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measures, as it is, for example, the number of apples in a basket. But, how can the
concept of measure be formalized?
7.2 The Concept of a Measure
Given a set
X
, and provided
X
• F ↂ[
0
,
1
]
is a family of fuzzy subsets of
X
, such that
μ
0
, μ
1
∈ F
, and that
is
a preorder in
F
translating a qualitative binary relation between the elements in
F
,
•
(
L
,
)
is a preordered set with first element 0,
we will say that
m
: F ₒ
L
is a
(
L
,
)
-measure, whenever
1.
m
(μ
0
)
=
0
2. If
μ
˃
, then
m
(μ)
m
(˃)
.
X
Example 7.2.1
Take
F =[
0
,
1
]
,(
L
,
)
=
(
[
0
,
1
]
,
)
the unit interval with the
partial linear order
of the real line, and the qualitative relation “
μ
is less fuzzy than
˃
”, translated by the so-called sharpened order
S
(
=
)
defined by
μ(
x
)
˃(
y
)
, f
˃(
x
)
1
/
2
μ
S
˃
⃔
˃(
x
)
μ(
x
)
, f
˃(
x
)>
1
/
2
,
that is a reflexive, transitive and antisymmetric, crisp relation. The fuzzy set
μ
0
.
5
is
X
are minimals in
the highest one, and all crisp sets
μ
∈{
0
,
1
}
(
F
,
S
)
. Any mapping,
X
m
:[
0
,
1
]
ₒ[
0
,
1
]
such that
•
If
μ
is crisp, then
m
(μ)
=
0
•
m
(μ
0
.
5
)
=
1
•
If
μ
S
˃
, then
m
(μ)
m
(˃)
,
X
. These measures
is a
(
[
0
,
1
]
,
)
-measure since
m
(μ
0
)
=
0 because of
μ
0
∈{
0
,
1
}
are called
measures of fuzziness
,or
fuzzy entropies
.
If
X
={
x
1
,...,
x
n
}
is finite, the following mappings are examples of fuzzy
entropies:
1
1.
m
(μ)
=
1
−
2
max
1
i
n
|
μ(
x
i
)
−
2
|
2.
m
(μ)
=
2
max
1
i
n
μ(
x
i
)
·
(
1
−
μ(
x
i
))
i
=
1
˃(μ(
3.
m
(μ)
=
x
i
))
, with
˃(
x
)
=
x
ln
x
−
(
1
−
x
)
ln
(
1
−
x
)
(logarithmic
entropy).
1, if
2
n
1
=
1
|
μ(
μ(
x
)>
0
.
5
1
4.
m
(μ)
=
x
i
)
−
μ
C
μ
(
x
i
)
|
, with
μ
C
μ
(
x
)
=
5
,the
0, if
μ(
x
)
0
.
closest crisp set to
μ
(linear index of fu
zziness)
2
n
i
=
1
(μ(
1
5.
m
(μ)
=
x
i
)
−
μ
C
μ
(
x
i
))
2
(quadratic index of fuzziness).
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