Information Technology Reference
In-Depth Information
Remark 1.3.4
It should be noticed that what has been presented is sufficient but not
necessary for the representation of
μ
PandQ
and
μ
Por Q
b means of
μ
P
and
μ
Q
.
P
)
,
Given
P
, if with M for
medium
,
MP
=
Not P
and
Not aP
=
and
(
aP
with a negation function
N
for
not
, and a symmetry
A
for the opposite, and
∗
for
and
, results
μ
MP
(
x
)
=
μ
P
and
(
aP
)
(
x
)
=
μ
P
(
x
)
∗
μ
(
aP
)
(
x
)
=
N
(μ
P
(
x
))
∗
N
(μ
P
(
A
(
x
))),
for all
x
in
X
.
1.4 Qualified, Modified, and Constrained Predicates
1.4.1 Qualified Predicates
Let
P
be a predicate on
X
, with L-degree
μ
P
:
X
ₒ
L
, and
˄
a predicate on
μ
P
(
X
)
ↂ
L
, with L-degree
μ
˄
:
μ
P
(
X
)
ₒ
L
. Suppose
ↂ
˄
, and consider the
qualified predicate
'
P
is
˄
',
'
x
is
(
P
is
˄ )
'
:=
x
is
P
is
˄ ,
∅ =
Pis
˄
ↂ
P
. On these conditions,
provided
μ
Pis
˄
=
μ
˄
ⓦ
μ
P
is an L-degree for '
P
is
˄
'in
X
, since:
x
Pis
˄
y
⃒
x
P
y
⃒
μ
P
(
x
)
μ
P
(
y
)
⃒
μ
P
(
x
)
˄
μ
P
(
y
)
⃒
μ
˄
(μ
P
(
x
))
μ
˄
(μ
P
(
y
)),
that is,
(μ
˄
ⓦ
μ
P
)(
x
)
(μ
˄
ⓦ
μ
P
)(
y
).
=
−
1
, and
For example, with
L
=[
0
,
1
]
,
P
=
small
in
[
0
,
10
]
, with
P
x
μ
P
(
x
)
=
1
−
10
,if
˄
=
large
in
[
0
,
1
]
is with
˄
=
, and
0
,
if
0
x
0
.
5
μ
˄
(
x
)
=
1
,
if
0
.
5
x
1
,
it results
0
,
if
5
x
<
10
μ
˄
ⓦ
μ
P
(
x
)
=
1
,
if
0
x
<
5
,
that allow the interpretation of
small is large
as
less than five
.
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