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In-Depth Information
Theorem of Decomposition
[40]. Any fuzzy set
~
with membership function
()
μ
x
can be decomposed on sets of
α
-level
~
~
A
=
∈
∪
α
A
α
[]
α
0
,
(1.11)
or
[
]
()
()
μ
x
=
α∈
∪
αμ
x
,
~
A
A
[]
0
,
α
()
()
1
μ
x
≥
α
;
⎩
⎨
⎧
()
A
μ
x
=
where
A
0
μ
x
<
α
.
α
A
According to (1.11), the fuzzy binary relation
R
can be presented as
R
=
∈
∪
α
R
α
[]
α
0
,
or
[
]
()
()
μ
x
=
α ∈
∪
αμ
x
;
R
R
[]
0
,
α
()
()
1
μ
x
≥
α
;
⎩
⎨
⎧
()
R
μ
x
=
.
where
R
0
μ
x
<
α
.
α
R
A reflective symmetric fuzzy binary relation is referred to as fuzzy binary
relation of similarity.
Fuzzy
binary
relation
R
is
referred
to
as
transitive,
if
(
)
(
)
(
)
μ
x
,
z
≥
μ
x
,
y
∧
μ
y
,
z
∀
x
,
y
,
z
∈
X
.
A transitive fuzzy binary relation of similarity is referred to as fuzzy binary
relation of conformity.
In actual practice the transitivity requirement is often difficult to meet. In order
to use expert survey for the purpose of constructing the similarity relation, the
transitive answers shall be demanded from these experts. Numerous practical
outcomes [16] are quite opposite: real outcomes of expert surveys are often
intransitive. However, in applications of fuzzy relations the transitive ones are of
great importance, because they possess many convenient properties and define
some correct structure of a set which they are set for. For example, if relation
R
over a set
X
characterizes similarity between objects, then transitivity of such
relation (the similarity relation) ensures a possibility of a partition of set
X
on
disjoint similarity classes (clusters).
Let
R
be similarity relation. Then according to the theorem of decomposition
for relations of similarity [15]
R
R
R
(
)
R
=
∪
α
α
×
R
,
α
(1.12)
(
)
α
,
if
R
x
,
y
=
1
⎩
⎨
⎧
(
)(
)
α
α
×
R
x
,
y
=
where
α
0
otherwise.
