..... and all o~e[oq, 1]. Proposition 3.2, it follows that IxY)
From
-
for all n~N~,~,o and cre[tr0, 1]. Hence (x~ ")) converges to x (~), uniformly on each interval [O~o, 1]. []
Proposition
3.5. Let x = (~, ~) • M(1). Then x(~)= ~(~)_ ~(o~) is of bounded variation on [C~o, 1] (• BV[ao, 1]) for all C~oe (0, 1).
Proof. From the definition of oc-cuts, it follows that ~(~) and ~(~) are nonascending functions of c~, and are bounded on each [OCo,1]. Hence ~*), ¢(*)•BV[cro, 1] (Theorem 2, [2, p. 84]). Since BV[cro, 1] is a vector space over the real numbers [2, p. 85], it follows that ~(~) - ~(") BV[o~0, 1]. [] By the definition of it-cuts, it follows that _~(~)
x (°0 = ~(~) - ~ ) = z(cr)
for all o~ • (0, 1).
Then by Proposition 3.5, z(cQ • BV[cro, 1] for all OCo•(0, 1). Also, since ~("), ~(o~) are right continuous functions of oc on (0, 1), so is z. ( i ) ~ ( i i ) : Suppose a function z satisfies (i). Using a technique similar to that of Theorem 2 [2, p. 84], we can construct two nonascending, nonnegative functions f, g on the interval (0, 1] such that on each closed subinterval [ao, 1], -
From Lemma 3.1, there exist ~, ~ e R*(1) such that ~<~)=f(a¢),
~(~) =g(aO
on [Olo, 11.
Hence =
for x = (~, ¢) e M(1), tx • (0, 1). This x is unique by Proposition 1.1. []
Proposition 3.8. Let ( x , ) be a sequence in M ( I ) such that (x~°0) converges uniformly to z
H.M. El-Hamouly, A,M. El-Abyad / Completenessof the standard fuzzy real line
102
BV[teo, 1] for all Oto • (0, 1). Then z ( a ) can be put as x ~) for some x • M(I).
Proof. Since x~~) is a right continuous function of a~ for all n and since (x~°')) converges uniformly to z(tr) on [fro, 1], it follows by Proposition 3.6 that z(a 0 is a right continuous function of a~ on (0, 1). Since z(a 0 • BV[a~o, 1],-for each a~0, and it is right continuous on (0,1), it follows by Proposition 3.7 that z ( a 0 = x <°° for some x • M ( I ) . []
Proof. Since ( x , ) is a fuzzy Cauchy sequence in M(I), then, by Theorem 4.1 and the completeness of R, the sequence of functions (x~~)) converges pointwise to a real function y. Also, using the fact that (x~) is a fuzzy Cauchy sequence, it follows that for 0%• (0, 1), e > 0 , there is a positive integer N~o,~ such that ~xm - x,~(e) < teo
Now for all o~ • [a~o, 1], for all m, n/> N~o,~, we have ~X m -- X n ] ( e ) < OC0 ~ OC.
Consequently. Theorem 3.1 and the definition of x~ ) yield
4. Completeness of R*(1) under the NEuclidean metric
IXm Theorem 4.1. Let ( x , ) be a f u z z y Cauchy sequence in (M(1), ~ ) . Then (x~~)) is a Cauchy sequence in R for all ot • (0, 1). Proof. Since ( x , ) is a fuzzy Cauchy sequence in M(I), we have lim
d(xm, xn) =
(m,n)-~(~,~)
lim
~Xm -- x~]l(e)
(m,n)-~(~, ~)
=0
re>0.
So for a given ~r•(0, 1), e > 0 , there is a positive integer N~.~ such that for all m, n/>
Nile,e, - xA(e) <
e
for all m, n/> No~o,~and all a~ • [ao, 1]. So, (x~~)) is a uniform Cauchy sequence on [a~o, 1], and hence converges uniformly to its pointwise limit y o n [ r i o , l]. []
Proposition 4.2. Let (x.) be a sequence in M(I), z • M(1). If (x~~)) converges to z (~), uniformly in ol on every interval [O~o, 1], O¢o• (0, 1), then ( x , ) converges to z in (M(1), ~ 1]). Proof. Let xn = ( ~ , ~ ) , z = (~, ~) e M(1). Suppose that (xn) does not converge to z. Then there exist e > 0 and a~o • (0, 1) such that
limsup d(x~, z)(2e) = limsup ~x, - zl](2e)
Hence, from Theorem 3.1 we get Ix(,,,~) - x(,~)[ ~< e
for all m, n >- N~o,~.
= 2tro>0.
for all m, n/> N~.,.
Therefore (x(,~)) is a Cauchy sequence in R.
[]
Therefore for all positive integers N there exists nN >I N such that x,N - z (e +) >
> 0.
It is easy to see that (cf. Definition 1.12) ~(1) = sup{t • R*: ~(t) = 1}
= inf{~(~): a~ • (0, 1)}, for ~ • R* (I) and
(~, ¢)(1)= ~(1)- ¢(1), for (~, ~) • M(I). Then we have:
Proposition 4.1. Let (x~) be a f u z z y Cauchy sequence in M(1). Then the sequence ( x ~ ) ) , as a sequence of functions of 06 converges uniformly to a function y, on [a~o, 1], for all Olo • (0, 1).
