Database Reference
In-Depth Information
Definition 14.3.
The membership function of the intersection of two
fuzzy sets
A
and
B
with membership functions
µ
A
and
µ
B
respectively
is defined as the minimum of the two individual membership functions
µ
A∩B
(
u
)=
min
{
µ
A
(
u
)
,µ
B
(
u
)
}
.
Definition 14.4.
The membership function of the complement of a fuzzy
set
A
with membership function
µ
A
is defined as the negation of the
specified membership function
µ
A
(
u
)=1
−
µ
A
(
u
).
To illustrate these fuzzy operations, we elaborate on the previous
example. Recall that John has a degree of youth of 0
.
875. Additionally,
John's happiness degree is 0
.
254. Thus, the membership of John in the set
Young
∪
Happy would be
max
(0
.
875
,
0
.
254) = 0
.
875, and its membership
in Young
Happy would be
min
(0
.
875
,
0
.
254) = 0
.
254.
It is possible to chain operators together, thereby constructing quite
complicated sets. It is also possible to derive many interesting sets from
chains of rules built up from simple operators. For example, John's
membership in the set
Young
∩
∪
Happy would be
max
(1
−
0
.
875
,
0
.
254) =
0
.
254.
The usage of the max and min operators for defining fuzzy union and
fuzzy intersection, respectively is very common. However, it is important to
note that these are not the only definitions of union and intersection suited
to fuzzy set theory.
14.5 Fuzzy Classification Rules
Definition 14.5.
The fuzzy subsethood
S
(
A, B
) measures the degree to
which
A
is a subset of
B
.
S
(
A, B
)=
M
(
A
B
)
M
(
A
)
∩
,
(14.3)
where
M
(
A
)isthe
cardinality
measure of a fuzzy set
A
andisdefinedas
M
(
A
)=
u∈U
µ
A
(
u
)
.
(14.4)
The subsethood can be used to measure the truth level of the rule
of classification rules. For example, given a classification rule such as “IF
Age is Young AND Mood is Happy THEN Comedy” we have to calculate
S
(
Hot
Sunny, Swimming
) in order to measure the truth level of the
classification rule.
∩
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