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1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
15
20
25
30
35
Age
Fig. 14.2 Membership function for the crisp young set.
objects and therefore either many or few objects in the set may have high
membership. However, an objects membership in a set (such as “young”)
and the sets complement (“not young”) must still sum to 1.
The main difference between classical set theory and fuzzy set theory
is that the latter admits to partial set membership. A classical or crisp
set, then, is a fuzzy set that restricts its membership values to
,
the endpoints of the unit interval. Membership functions can be used to
represent a crisp set. For example, Figure 14.2 presents a crisp membership
function defined as:
µ
CrispY oung
(
u
)=
0
{
0
,
1
}
age
(
u
)
>
22
22
.
(14.2)
1
age
(
u
)
≤
14.3 Fuzzy Classification Problems
All classification problems we have discussed so far in this chapter assume
that each instance takes one value for each attribute and that each instance
is classified into only one of the mutually exclusive classes
[
Yuan and Shaw
(1995)
]
.
To illustrate the idea, we introduce the problem of modeling the
preferences of TV viewers. In this problem, there are three input attributes:
A
=
{
Time of Day, Age Group, Mood
}
and each attribute has the following values:
•
dom
(Time of Day) =
{
Morning
,
Noon
,
Evening
,
Night
}
•
dom
(Age Group) =
{
Young
,
Adult
}
•
dom
(Mood) =
{
Happy
,
Indifferent
,
Sad
,
Sour
,
Grumpy
}
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