So, Theorem 3.1 yields that for all N, [x~ ) - z(°')[ > e
for some tr • (re0, 1).
This contradicts the fact that (x~~)) converges to z (") uniformly in a~ on [tro, 1]. Hence ( x , ) converges to z in M(1). []
Proposition 4.3. Let (xn) be a sequence in M(1) such that each (x~~)) is a Cauchy sequence in R for all a~ e (0, 1], and ( x ~ ) ) , as a function in o:, is uniformly Cauchy on every interval [Olo, 1], Croe(0, 1). Then (xn) is a f u z z y Cauchy sequence in (M(1), [ ]).
H.M. EI-Hamouly, A.M, El-Abyad / Completeness of the standard fuzzy real line Proof. The result follows by a p r o o f similar to that of Proposition 4.2. []
Example 4.1. T a k e fn(tr) = ~
(1/m2)sin m6tr,
c~ e [0:0, 1].
m=l
Since the set of all functions of b o u n d e d variation on [a~0, 1] is a linear vector space, each f~ is of b o u n d e d variation. It is obvious that (f~ converges uniformly to
Proposition 3.6, z is a right continuous function of a on (0, 1). Also, f r o m the fact that ~ ) are nonnegative and nonascending functions of o~ for all n ~> 1, we get that z is a nonnegative and nonascending function. Hence, L e m m a 3.1 yields that there exists ~ R * ( I ) such that ~ ) = z(tr) for all tr ~ (0, 1] (at tr = 1, we use the left continuity of all functions considered). So Proposition 4.2 yields ~n---> ~
in (R*(1), 9]).
This proves that (R*(I), ~ ~) is complete.
f(tr) = ~
(1/mZ)sinm6tr
on [a~0, 1]. H o w e v e r , the limit function f is not of bounded variation.
Theorem 4.2. The N-Euclidean space (M(1), ~ ]) is incomplete. Proof. T a k e the sequence (X n ) in M(1) with =
= ~
(1/m2)sin m6tr = f , ( t r ) ,
[]
References
m=l
=
103
cre (a~0, 1],
m=l
of the previous example. This sequence is well defined, due to Propositions 3.7 and 4.1. It is clear that this sequence of functions in oc is uniformly Cauchy on all intervals [a~0, 1], and its limit f is not of b o u n d e d variation on [re0, 1] for any a~0 ~ (0, 1). Consequently, the conjunction of Proposition 3.4 and Proposition 3.7 entails that ( x , ) does not converge in M(I). H o w e v e r , by Proposition 4.3, ( x , ) is a fuzzy Cauchy sequence in M(1). This proves that M(1) is incomplete. [] If we restrict ourselves to R*(I) as a subset in M(1), we avoid the p r o b l e m of the boundedness of variation of a limit of real functions. This enables us to get:
Theorem 4.3. The set of nonnegative f u z z y real numbers R*(I) is complete under the NEuclidean metric. Proof. Let ( ~ , ) be a fuzzy Cauchy sequence in R*(1) c M(1). Then, by Proposition 4.1, ( ~ " ) ) converges uniformly to z (~) on [o~0, 1] for all tr0e(0, 1). Since each m e m b e r of ( ~ ) ) , as a function of o~, is right continuous, then, by
]1] T.M.G. Ahsanullah and S. Ganguly, On fuzzy neighborhood rings, Fuzzy Sets and Systems 34 (1990) 255-262. [2] G. Debarra, Introduction to Measure Theory (Van Nostrand Reinhold, New York, 1975). [3] M. Grabiec, Fixed point in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988) 385-389. [4] A.M. E1-Abyad and H.M. E1-Hamouly, Fuzzy inner product spaces, Fuzzy Sets and Systems 44 (1991) 309-326. [5] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74-79. [6] R. Lowen, Fuzzy neighborhood spaces, Fuzzy Sets and Systems 7 (1982) 370-385. [7] R. Lowen, On (R(L), ~9), Fuzzy Sets and Systems 10 (1983) 203-209. [8] D. Luenberger, Optimization by Vector Spaces Methods (John Wiley, New York, 1969). [9] A.S. Mashhour and N.N. Morsi, Fuzzy metric neighbourhood spaces, Fuzzy Sets and Systems 45 (1992) 367-388. [10] N.N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems 27 (1988) 351-372. [11] N.N. Morsi, The Urysohn Lemma for fuzzy neighbourhood spaces, Fuzzy Sets and Systems 39 (1991) 347-360. [12] N.N. Morsi and S.E. Yehia, Continuity of fuzzy multiplication in the N-Euclidean space, Fuzzy Sets and Systems 46 (1992) 97-106. [13] S.E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems 8 (1982) 39-52. [14] S.E. Rodabaugh, Complete fuzzy topological hyperfields and fuzzy multiplication in the L-fuzzy real line, Fuzzy Sets and Systems 15 (1985) 285-310. [15] Rudin, Principles of Mathematical Analysis (McGraw Hill, New York, 1975). [16] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960) 314-334. [17] M.D. Weiss, Fixed points, separation and induced topologies for fuzzy sets, J. Math. Anal, Appl. 50 (1975) 142-150. [18] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part 1, Inform. Sci. 8 (1975) 199-249.
