Information Technology Reference
In-Depth Information
1.3 AND/OR
Let
P
,
Q
be two predicates in
X
. Consider the new predicates '
P
and
Q
', and '
P
or
Q
', used by means of:
•
'
x
is
P
and
Q
'
⃔
'
x
is
P
'
and
'
x
is
Q
'
Not ('
x
is
P
'
and
'
x
is
•
'
x
is
P
or
Q
'
⃔
Not (Not '
x
is
P
'
and
Not '
x
is
Q
')
⃔
x
is '(
P
and Q
)',
with respective primary uses
Q
')
⃔
PandQ
ↂ
P
∩
Q
, and
Por Q
. Take L-degrees
μ
P
, μ
Q
.
1.3.1 AND
Given
(
L
,
)
,let
∗:
L
×
L
ₒ
L
be an operation, verifying the properties
•
a
b
,
c
d
⃒
a
∗
c
b
∗
d
•
a
∗
c
a
,
a
∗
c
c
,
Then,
μ
P
(
x
)
∗
μ
Q
(
x
)
is an L-degree for
P
and
Q
in
X
, since:
x
PandQ
y
⃒
x
P
yandx
Q
y
⃒
μ
P
(
x
)
μ
P
(
y
)
and
μ
Q
(
x
)
μ
Q
(
y
)
⃒
μ
P
(
x
)
∗
μ
Q
(
x
)
μ
P
(
y
)
∗
μ
Q
(
y
)
.
Hence, defining
μ
P
(
x
)
∗
μ
Q
(
x
)
=
μ
PandQ
(
x
)
,anL-degreefor'
P
and
Q
'in
∗
X
is obtained, and the operation
can be called an
and-operation
or a conjunction.
Notice that it is,
μ
PandQ
(
x
)
μ
P
(
x
),
and
μ
PandQ
(
x
)
μ
Q
(
x
).
In the case
(
L
,
)
is a lattice (see Sect.
1.6
) with the minimum operation
·=
min
,
and
∗ ·
,itimplies
μ
PandQ
(
x
)
μ
P
(
x
)
·
μ
Q
(
x
).
Then, if
(
L
,
·
,
+
,
)
is a lattice, then the
and
-operation is at least
∗=·
(min), with
which we have the L-degree
μ
PandQ
(
x
)
=
μ
P
(
x
)
·
μ
Q
(
x
),
∀
x
∈
X
.
1.3.2 OR
If
∗
is an and-operation, and
N
:
L
ₒ
L
is a strong negation, define
a
↕
b
=
N
(
N
(
a
)
∗
N
(
b
)),
for all
a
,
b
∈
L
.
Since
a
b
,
c
d
⃒
N
(
b
)
N
(
a
),
N
(
d
)
N
(
c
)
⃒
N
(
b
)
∗
N
(
d
)
(
)
∗
(
)
⃒
(
(
)
∗
(
))
(
(
)
∗
(
))
↕
↕
N
a
N
c
N
N
a
N
c
N
N
b
N
d
, it results
a
c
b
d
.
Search WWH ::
Custom Search