A countable set derived by fuzzy set

5 Pages·0.082 MB·English

by  Huang H.

A countable set derived by fuzzy set
Huan Huang
5
Department of Mathematics, Jimei University, Xiamen 361021, China.
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t
c Abstract
O
9 In this paper, it shows that for each fuzzy set u on Rm, the set D(u) is at most
1
countable. Based on this, it modifies the proof of assertion (I) in step 2 of the

sufficiency part of Theorem 4.1 in paper:Characterizations of compact sets in fuzzy
]
M sets spaces with L metric, http://arxiv.org/abs/1509.00447.
p
G
.
h Key words: Fuzzy sets; Level cut sets; Countable;
t
a
m
[

1
v
1 Introduction
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5
9
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In this paper, we show that for each fuzzy set u on Rm, the set D(u) = {α ∈
0
. (0,1) : [u] * {u > α} } is at most countable.
0 α
1
5
1
: 2 Preliminaries
v
i
X
r
a We introduced some basic concepts about fuzzy sets. For details, we refer
to [1,2].
We use F(Rm) to represent all fuzzy subsets on Rm, i.e. functions from Rm to
[0,1]. For u ∈ F(Rm), let [u] denote the α-cut of u, i.e.
α
{x ∈ Rm : u(x) ≥ α}, α ∈ (0,1],
[u] =
α suppu = {x ∈ Rm : u(x) > 0}, α = 0.

Let {u > α} denotethe strong α-cut of u, i.e., {u > α} = {x ∈ Rm : u(x) >
α}.
Email address: [email protected] (H. Huang).
Preprint submitted to Elsevier
3 Main results
Let u ∈ F(Rm), t ∈ Rm and r be a positive number in R. Define a function
S (·,·) : Sm−1 ×[0,1] → {−∞}∪R by
u,t,r
−∞, if [u] ∩B(t,r) = ∅,
α
S (e,α) =
u,t,r 
sup{he,x−ti : x ∈ [u]α ∩B(t,r)}, if [u]α ∩B(t,r) 6= ∅,