95
Completeness of the standard fuzzy real line for FNS Hassan M. EI-Hamouly and Abdelwahab M. E1-Abyad
Dept. of Mathematics', Military TechnicalCollege, Cairo, Egypt Received February 1990 Revised March 1991
Abstract: We consider a special class of fuzzy metric neighbourhood spaces, where the fuzzy distance is induced by a fuzzy pseudo-norm. This class is contained in Lowen's well known category FNS of Fuzzy Neighbourhood Spaces. We define the fuzzy limit of a sequence, a fuzzy Cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces. We prove that the set of all nonnegative fuzzy real numbers R*(I) under the N-Euclidean metric (the standard fuzzy real line for FNS) is complete, but that the smallest fuzzy neighbourhood real algebra M(1) including R*(1) is not complete. We also characterize the fuzzy Cauchy sequences in M(I) in terms of their corresponding (real) sequences of tr-cuts.
incomplete and R*(1) is complete as a subset of M(1) over the N-Euclidean metric, are developed in Section 4. Let us start with some definitions, in which R* will denote the set of nonnegative real numbers. Definition 1.1 [5, 13]. A nonnegative fuzzy real number ~ is a nonascending, left continuous function from R* into I = [0, 1] with ~ ( 0 ) = 1 and ~(+o0-) = 0. The set of all n0nnegative fuzzy real numbers is denoted by R*(I). The partial order >! on R*(I) is the natural ordering of real functions. R* is canonically embedded in R*(I) in the following fashion; for every r • R * we associate the fuzzy real number f defined by ~(t) = {~
if t • [0, r], if t • (r, o0).
Then 13 is the smallest element in R*(I).
Keywords: Fuzzy metric neighbourhood spaces; fuzzy convergence; completeness.
1. Introduction In many application problems in optimization and approximation theories, the completeness of the underlying space is crucial to guarantee the existence of a solution. For example, the minimum norm problem [8] and the fixed point theorerla [3] need the underlying space to be complete. In this section we give some premiminary definitions that we will need in our study. Definitions of fuzzy Cauchy sequence, fuzzy convergence and fuzzy limit of a sequence in fuzzy metric neighbourhood spaces are given in Section 2. We characterize the fuzzy Cauchy sequences in M(1) in terms of their corresponding (real) sequences of a~-cuts in Section 3. The main results of this research, that M(1) is
Correspondence to: Dr. A. M. E1-Abyad, Dept. of Mathematics, Military Technical College, Cairo, Egypt.
The three main operations on R * ( 1 ) addition, nonnegative scalar multiplication and multiplication - are defined through Zadeh's Extension Principle [18] as follows: Definition 1.2 [13, 14]. Let ~, ~ e R*(1), t • R* Then: (i) Addition: (~ ~ ~)(t) = sup{~(a) ^ ~(b): a + b = t}. (ii) Scalar multiplication by a nonnegative real number r t> 0: {()(
(r~)(t) =
t/r)
if r = 0 , if r > 0 .
(iii) Multiplication: (~¢)(t) = sup{~(a) ^ ¢(b): ab = t}. It is well known [14] that the above operations are well defined on R*(1), and preserve the order on R*(I), and R*(1) is closed under these operations. The algebraic properties of addition
0165-0114/92/$05.(10t~) 1992--Elsevier SciencePublishers B.V. All rights reserved
96
H.M. EI-Hamouly, A.M. EI-Abyad / Completenessof the standardfuzzy real line
and scalar multiplication on R*(1) enable us to embed R*(I) in a smallest real vector space M(1) as follows: Definition 1.3 [9]. The set M(1) is the Cartesian product R * ( I ) x R * ( I ) modulo an equivalence relation - defined by (~,¢)-(~,#)
iff
iff
Definition 1.6 [9]. The N-Euclidean norm on M(1) is the fuzzy norm defined by" For (~, ~) ~ M(I), ~(~, ¢)] = inf{Z ¢ R*(I): Z/> (~, ¢) and 3,/> (~, ~)} = inf{A ~ R*(I): 3. ~ ~/> and ~ . ~ >
~#~¢~..
The set M*(I) is defined by
M*(I) = {(~, ¢) e M(I): (~, ¢) ~ (0, 0)} = {(~, ¢) e M(I): ~ I> ¢}. R*(1) is canonically embedded in M*(I) by representing each ~ ~ R*(I) as (~, 0) e g ( l ) . Also, R is embedded in M(1) as follows: If r ~>0 it is identified with (f, 0), and with (0, ( - f ) ) if r < 0. Definition 1.4 [9]. Addition and scalar multiplication are defined on M(I) as follows: For (~, ~), (3., #) ~ M(1) and t e R: (i) Addition: (~, ~) ¢ (Z, ~,) = (~ • ~, ~ ¢ #). (ii) Scalar multiplication:
t(~,
(iii) Ilxll ~:O for all x ~e0 in X.
¢~#=¢~.
The partial order I> on M(I) is defined by (~,~)~>(X,#)
further satisfies:
t >I 0, ~) j"[ (t~, t~), (It[ ~, It[ ~5), t < O.
¢}.
