# Báo cáo nghiên cứu khoa học: " Liên hệ giữa không gian metric mờ với không gian Menger và không gian metric xác suất" pptx

α
α
α
1.
X A X µ
A
: X →
[0, 1] X [0, 1] A = {(a, µ
A
(a))|a ∈ X} µ
A
µ
A
(a) ∈ [0, 1] a
A µ
A
[0, 1] 0
1
A = {(a, µ
A
(a))|a ∈ X} µ

A
A µ
A
(a) = 0
a ∈ X. A µ
A
(a) = 1,
a ∈ X.
µ ν X µ  ν
µ  ν µ = ν
µ  ν µ(x)  ν(x), x ∈ X,(1.0.1)
µ  ν µ(x)  ν(x), x ∈ X,(1.0.2)
µ = ν µ(x) = ν(x), x ∈ X.(1.0.3)
1
x : R−→[0, 1] t∈R x(t)
x t
0
x : R → [0, 1],
t
0
x s  t  r
x(t)  min{x(s), x(r)}.(1.0.4)
t
0
∈ R x x(t
0
) = 1 x
x x(t) = 0 t < 0
E
G
G ⊆ E
x ∈ R x x(t) =

1 t = x,
0 t = x.
+, −, ·, / E × E
(x + y)(t) = sup
s∈R
min{x(s), y(t − s)}, ∀x, y ∈ E, ∀t ∈ R,(1.0.5)
(x − y)(t) = sup

s∈R
min{x(s), y(s − t)}, ∀x, y ∈ E, ∀t ∈ R,(1.0.6)
(x · y)(t) = sup
s∈R
min

x(s), y

t
s

, ∀x, y ∈ E, ∀t ∈ R,(1.0.7)
(x/y)(t) = sup
s∈R
min{x(ts), y(s)}, ∀x, y ∈ E, ∀t ∈ R.(1.0.8)
x ∈ R
0 1 E
0(t) =

1 t = 0,
0 t = 0.
1(t) =

1 t = 1,
0 t = 1.
x ∈ R x(t) = 0(t − x) t ∈ R.
x ∈ R t ∈ R
0(t − x) =

1 t − x = 0,
0 t − x = 0.
=

1 x = t,
0 x = t.
= x(t) 
x x
|x|
|x|(t) =

max{x(t), x(−t))} t  0,
0 t < 0.
(1.0.9)
y ∈ E 0 − y ∈ E −y
x α ∈ (0, 1] α α
x [x]
α
[x]
α
= {t ∈ R|x(t)  α}.(1.0.10)
α

a
α
, b
α

[x]
α
=

a
α
, b
α

a
α
∞ b
α

− ∞, b
α
 
a
α
, +∞

X ∅ λ
α
: X × X → R X × X
R α ∈ (0, 1] α
1
, α
2
∈ (0, 1] α
1
< α
2
λ
α
1
(x, y)  λ
α
2
(x, y) (x, y) ∈ X × X
X ∅ ρ
α
: X × X → R X × X
R α ∈ (0, 1] α
1
, α
2
∈ (0, 1] α
1
< α
2
ρ
α
1
(x, y)  ρ
α
2
(x, y) (x, y) ∈ X × X
X ∅ d : X × X → G X × X
G L, R : [0, 1] × [0, 1] → [0, 1] x
y L(0, 0) = 0 R(1, 1) = 1

d(x, y)

α
=

λ
α
(x, y), ρ
α
(x, y)

,(1.0.11)
x, y ∈ X λ ρ (X, d, L, R)
(i.) d(x, y) = 0 x = y,
(ii.) d(x, y) = d(y, x) x, y ∈ X,
(iii.) x, y, z ∈ X
(1.) d(x, y)(s + t)  L

d(x, z)(s), d(z, y)(t)

s  λ
1
(x, z) t  λ
1
(z, y) s + t  λ
1
(x, y)
(2.) d(x, y)(s + t)  R

d(x, z)(s), d(z, y)(t)

s  λ
1
(x, z) t  λ
1
(z, y) s + t  λ
1
(x, y)
λ
α
, ρ
α
λ
α
α ρ
α
α
d
(X, d) L R
L(a, b) = 0 a, b ∈ [0, 1] R(a, b) =

0 a = b = 0,
1 .
∆ [0, 1] × [0, 1] → [0, 1] ∆
∆(a, 1) = a, a ∈ [0, 1];
∆(a, b) = ∆(b, a), a, b ∈ [0, 1];
∆(c, d)  ∆(a, b), c  a d  b a, b, c, d ∈ [0, 1];
∆(a, ∆(b, c)) = ∆(∆(a, b), c) a, b, c ∈ [0, 1].
F : R −→ R
+
F
inf
t∈R
F (t) = 0 sup
t∈R
F (t) = 1
D
X ∅ F : X × X → D
X × X D x, y ∈ X
F
xy
= F(x, y) (X, F )
F
xy
F
xy
(t) = 1 t > 0 x = y,
F
xy
(0) = 0 x, y ∈ X,
F
xy
(t) = F
yx
(t) t ∈ R x, y ∈ X,
F
xz
(t) = 1 F
zy
(s) = 1 F
xy
(s + t) = 1 x, y, z ∈ X
X ∅ (X, F, ∆)
(X, F ) ∆ : [0, 1] ×
[0, 1] −→ [0, 1] t (x, y) ∈ X × X F
xy
F
xy
(0) = 0
(i)

t > 0 F
xy
(t) = 1 x = y,
(ii)

