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

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,


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


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)


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

(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]

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

,(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

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

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

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 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

∆ ∆(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

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