It is proved in [9] that this function [] is indeed a fuzzy norm on M(1), Definition 1.7 [12]. Multiplication on M(1) is defined by: For (~, ~), (~., #) ~ M(I),
(~, ¢)(L ~,) = (~x • ~ , ~, ¢ ~z). Theorem 1.2 [12]. (i) Multiplication on M(I) is
well defined. (ii) The canonical embedding of R*(I) into M (1) preserves multiplication. (iii) Under addition, scalar multiplication and multiplication, M(1) is a real commutative and associative algebra with unit element i = (i, 0). (iv) M(I) is not an integral domain. (v) (M(I), ~ ~) is a fuzzy normed vector space and is a fuzzy normed algebra under multiplication i.e. for every x, y ~ M(I) we have
[xy~<~x]~y]. (vi) M*(I) is closed under multiplication.
[]
Theorem 1.1 [9]. The above addition and scalar
multiplication are well defined on M(1). Under these two operations, M(I) is the smallest real vector space containing R*(I). In particular, the canonical embedding of R *(1) into M (1) preserves addition and scalar multiplication, while the canonical embedding of R into M(1) is a vector space embedding. [] Definition 1.5 [9]. A fuzzy pseudo-norm on a vector space X is a mapping l[ II :X-->R*(1) which satisfies: For x, y e X and r e R: (i) I}rx[} = Ir[ [Ix}l, (ii) IIx + y II ~< IIx II ~ IIY II (triangle inequality). (X, 11II) is called a fuzzy pseudo-normed space, and it is called a fuzzy normed space if it
Definition 1.8 [10]. A fuzzy pseudo-metric on a nonempty set X is a mapping d : X × X-->R*(1) which satisfies the following: For x, y e X: (i) d(x, x) = O, (ii) d(x, y) = d(y, x) (symmetry), (iii) d(x, z) <~d(x, y) ~) d(y, z) (triangle inequality). (X, d) is called fuzzy pseudo-metric space, and it is called a fuzzy metric space if it further satisfies: (iv) d(x, y) ~ 0 for all x :/:y in X. The fuzzy pseudo-metric associated with a fuzzy pseudo-normed space (X, II II) is given by: For x, y e X,
d(x, y ) = IlY -xll.
H.M. EI-Hamouly,A.M. EI-Abyad / Completenessof the standardfuzzy realline
97
In particular, the fuzzy metric on M(1), induced in this manner from the N-Euclidean norm on M(I), is called the N-Euclidean metric. Same terminology applies to the restriction of this fuzzy metric to R*(1)~ M(I).
The ring M(I) is clearly a fuzzy neighbourhood ring, as defined in [1].
Definition 1.9 [5]. For every r • R, the fuzzy subset Lr of R*(I) is defined by: For every
U (~) = {x ~ X : U ( x ) > o~}.
Definition I . U [17]. Let U be a fuzzy subset of a universe X and let tr • (0, 1). The it-cut of U is the crisp subset of X
Therefore, for each ~ • R * ( I ) its a~-cut is a bounded interval of reals
• R*(1), Lr(~) = 1 - ~(r).
~(~) = [0, t) or [0, t] It is obvious that Lr(~) is a real left continuous nondescending function in r and is nonascending in ~.
Definition 1.10 [10]. Let (X, d) be a fuzzy pseudo-metric space. The fuzzy open ball with center x • X and radius r > 0 is the fuzzy subset B(x, r) of X defined by: For every y • X
B(x, r)(y) = Lr(d(x, y)). In [9], it is proved prefilterbases
fl(x)={B(x,r):r>O},
that
the
family
of
x•X,
is a fuzzy neighbourhood basis, in the sense of Lowen [6], for a fuzzy neighbourhood topology t(d) on X. The fuzzy neighbourhood space (fns) (X, t(d)) is called fuzzy metric meighbourhood space. The fns (M(l),t(~])) is called the
N-Euclidean space. The N-Euclidean space is the fuzzy analogue, for FNS, of the real line and its usual topology for the following reasons: 1. The N-Euclidean space is used [9] to introduce a property on fuzzy topological spaces (fts's) called N-complete regularity. This property characterizes the well known Lowen fuzzy uniformizable fts's. 2. It is also used [11] in formulating a fuzzy Urysohn lemma for the characterization of the axiom of fuzzy normality on fns's. 3. It is shown in [12] that the three operations, addition, scalar multiplication and multiplication, are continuous in the NEuclidean space. 4. The N-Euclidean space plays an essential role in our theory of fuzzy inner product spaces [4].
where t -- sup{x • R*: ~(x) > oc}. We identify ~(") with this real number t. It is easily proved that taking a~-cuts preserves the three operations on R*(I), and its order, in the following sense: For ~, ~ • R*(1), tr • (0, 1) and r/> 0 we have (i) (~ ~ ~)(~) -- ~(~) + ~("), (ii) (r~) (") = r~ (~), (iii) ( ~ ) ( ~ ) = ~(")~("), (iv) ~ ~ ~ iff ~(~) ~ ~(") for every tre (0, 1). Definition 1.12 [4]. Let (~, ~ ) • M(I) and ce • (0, 1). We define the ol-cut of (~, ~) to be the real number
We also define the 1-cut of (~, ~ ) • M(1) to be the real number (~, ~ ) l = lim (~, ~)("). o~+1
This 1-cut exists, since clearly we have (~, ~)1= lim (~(~)-~(~)) ac~
1-
= inf ~(oo_ inf ~("). a'>O
~>0
Also, (~, ~)l is unambiguously defined because so are all a-cuts of (~, ~).
