Information Technology Reference
In-Depth Information
Based on these known definitions, let us define operations for set
elements,
k
Ξ
which form semantic space with term-sets member
shi
p fun
ctio
ns.
With intersection of two elements
X
∩
X
;
;
i
=
1
k
j
=
1
k
i
j
{
}
[
]
()
()
()
μ
x
=
min
μ
x
,
μ
x
;
∀
x
;
l
=
1
m
.
l
il
jl
With intersection of elements
X
∩
...
∩
X
1
k
[
()
()
]
{
}
min
μ
x
,...,
μ
x
;
∀
x
.
1
l
kl
i
=
1
k
X
∪
X
With union of two elements
i
j
[
]
⎩
⎨
⎧
⎭
⎬
⎫
()
()
()
μ
x
=
max
1
μ
x
,
μ
x
;
∀
x
.
l
il
jl
i
=
k
X
∪ ...
∪
X
With union of elements
1
k
⎩
⎨
⎧
⎭
⎬
⎫
()
[
()
()
]
μ
x
=
max
μ
x
,...,
μ
x
;
∀
x
.
l
1
l
kl
i
=
1
k
X
+
X
With generalized sum of two elements
i
j
{
()
(
)
}
il
jl
il
jl
il
L
jl
L
il
R
jl
R
μ
x
≡
a
+
a
,
a
+
a
,
a
+
a
,
a
+
a
.
l
1
1
2
2
X
with a positive number
c
for COSS
With generalized product of an element
{
()
(
)
}
il
il
il
L
il
R
μ
x
≡
ca
,
ca
,
ca
,
ca
.
l
1
2
k
With generalized linear combination
∑
=
,
for COSS
c
i
X
c
>
0
i
i
i
1
⎧
⎫
⎛
k
k
k
k
⎞
()
∑
∑
∑
∑
il
il
il
L
il
R
μ
x
≡
⎜
⎝
c
a
,
c
a
,
c
a
,
c
a
⎟
⎠
.
⎩
⎨
⎭
⎬
l
i
1
i
2
i
i
i
=
1
i
=
1
i
=
1
i
=
1
Let us assume that semantic space
Y
with membership functions of term-set
()
η
x
belongs to semantic space
Z
with membership functions of term-set
()
l
v
l
x
, if the following conditions are satisfied
()
()
≤η
Let us define the quantity indexes characterizing distinctions or
sim
ilarities of two
elements of set
x
v
x
,
∀
l
=
1
m
.
l
l
()
{
()
}
{
}
μ
x
k
with membership functions
;
for
Ξ
μ
x
,
l
=
1
m
jl
il
[
]
[]
universal set
, then it is
necessary to primarily reduce parameters of membership functions of elements to
[0.1] by the formula
. If universal set is a segment
U
=
a
,
b
U
=
0
⎛
il
il
il
L
il
R
⎞
a
−
a
a
−
a
a
a
()
μ
x
≡
⎜
⎜
⎝
1
;
2
;
;
⎟
⎟
⎠
.
il
b
−
a
b
−
a
b
−
a
b
−
a
