Information Technology Reference
In-Depth Information
μ
small
(
)
=
−
μ
not small
(
)
=
Let's take
x
1
x
,
x
x
.Itis:
•
ˀ
μ
(
Fissmall
)
=
Sup
x
∈[
min
(μ(
x
), μ
small
(
x
))
=
Sup
x
∈[
min
(μ(
x
),
1
−
x
)
=
0
,
1
]
0
,
1
]
1
/
2
⇒
N
ˀ
μ
(
Fissmall
)
=
1
−
ˀ
μ
(
Fisnotsmall
)
•
ˀ
μ
(
Fisnotsmall
)
=
Sup
min
(μ(
x
), μ
not small
(
x
))
=
Sup
min
(μ(
x
),
x
)
=
x
∈[
0
,
1
]
x
∈[
0
,
1
]
1
⇒
N
ˀ
μ
(
Fissmall
)
=
1
−
ˀ
μ
(
Fissmall
)
.
Hence:
-
ˀ
μ
(
Fissmall
)
=
0
.
5, and
N
ˀ
μ
(
Fissmall
)
=
1
−
1
=
0.
-
ˀ
μ
(
Fisnotsmall
)
=
1, and
N
ˀ
μ
(
Fisnotsmall
)
=
1
−
0
.
5
=
0
.
5.
Remark 7.6.3
Notice that this example is, like the following, not with questions
related to precise or crisp sets, but to impreciseness (fuzzy sets). Although Possibility
Theory is introducedwith crisp sets, it is also applicable to fuzzy setswithin the theory
given by the triplet
(
min
,
max
,
1
−
id
)
.
Example 7.6.4
John is a member of a community where the predicate
Y
=
young
is used following
⊧
⊨
1,
if
x
30
μ
Y
(
0,
if
x
>
40
x
)
=
⊩
40
−
x
, f 0
<
x
40
10
Find the possibility and the necessity of the statement “John is around 35 years
old”.
Solution.
The graphics of
μ
Y
and
μ
P
, with
P
=
Ar ound
35, are
Hence,
ˀ
μ
(
John is around 35 years old
)
=
Sup
x
∈[
0
,
100
]
min
(μ
Y
(
x
), μ
P
(
x
))
=
2
3
, since it does correspond with the intersection of
μ
Y
and
μ
P
, that is, of the straight
40
−
x
x
−
3
100
3
2
lines respectively given by
y
=
,
y
=
.Itresults
x
=
and
y
=
3
. Since
10
5
ˀ
μ
(
John is around 35 years old
)<
1, it results
N
ˀ
μ
(
John is around 35 yearsold
)
=
0.
Search WWH ::
Custom Search