Proposition 1.1 [4]. (i) The o:-cut (~, ~)(~) is well defined on M(I). (ii) (~, ~)--()., ~u) iff they have the same indexed family of re-cuts. (iii) (~, ~) • M*(1) iff (~, ~)(~) >10. (iv) Taking re-cuts is an order preserving real algebra homomorphism from M(I) onto R, for every fixed (r e (0, 1). []
H.M. El-Hamouly, A.M. EI-Abyad / Completeness of the standard fuzzy real line
98
2. Fuzzy convergence in fuzzy metric spaces
Grabiec in [3] gave a definition for a Cauchy sequence and a convergent sequence in a fuzzy metric space. Also, he defined a complete fuzzy metric space. In this section, we will modify these definitions in terms of fuzzy pseudometrics. Grabiec used the Schweizer and Sklar distance [16] which is the probabilistic metric (a probability nondescending distribution function). In contrast, we will use as a distance the fuzzy pseudo-metric as used in [4, 9, 10, 12], which is a nonnegative fuzzy real number [Definition 1.1]. The advantage of using the latter metric is that the triangle inequality takes the usual sense, while it is reversed in the former one. Definition 2.1.
Let (X, d) be a fuzzy pseudometric space. Let (x,)~=l be a sequence in X. Then the f u z z y limit of (x,)~=1 is a fuzzy set in I x denoted by L
d(x, xn)(e)]
[7, Lemma 2.2.1]. Hence, taking limsup of both sides we get
d(x, y)(2e) ~
d(x, y)(e) = 0
Ve > O.
So, from Definition 1.8 we have x =y, hence the limit is unique. []
and
Definition 2.3. A sequence (xn) in a fuzzy pseudo-metric space (X, d) is called a f u z z y Cauchy sequence if 1 = inf liminf Le[d(xm, x,)] = 1 - sup limsup d(xm, xn)(e), where liminf, limsup are taken as m, n tend to infinity, i.e. (x,)~=~ is fuzzy Cauchy sequence iff lim d(Xm, Xn)(e)
= 0
for all e > 0.
Theorem 2.2. Let (x~)~=l be a sequence in a fuzzy metric space (X, d) which converges to x e X . Then (xn)~=l is a fuzzy Cauchy sequence in (X, d)
= 1 - sup limsup d(x, xn)(e). Inf and sup will always be taken over all positive e and liminf and limsup will be taken as n tends to infinity. This convention will be followed throughout. We should keep in mind that 0_(~.) measures the degree of convergence of ( x , ) to x. For instance, in the classical limit, d(x, x,) tends to zero and 0_<~,>goes to 1 as x, tends to x. Definition 2.2. A sequence in a fuzzy pseudo-
metric space (X, d) is said to be convergent to x • X if 0_
Proof. Since (x~)~=l converges to x e X ,
we
have Q-(x.)(x) = 1 - sup limsup d(x, x,)(e) = 1. Therefore limsup d(x, x,)(e) = 0
Ve > O.
But, from the triangle inequality,
d(xm, x.)(2e) -< (d(Xm, x) ~ d(x, x.))(2e) <<-d(xm, x)(e) v d(x, x.)(e). Taking limsup as m, n tend to infinity, we get limsup d(xm, x~)(e) = 0
Ve > O,
i.e. {x~)~=l is a fuzzy Cauchy sequence in X. []
This yields, for all e > O,
From the triangle inequality, we have
Definition 2.4. A fuzzy metric space (X, d) is complete iff for every fuzzy Cauchy sequence (xn)~=l in X there exists x • X such that
d(x, y)(Ze) ~< (d(x, x,,) ~ d(x,,, y))(Ze)
0_(x.>(x) = 1.
limsup d(x, x,,)(e) = limsup d(y, x,,)(e) = O.
d(x, x,)(e) v d(x,, y)(e),
If (X, d) is a fuzzy normed space, then it is
99
H.M. EI-Hamouly, A.M. El-Abyad / Completenessof the standard fuzzy real line
called fuzzy Banach space, and if it is a fuzzy inner product space [4] it is called f u z z y Hilbert space.
Proposition 1.1. Since z(a:) is right continuous, and nonascending on (0, 1) and using the above definition of ~, we get ~<°°=z(a:) Va:e (0, 1). []
3. a-cuts in M(1)
In the next proposition, we characterize the set of nonnegative fuzzy real numbers R*(1) as a subset of M(1), by means of a:-cuts.
In this section, we characterize the fuzzy Cauchy sequences in M ( I ) in terms of their corresponding (real) sequences of a:-cuts. If we consider R*(1) on its own, it is just an Abelian cancellation monoid under fuzzy addition. We do not have subtraction, and so we cannot speak of a fuzzy norm. However, if we consider R*(1) as a subset canonically embedded in M(I), we can define the above three concepts as follows: For ~, ~ ~ R*(1) ~ M(I): (i) ~ - ¢ = (~, 13) - (¢, 13) = (~, 13) ~ (13, ¢) = (~, ¢)
in M(I),
(ii) d(~, ~) = [[(~, ¢)] ~ R*(1),
(iii) ~ ] = [I(~, 13)]] = inf{¢ e R*(I): ~ ~) ~/> 13 and ~ 1 3 ~ > ~} --~. The main objective of this research is to show that the N-Euclidean space (M(I), ~ ) is an incomplete fuzzy metric space, while R*(1), as a subset of M ( I ) and under the same N-Euclidean metric, is complete. To prove this, we now build the needed mathematical machinery. L e m m a 3.1. Let z be a real nonnegative function
on the interval (0, 1]. Then the following two statements are equivalent: (i) z is right continuous and nonascending on (0, 1). (ii) There is ~ ~ R*(1) such that ~(~) = z(a:)
for all a: ~ (0, 1).