F
xy
= F
yx
, x, y ∈ X,
(iii)

F
xy
(s + r)  ∆

F
xz
(s), F
zy
(r)

, x, y, z ∈ X, s, r  0
L R ∆
T
1
(a, b) = max(a + b − 1, 0) (max( − 1, 0))
T
2
(a, b) = ab ( )
T
3
(a, b) = min(a, b) (min)
T
4
(a, b) = max(a, b) (max)
T
5
(a, b) = a + b − ab ( )
T
6
(a, b) = min(a + b, 1) (min( , 1))
T
i
, i = 1, 2, , 6 i  j T
i
(a, b) 
T
j
(a, b) a, b ∈ [0, 1]
2. α
α
R
x ∈ R, x(t) = 1 t = x x(t) = 0
t = x x x
s  t  r, s, r ∈ R.
x = t x(t) = 1
x(t)  min{x(s), x(r)}
x = t, x(t) = 0 x = s x = r
x = r x = s x = s x = r
x(t)  min{x(s), x(r)} = 0
x R 
x, y ∈ E (−y)(t) = y(−t)
t ∈ R x − y = x + (−y)
x α ∈ (0, 1] α [x]
α
R
x, y ∈ E [x]
α

a
α
1
, b
α
1

[y]
α

a
α
2
, b
α
2

[x + y]
α
=

a
α
1
+ a
α
2
, b
α
1
+ b
α
2

,(2.0.12)
[x.y]
α
=

a
α
1
.a
α
2
, b
α
1
.b
α
2

, x, y ∈ G,(2.0.13)
[x − y]
α
=

a
α
1
− b
α
2
, b
α
1
− a
α
2

,(2.0.14)
[1 x]
α
=

1
b
α
1
,
1
a
α
1

a
α
1
> 0,(2.0.15)

|x|

α
=

max{0, a
α
1
, −b
α
1
}, max{|a
α
1
|, |b
α
1
|}

.(2.0.16)
(X, d, L, R)
(1) t ∈ R
+
α ∈ (0, 1] d(x, y)(t)  α
λ
α
(x, y)  t  ρ
α
(x, y).
(2) t ∈ R
+
α ∈ (0, 1] d(x, y )(t) < α
λ
α
(x, y) > t ρ
α
(x, y) < t.
(1) t ∈ R
+
α ∈ (0, 1] d(x, y)(t)  α
t ∈ [d(x, y)]
α
λ
α
(x, y)  t 
ρ
α
(x, y).
(2) t ∈ R
+
α ∈ (0, 1] d(x, y)(t) < α.
t ∈ [d(x, y)]
α
t ∈ [λ
α
(x, y), ρ
α
(x, y)] λ
α
(x, y) > t
ρ
α
(x, y) < t. 
d(x, y) ∈ G
x, y ∈ X G
0  d(x, y)(t)  1 x, y ∈ X t  0
3.
(X, d, L, R) L R
T
i
, i = 1, 2, , 6 F : X × X → G
F(x, y)(t) =

0
1 − d(x, y)(t) =
(X, F)
F
F
xy
(t) = 1, t > 0 d(x, y)(t) = 0 t > 0
d(x, y) =
0
x = y. F
xy
(t) = 1 t > 0 x = y
F
xy
(0) = 0, x, y ∈ X.
x, y ∈ X t ∈ R
F
xy
(t) = 1 − d(x, y)(t) = 1 − d(y, x)(t) = F
yx
(t).
F
xy
(t) = F
yx
(t), t ∈ R x, y ∈ X.
t, s ∈ R, x, y, z ∈ X F
xz
(t) = 1 F
zy
(s) = 1
d(x, z)(t) = 0 d(z, y)(s) = 0.
d(x, y)(t + s)  L

d(x, z)(t), d(z, y)(t)

= L(0, 0) = 0,
t  λ
1
(x, z), s  λ
1
(z, y) t + s  λ
1
(x, y).
d(x, y)(t + s)  R

d(x, z)(t), d(z, y)(t)

= R(0, 0) = 0,
t  λ
1
(x, z), s  λ
1
(z, y) t + s  λ
1
(x, y).
d(x, y) ∈ G d(x, y)(t + s) = 0, t, s ∈ R F
xy
(t + s) = 1
F
xy
(t) = 1 F
yz
(t) = 1 F
xz
(t + s) = 1 t, s ∈ R
(X, F) 
(X, F) d : X × X → G
X × X G
d(x, y)(t) = 1 − F
xy
(t), t ≥ 0 d(x, y)(t) = 0 t < 0.
(X, d, L, R) L R
L(a, b) = 0 a, b ∈ [0, 1] R(a, b) =

