Databases Reference
In-Depth Information
∀
a
∈
U
,andthat
P
B∩A
(
a
)
≤
P
B
(
a
)
,
∀
a
∈
U
.Thus,
P
A∩B
(
a
)=
min
{
P
A
(
a
)
,P
B
(
a
)
}
.
It
further
follows
from
(16)
that
P
B|A
(
a
)=
P
A∩B
(
a
)
/P
A
(
a
)=
min
{
P
A
(
a
)
,P
B
(
a
)
}
/P
A
.Since
P
A
(
a
)
≤
P
B
(
a
), we
have
min
{
P
A
(
a
)
,P
B
(
a
)
}
=
P
A
(
a
)and
P
B|A
(
a
)=1.
It is straightforward that the above Lemma still holds by exchanging A
and B.
The
Z
−
-system may still work approximately even when the condition
of Theorem 1 does not hold exactly. This analysis explains some successful
applications of the Z-system, e.g., those in control systems.
3.2 Examples: Z-System and B-System Fail to Work Well
In addition to the above cases, the “MIN-MAX” operations may not yield
reasonable results. Let us to observe the following examples.
Example 1.
Consider a set for youth
B
and a set for juvenile
A
. It is partially
true to regard a person
a
= 17-year-old who is 17 years old as a youth (e.g., a
degree of 0
.
7), but it is also partially true to regard this person as a juvenile
(e.g., a degree 0
.
4). On the other hand, based on common sense, we can
be 100% sure that this person belongs to the union set
C
of youths and
juveniles. However, according to the
Z
−
-system by (7), we have
µ
B
(17) =
0
.
7
,µ
A
(17) = 0
.
4, and
µ
C
(17) = max
=0
.
7. In a contrast, according
to the P-theory, by (15) we have
P
B
(17) = 0
.
7
,P
A
(17) = 0
.
4, and
P
C
(17) =
0
.
7+0
.
4
{
0
.
7
,
0
.
4
}
1, which becomes 1 if we have additional information
that
P
A∩B
(17) = 0
.
1. In other words, the P-theory can give a result that is
consistent with our common sense, but the
Z
−
-system does not.
For the B-system given by (11), we also get
µ
C
(17) = min
−
P
A∩B
(17)
≤
{
0
.
7+0
.
4
,
1
}
=1
correctly. However, for the cases where
A
A
, the B-system can
not give a reasonable result, while the
Z
−
-system works well, as illustrated
by the following example:
⊆
B
or
B
⊆
Example 2.
Again, based on common sense, saying
B=
−
years − old is a youth}
is only partially true with a degree of 0
.
2 and say-
ing
C={ a person who is either a youth or a juvenile}
is also partially true
with a degree of 0
.
2. Obviously,
C
{
a person a
=30
B
=
C
still represents a 0
.
2degree
of truth. Based on the
Z
−
-system, we can correctly calculate
µ
C∨B
(30) =
max
∨
=0
.
2. According to the P-
theory, by (15) we can also correctly calculate
P
C∨B
(30) = 0
.
2+0
.
2
{
0
.
2
,
0
.
2
}
=0
.
2
,µ
C∧B
(30) = min
{
0
.
2
,
0
.
2
}
0
.
2=0
.
2
and
P
C∧B
(30) = 0
.
2. However, according to the B-system, we get
µ
C∨B
(30) =
min
−
{
0
.
2+0
.
2
,
1
}
=0
.
4and
µ
C∧B
(30) = max
{
0
.
2+0
.
2
−
1
,
0
}
=0,bothof
which are incorrect.
There are also cases where both the B-system by (11) and the
Z
−
-system
by (7) fail to produce a reasonable result. The following is an example.