Proof. ( i i ) ~ (i): Is obvious from the definition of a:-cuts. (i) ~ (ii): Let ~ : R* --~ I be defined as follows:
~(t) = sup{a:: z(a:) I> t}
for all t/> 0.
Since z(a:) is nonascending on (0, 1), it follows that ~ is nonascending and left continuous on R*. Hence ~ e R * ( l ) . This ~ is unique by
P r o p o s i t i o n 3.1. Let x e M(1). Then x e R*(I) iff x (~), as a function of a:, is nonnegative, nonascending and right continuous on (0, 1). Proof. This follows from the
Lemma 3.1 and Proposition 1.1.
conjunction []
of
3.1. Let x e M ( I ) , e > O and a:o (0, 1), and let Q be a dense subset of [0, 1]. Then:
Theorem
[x(~)[ ~< e for all a: ~ [a:o, 1] fl Q
if and only if IM(~ +) ~< a:o. We break the proof of this theorem into the following two propositions, in which the above notations are maintained: 3.2. If x • M(1), (0, 1) are such that Proposition
e > 0 and a:o
M ( ~ + ) <~ a:o,
then
Ix<~)l ~< e for all a: 6 [a:0, 1]. Proof. By the right continuity of x (~) at a:o, and
by its left continuity at 1, it suffices to prove that Ix(~')l<~e for all a:e(a:o, 1).
(3.1)
Let x -- ()., ~) and suppose that (3.1) does not hold. Then, Ix<~) I = IZ <~) - u<~)l > e,. for some a: z (O~o, 1) and some el > e. Hence, for that pair (a:, el), we have either Z(~) -/~(~) > el,
(3.2)
or / ~ ) _ 3(,0 > el.
(3.3)
H.M. EI-Hamouly, A.M. EI-Abyad / Completeness of the standard fuzzy real line
100
If (3.2) holds, then
4, # e R * ( I ) . Then our assumption becomes
4 (~) > #(~) + el. for all a~ e (Cro, 1) f3 Q.
By definition of a~-cut, 4 ~), we have 4(# (~) + el) > re.
(3.4)
Now, for all r / • R * ( I ) such that
Define a function ¢ : R* --~ I by ¢(t)
T/~#~>4
(3.5)
(3.9)
1 [ a~o A (A ~ #)(t)
if t • [0, el,
if t • (e, ~).
Then it is easily seen ¢(e +) ~ ~o. Now, for all s • (e, oo),
we get = sup{q(b) A #(c): b + c = #(~) + e~}.
that
~•R*(1)
and
(3.6)
But, whenever c > #(oo we get by definition of
#(a),
= sup{ ¢(t) A 4(S -- t): t • [0, S]} = sup {1 ^ 4(s - t): t • [0, e]}
u(c) ~< a.
v sup{o~o ^ (4 ~ #)(t) ^ 4(s - t): t • (e, S]}
Therefore, (3.6) yields
= 4 ( s - e)
a~
v sup{o~o A 4(b) A #(C) A 4(S -- b - c): b + c • (e, s]}
Hence 3b/> e~ such that
(because 4 is nonascending). ~(b) >
In particular, by taking b = 0 and c - - s above, we get for all s • (e, ~),
Consequently, (3.7)
~(E1) > ¢:lf.
(¢ ~ 4)(s) 1> 4(s - e) v [fro A #(s)].
(3.10)
We have the following three cases: If s e [0, e], then since ~ @ 4 1> ~ we get
Thus, taking r / = Ux]]we get [x](e,) = [[4 - / z ] ( e , ) > a~.
(3.8)
If (3.3) holds, an exactly similar argument again yields (3.8). But, for all i r e (O~o, 1) and el • (e, o~), the assumption and the nonascendency of Ix] yield
~X]](EI)< OE,
(¢ ~) 4)(s)/> ¢(s) = 1 ~/~(s).
(3.11)
If s • (e, oo) and #(s) ~< tXo, then from (3.10), (3.12)
( ¢ ( ~ 4 ) ( s ) > ~ 4 ( s - e) v # ( s ) > ~ ( s ) .
If s e (e, ~) and #(s) > n~o, then the definition of a-cuts ensures that for all cr • (a~o, #(s)), /~('~) I> S.
which contradicts (3.8). So, (3.1) must hold. This completes the proof. []
This, coupled with (3.9), yields for all n~e (fro, #(s)) O Q, 4 (`0/>//(~) - e i> s -- e.
Proposition 3.3.
If x • M(I),
e > 0 and
O:o •
By the definition of or-cuts again, we get
(0, 1) are such that
4(s - e) = 4(s - e - ) I>
Ix(")l ~< ~ f o r all te • (teo, 1) tq Q,
then
for all a • (ao, #(s)) N Q, since Q is dense in L
~x](~ +) ~< ao-
4(s - e) I> u(s).
and
equivalently,
Using this, we deduce from (3.10) that Proof. We
have x = ( 4 ,
#)eM(1)
for some
(¢ ~) 4)(s) I> 4(s - e) 1> #(s).