0
1
d
d(x, y) = 0 d(x, y)(t) = 0, t > 0
F
xy
(t) = 1, t > 0 x = y. x = y F
xy
(t) = 1
t > 0 d(x, y)(t) = 0 t > 0 F
xy
(0) = 0 d(x, y)(0) = 1
d d(x, y ) = 0. d(x, y) = 0 x = y.
d(x, y) = d(y, x), x, y ∈ X.
x, y, z ∈ X L(a, b) = 0 a, b ∈ [0, 1]
d(x, y)(t + s)  0 = L(d(x, z)(t), d(z, y)(s)), s, t.
d(x, y)(t + s)  L(d(x, z)(t), d(z, y)(s)),
s  λ
1
(z, y), t  λ
1
(x, z) s + t  λ
1
(x, y).
s  λ
1
(z, y), t  λ
1
(x, z) s + t  λ
1
(x, y) d(x, z)(t) > 0
d(z, y)(s) > 0 d(x, y)(t + s) > 0 R(a, b) =

0
1
d(x, y)(t + s)  R(d(x, z)(t), d(z, y)(s)),
s  λ
1
(z, y), t  λ
1
(x, z) s + t  λ
1
(x, y).
(X, d, L, R) 
(X, F, ∆) d : X × X → G X × X
G
d(x, y)(t) =

0 t < t
xy
1 − F
xy
(t) t  t
xy
t
xy
= sup{t|F
xy
(t) = 0} (X, d, L, R)
L, R L(a, b) = 0 R(a, b) = 1 − ∆(1 − a, 1 − b), a, b ∈ [0, 1].
(X, d, L, R)
F
xy
d(x, y) ∈ G
x, y ∈ X L, R L, R : [0, 1]×[0, 1] → [0, 1]
x y L(0, 0) = 0 R(1, 1) = 1
d(x, y) = 0 d(x, y)(t) = 0, t = 0 d(x, y)(0) = 1
F
xy
(t) = 1, t > 0 0 = t
xy
x = y
x = y F
xy
(t) = 1 t > 0 t = 0 F
xy
(0) = 0
t
xy
= 0 d(x, y)(t) = 0 t = 0 d(x, y)(0) = 1 d(x, y) = 0
x = y.
x, y ∈ X t ∈ R F F
xy
(t) = F
yx
(t)
d(x, y) = d(y, x), x, y ∈ X.
x, y ∈ X
d(x, y)(t + s)  0 = L

d(x, z)(t), d(z, y)(t)

, t, s ∈ R.
x, y, z ∈ X t, s  t
xy
≥ 0
∆(1 − d(x, z)(t), 1 − d(z, y )(s)) = ∆(F
xz
(t), F
zy
(s))  F
xy
(t + s)
⇐⇒ 1 − ∆(1 − d(x, z)(t), 1 − d(z, y)(s))  1 − F
xy
(t + s)
⇐⇒ d(x, y)(t + s)  R(d(x, z)(t), d(z, y)(s)), x, y ∈ X.
s  λ
1
(z, y), t  λ
1
(x, z) s + t  λ
1
(x, y)  t
xy
d(x, y)(t + s)  R(d(x, z)(t), d(z, y)(s)).
(X, d, L, R) 
(X, d, L, R)
lim
t→+∞
d(x, y)(t) = 0, x, y ∈ X R(1, a) = R(a, 1) = 1, a ∈ [0, 1]
F
F
xy
(t) =

0 t < λ
1
(x, y),
1 − d(x, y)(t) t  λ
1
(x, y).
∆ ∆(a, b) = 1 − R(1 − a, 1 − b) a, b ∈ [0, 1] (X, F, ∆)
∆ t
d(x, y) ∈ G d(x, y)(t) F
xy
F
xy
F
xy
F
xy
(t) = 1 t > 0 d(x, y)(t) = 0 t > 0
d(x, y)(0) = 1 d(x, y)(t) = 0 t < 0 d(x, y) = 0
x = y
F
xy
(t) = 1 t > 0 x = y.
F
xy
(t) = F
yx
(t) t ∈ R
x, y, z ∈ X s, t  0 d, R

F
xy
(t + s) = 1 − d(x, y)(t + s)  1 − R

d(x, z)(t), d(z, y)(s)

= 1 − R

1 − (1 − d(x, z)(t)), 1 − (1 − d(z, y)(s))

= 1 − R

1 − F
xz
(t), 1 − F
zy
(s)

= ∆

F
xz
(t), F
zy
(s)

.
F
xy
(t + s)  ∆

F
xz
(t), F
zy
(s)

x, y, z ∈ X s, t  0
(X, F, ∆) 
124 8
12
α

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