Information Technology Reference
In-Depth Information
(μ
∨
˃)(
)
=
(
(μ(
), ˃(
)),
(μ(
), ˃(
)),
(μ(
), ˃(
)),
-
3
max
min
1
3
min
3
1
min
2
3
(μ(
), ˃(
)),
(μ(
), ˃(
)))
=
(
(
.
,
.
),
(
,
.
),
min
3
2
min
3
3
max
min
0
8
0
6
min
1
0
9
min
(
0
.
7
,
0
.
6
),
min
(
1
,
1
),
min
(
1
,
0
.
6
))
=
1,
that is
μ
∨
˃
=
0
.
8
|
1
+
0
.
8
|
2
+
1
|
3.
This fuzzy set is different from
μ
+
˃
=
0
.
9
|
1
+
1
|
2
+
1
|
3, with
+
the t-conorm
max
.
Remark 6.3.2
It is easy to check that, although
it is not
μ
≤
˃
pointwise,
it is
μ
∧
˃
≤
∗
μ
∨
˃
.
Remark 6.3.3
Since
t
=
min
(
x
,
y
)
means
•
t
=
x
,if
x
≤
y
,
•
t
=
y
,if
y
≤
x
,
it is immediate that
(μ
∧
˃)(
t
)
=
max
[
Sup
t
min
(μ(
x
), ˃(
t
)),
Sup
t
min
(μ(
t
), ˃(
y
))
]
,
≤
x
≤
y
a formula that facilitates to obtain
μ
∧
˃
,given
μ
and
˃
. Analogously, since
t
=
max
(
x
,
y
)
means
•
t
=
x
,
if
y
≤
x
,
•
t
=
y
,
if
y
≥
x
,
there is a similar formula for
μ
∨
˃
.
For example, in the case of
μ
and
˃
given in the figure
it results:
1. If
t
≤
a
1
,(μ
∧
˃)(
t
)
=
0
2. If
a
2
≤
t
,(μ
∧
˃)(
t
)
=
0
3. If
a
1
≤
t
≤
b
1
,(μ
∧
˃)(
t
)
=
μ(
t
)
4. If
b
1
≤
t
≤
c
,(μ
∧
˃)(
t
)
=
˃(
t
)
5. If
c
≤
t
≤
a
2
,(μ
∧
˃)(
t
)
=
μ(
t
)
.
μ
≤
∗
˃
Hence,
μ
∧
˃
=
μ
, and
, although it is not
μ
≤
˃
.
Search WWH ::
Custom Search