(3.13)
H.M. EI-Hamouly, A.M. EI-Abyad / Completeness of the standard fuzzy real line
The above three inequalities (3.11), (3.12) and (3.13) complete the proof that
101
is a right continuous nonascending function of tr on (0, 1], and is left continuous at tr = 1.
Proposition 3.6. Let ( y , ) be a sequence of right But, the definition of ~ is symmetric in ~. and ~. Hence also, ~#~>~. Consequently, the definition of ~(3.,/z)~ ensures that
Ix] = ~(Z, U)B ~< .~Therefore, Ux](e +)
+)
continuous functions of tr which converges uniformly to a function y on [Cro, 1], for some o¢0e (0, 1). Then y is a right continuous function of tr on [Cro, 1].
Proof. This is an immediate consequence of Theorem 7.11, [15, p. 135]. [] The following proposition is a generalization of Lemma 3.1 to M(1).
C o,
which proves the proposition, and completes the proof of Theorem 3.1. []
Proposition 3.7. Let z be a real function on the
which converges to x • M ( l ) . Then (x~~)) converges to x (°0, uniformly on each interval [a~o, 1], a~o e (0, 1).
interval (0, 1]. Then the following two statements are equivalent: (i) z is right continuous and is of bounded variation on every interval [OCo, 1], Cro • (0, 1). (ii) There is x • M ( I ) such that x (~) = z(o 0 for all cr • (0, 1).
ProoL Since (xn) converges to x, we have
Proof. (ii)ff(i): Let x = (~, ~ ) • M(1) be such
lim Uxn - x](e) = 0
that
Proposition 3.4. Let (x.) be a sequence in M(1)
for all e > 0.
That is, for each or0 • (0, 1) and e > 0, there is a positive integer N~,~o such that -
<
6,
for all n>~N ..... and all o~e[oq, 1]. Proposition 3.2, it follows that IxY)
From
-
for all n~N~,~,o and cre[tr0, 1]. Hence (x~ ")) converges to x (~), uniformly on each interval [O~o, 1]. []
Proposition
3.5. Let x = (~, ~) • M(1). Then x(~)= ~(~)_ ~(o~) is of bounded variation on [C~o, 1] (• BV[ao, 1]) for all C~oe (0, 1).
Proof. From the definition of oc-cuts, it follows that ~(~) and ~(~) are nonascending functions of c~, and are bounded on each [OCo,1]. Hence ~*), ¢(*)•BV[cro, 1] (Theorem 2, [2, p. 84]). Since BV[cro, 1] is a vector space over the real numbers [2, p. 85], it follows that ~(~) - ~(") BV[o~0, 1]. [] By the definition of it-cuts, it follows that _~(~)
x (°0 = ~(~) - ~ ) = z(cr)
for all o~ • (0, 1).
Then by Proposition 3.5, z(cQ • BV[cro, 1] for all OCo•(0, 1). Also, since ~("), ~(o~) are right continuous functions of oc on (0, 1), so is z. ( i ) ~ ( i i ) : Suppose a function z satisfies (i). Using a technique similar to that of Theorem 2 [2, p. 84], we can construct two nonascending, nonnegative functions f, g on the interval (0, 1] such that on each closed subinterval [ao, 1], -
From Lemma 3.1, there exist ~, ~ e R*(1) such that ~<~)=f(a¢),
~(~) =g(aO
on [Olo, 11.
Hence =
for x = (~, ¢) e M(1), tx • (0, 1). This x is unique by Proposition 1.1. []
Proposition 3.8. Let ( x , ) be a sequence in M ( I ) such that (x~°0) converges uniformly to z
H.M. El-Hamouly, A,M. El-Abyad / Completenessof the standard fuzzy real line
102
BV[teo, 1] for all Oto • (0, 1). Then z ( a ) can be put as x ~) for some x • M(I).
Proof. Since x~~) is a right continuous function of a~ for all n and since (x~°')) converges uniformly to z(tr) on [fro, 1], it follows by Proposition 3.6 that z(a 0 is a right continuous function of a~ on (0, 1). Since z(a 0 • BV[a~o, 1],-for each a~0, and it is right continuous on (0,1), it follows by Proposition 3.7 that z ( a 0 = x <°° for some x • M ( I ) . []
Proof. Since ( x , ) is a fuzzy Cauchy sequence in M(I), then, by Theorem 4.1 and the completeness of R, the sequence of functions (x~~)) converges pointwise to a real function y. Also, using the fact that (x~) is a fuzzy Cauchy sequence, it follows that for 0%• (0, 1), e > 0 , there is a positive integer N~o,~ such that ~xm - x,~(e) < teo
Now for all o~ • [a~o, 1], for all m, n/> N~o,~, we have ~X m -- X n ] ( e ) < OC0 ~ OC.