where B(t,r) denote the closed ball {x ∈ Rm : d(t,x) ≤ r}.
We say α ∈ (0,1) is a discontinuous point of S (e,·) if
u,t,r
(i) S (e,α) ∈ R, and
u,t,r
(ii)S (e,β) = −∞forallβ > αor−∞ < lim S (e,β) < lim S (e,β).
u,t,r β→α+ u,t,r β→α− u,t,r
Denote the set of all discontinuous points of S (e,·) by D . Then D
u,t,r u,t,r,e u,t,r,e
is at most countable because S (e,·) is a monotone function on [0,1].
u,t,r
Theorem 3.1 Let u be a fuzzy set on Rm, t be a point in Rm, and r be a
positive real number. Then D := D is at most countable.
u,t,r e∈Sm−1 u,t,r,e
S
Proof Let ̟ be a countable dense subset of Sm−1. Then
D = D (1)
u,t,r u,t,r,e
e∈̟
[
In fact, suppose that α ∈ D , then there exists e ∈ Sm−1 such that α ∈
u,t,r
D . Hence S(u,t,r)(e,α) > −∞. This is equivalent to [u] ∩B(t,r) 6= ∅.
u,t,r,e α
Therefore
S(u,t,r)(f,α) > −∞ for all f ∈ Sm−1. (2)
To show α ∈ D , we divide the proof into two cases.
e∈̟ u,t,r,e
S
Case 1. S(u,t,r)(e,β) = −∞ for all β > α.
In this case, [u] ∩ B(t,r) = ∅ for any β > α, and so S(u,t,r)(f,β) = −∞
β
when f ∈ Sm−1 and β > α. Combined with (2), we know α ∈ D for each
u,t,r,f
f ∈ Sm−1. Thus α ∈ D .
e∈̟ u,t,r,e
S
Case 2. −∞ < lim S (e,β) < lim S (e,β).
β→α+ u,t,r β→α− u,t,r
In this case, there is an α > α such that [u] ∩B(t,r) 6= ∅ when λ ∈ [0,α ].
0 λ 0
Set
ξ := lim S (e,β)− lim S (e,β) > 0. (3)
u,t,r u,t,r
β→α− β→α+
Notice that, for all β ∈ [0,1] with [u] ∩B(t,r) 6= ∅,
β
|S (e,β)−S (f,β)|
u,t,r u,t,r
2
= |sup{he,x−ti : x ∈ [u] ∩B(t,r)}−sup{hf,x−ti : x ∈ [u] ∩B(t,r)}|
β β
≤ sup{| he−f,x−ti |: x ∈ [u] ∩B(t,r)}
β
≤ ke−fk·r,
hence, for any λ ∈ [0,α ],
0
| S (e,λ)−S (f,λ) |≤ ke−fk·r,
u,t,r u,t,r
and so, combined with (3), we know there exists δ > 0 such that, for all
f ∈ Sm−1 ∩B(e,δ),
lim S (f,β)− lim S (f,β) > ξ/2,
u,t,r u,t,r
β→α− β→α+
this means that α ∈ D when f ∈ Sm−1∩B(e,δ). Thus there exists g ∈ ̟
u,t,r,f
such that α ∈ D , i.e., α ∈ D .
u,t,r,g e∈̟ u,t,r,e
S
Now we obtain (1). Since ̟ is countable and D is at most countable, we
u,t,r,e
know D is at most countable. ✷
u,t,r
Remark 3.1 In the proof of Theorem 3.1, in order to show D is at most
u,t,r
countable, it proves that D = D . This kind of trick was used in
u,t,r e∈̟ u,t,r,e
the proof of Lemma 4 in [3] to show a set is at most countable.
S
Theorem 3.2 Let u be a fuzzy set on Rm, then D(u) = {α ∈ (0,1) : [u] *
α
{u > α} } is at most countable.
Proof Ifα ∈ D(u),thenthereisay ∈ Rm suchthaty ∈ [u] buty ∈/ {u > α}.
α
Thus,
d(y,{u > α}) > ε > 0. (4)
Choose a q ∈ Qm = {(z1,z2,...,zm) ∈ Rm : zi ∈ Q, i = 1,2,...,m} which
satisfies that ky − qk > 0. We assure that α ∈ D for some r ∈ Q with
u,q,r
r ≥ ky −qk.
In fact, let e = (y −q)/ky−qk. Then
S (e,α) ≥ he,y −qi = ky −qk. (5)
u,q,r
If [u] ∩B(q,r) = ∅ for any β > α, then S (e,β) = −∞ for all β > α, and
β u,q,r
thus α ∈ D .
u,q,r,e
If there exists β > α such that [u] ∩ B(q,r) 6= ∅. Pick an arbitrary x ∈
β
{u > α}∩B(q,r), then, by (4),
kx−qk ≤ r,
kx−yk > ε.
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If x = q, then he,x−qi = 0. Suppose that x 6= q. Notice that
hy −q,x−qi kx−qk2 +ky −qk2 −kx−yk2
= cosα = ,
ky −qk·kx−qk 2ky −qk·kx−qk
where α is the angle between two vectors x−q and y −q. Thus
1 kx−qk2 kx−yk2
he,x−qi = +ky −qk−
2 ky −qk ky −qk !
1 r2 ε2
≤ +ky −qk− ,
2 ky −qk ky −qk!
and so there exists a δ > 0 such that for all r ∈ [ky −qk,ky−qk+δ),
1 ε2
he,x−qi ≤ ky −qk− . (6)
4ky −qk
Combined with (5) and (6), it then follows from the arbitrariness of x ∈
{u > α}∩B(q,r) that
lim S (e,β) < S (e,α)
u,q,r u,q,r
β→α+
when r ∈ [ky−qk,ky−qk+δ). This implies that there exists r ∈ Q such that
α ∈ D ⊂ D .
u,q,r,e u,q,r
Now we know
D(u) ⊆ D .
u,q,r
q∈Q[m,r∈Q
It then follows from Theorem 3.1 that D(u) is at most countable. ✷
4 An application
In this section, we give an application of the main result.
The following is assertion (I) in step 2 of the sufficiency part of the proof of
Theorem 4.1 in paper: Characterizations of compact sets in fuzzy sets spaces
with L metric, see http://arxiv.org/abs/1509.00447.
p
(I) P(v) is at most countable.
We can show this assertion in the following way.
In fact, given α ∈ P(v), it holds that {v > α} & [v] , and hence α ∈ D(v) :=
α
{γ ∈ (0,1) : [v] * {v > γ} }. Thus P(v) ⊆ D(v). By Theorem 3.2, D(v) is
γ
at most countable. So P(v) is at most countable.
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References
[1] C. Wu, M. Ma, The Basic of Fuzzy Analysis (in Chinese), National Defence
Industry press, Beijing, 1991.
[2] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets–Theory and Application,
World Scientific, Singapore, 1994.
[3] W. Trutschnig, Characterization of the sendograph-convergence of fuzzy sets
by means of their L - and lelewise convergence, Fuzzy Sets and Systems 161
p
(2010) 1064-1077.
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