Consequently. Theorem 3.1 and the definition of x~ ) yield
4. Completeness of R*(1) under the NEuclidean metric
IXm Theorem 4.1. Let ( x , ) be a f u z z y Cauchy sequence in (M(1), ~ ) . Then (x~~)) is a Cauchy sequence in R for all ot • (0, 1). Proof. Since ( x , ) is a fuzzy Cauchy sequence in M(I), we have lim
d(xm, xn) =
(m,n)-~(~,~)
lim
~Xm -- x~]l(e)
(m,n)-~(~, ~)
=0
re>0.
So for a given ~r•(0, 1), e > 0 , there is a positive integer N~.~ such that for all m, n/>
Nile,e, - xA(e) <
e
for all m, n/> No~o,~and all a~ • [ao, 1]. So, (x~~)) is a uniform Cauchy sequence on [a~o, 1], and hence converges uniformly to its pointwise limit y o n [ r i o , l]. []
Proposition 4.2. Let (x.) be a sequence in M(I), z • M(1). If (x~~)) converges to z (~), uniformly in ol on every interval [O~o, 1], O¢o• (0, 1), then ( x , ) converges to z in (M(1), ~ 1]). Proof. Let xn = ( ~ , ~ ) , z = (~, ~) e M(1). Suppose that (xn) does not converge to z. Then there exist e > 0 and a~o • (0, 1) such that
limsup d(x~, z)(2e) = limsup ~x, - zl](2e)
Hence, from Theorem 3.1 we get Ix(,,,~) - x(,~)[ ~< e
for all m, n >- N~o,~.
= 2tro>0.
for all m, n/> N~.,.
Therefore (x(,~)) is a Cauchy sequence in R.
[]
Therefore for all positive integers N there exists nN >I N such that x,N - z (e +) >
> 0.
It is easy to see that (cf. Definition 1.12) ~(1) = sup{t • R*: ~(t) = 1}
= inf{~(~): a~ • (0, 1)}, for ~ • R* (I) and
(~, ¢)(1)= ~(1)- ¢(1), for (~, ~) • M(I). Then we have:
Proposition 4.1. Let (x~) be a f u z z y Cauchy sequence in M(1). Then the sequence ( x ~ ) ) , as a sequence of functions of 06 converges uniformly to a function y, on [a~o, 1], for all Olo • (0, 1).
So, Theorem 3.1 yields that for all N, [x~ ) - z(°')[ > e
for some tr • (re0, 1).
This contradicts the fact that (x~~)) converges to z (") uniformly in a~ on [tro, 1]. Hence ( x , ) converges to z in M(1). []
Proposition 4.3. Let (xn) be a sequence in M(1) such that each (x~~)) is a Cauchy sequence in R for all a~ e (0, 1], and ( x ~ ) ) , as a function in o:, is uniformly Cauchy on every interval [Olo, 1], Croe(0, 1). Then (xn) is a f u z z y Cauchy sequence in (M(1), [ ]).
H.M. EI-Hamouly, A.M, El-Abyad / Completeness of the standard fuzzy real line Proof. The result follows by a p r o o f similar to that of Proposition 4.2. []
Example 4.1. T a k e fn(tr) = ~
(1/m2)sin m6tr,
c~ e [0:0, 1].
m=l
Since the set of all functions of b o u n d e d variation on [a~0, 1] is a linear vector space, each f~ is of b o u n d e d variation. It is obvious that (f~ converges uniformly to
Proposition 3.6, z is a right continuous function of a on (0, 1). Also, f r o m the fact that ~ ) are nonnegative and nonascending functions of o~ for all n ~> 1, we get that z is a nonnegative and nonascending function. Hence, L e m m a 3.1 yields that there exists ~ R * ( I ) such that ~ ) = z(tr) for all tr ~ (0, 1] (at tr = 1, we use the left continuity of all functions considered). So Proposition 4.2 yields ~n---> ~
in (R*(1), 9]).
This proves that (R*(I), ~ ~) is complete.
f(tr) = ~
(1/mZ)sinm6tr
on [a~0, 1]. H o w e v e r , the limit function f is not of bounded variation.
Theorem 4.2. The N-Euclidean space (M(1), ~ ]) is incomplete. Proof. T a k e the sequence (X n ) in M(1) with =
= ~
(1/m2)sin m6tr = f , ( t r ) ,
[]
References
m=l
=
103
cre (a~0, 1],
m=l
of the previous example. This sequence is well defined, due to Propositions 3.7 and 4.1. It is clear that this sequence of functions in oc is uniformly Cauchy on all intervals [a~0, 1], and its limit f is not of b o u n d e d variation on [re0, 1] for any a~0 ~ (0, 1). Consequently, the conjunction of Proposition 3.4 and Proposition 3.7 entails that ( x , ) does not converge in M(I). H o w e v e r , by Proposition 4.3, ( x , ) is a fuzzy Cauchy sequence in M(1). This proves that M(1) is incomplete. [] If we restrict ourselves to R*(I) as a subset in M(1), we avoid the p r o b l e m of the boundedness of variation of a limit of real functions. This enables us to get:
Theorem 4.3. The set of nonnegative f u z z y real numbers R*(I) is complete under the NEuclidean metric. Proof. Let ( ~ , ) be a fuzzy Cauchy sequence in R*(1) c M(1). Then, by Proposition 4.1, ( ~ " ) ) converges uniformly to z (~) on [o~0, 1] for all tr0e(0, 1). Since each m e m b e r of ( ~ ) ) , as a function of o~, is right continuous, then, by
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