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14.2 Membership Function
In classical set theory, a certain element either belongs or does not belong to
a set. Fuzzy set theory, on the other hand, permits the gradual assessment
of the membership of elements in relation to a set.
Definition 14.1.
Let
U
be a universe of discourse, representing a col-
lection of objects denoted generically by
u
. A fuzzy set
A
in a universe
of discourse
U
is characterized by a membership function
µ
A
which takes
values in the interval [0, 1], where
µ
A
(
u
)=0meansthat
u
is definitely not
amemberof
A
and
µ
A
(
u
)=1meansthat
u
is definitely a member of
A
.
The above definition can be illustrated on a vague set, that we will label
as
young
. In this case, the set
U
is the set of people. To each person in
U
,we
define the degree of membership to the fuzzy set
young
.Themembership
function answers the question: “To what degree is person
u
young?”. The
easiest way to do this is with a membership function based on the person's
age. For example, Figure 14.1 presents the following membership function:
0
age
(
u
)
>
32
1
age
(
u
)
<
16
µ
Young
(
u
)=
.
(14.1)
32
−
age
(
u
)
16
otherwise
Given this definition, John, who is 18 years old, has degree of youth of
0
.
875. Philip, 20 years old, has degree of youth of 0
.
75. Unlike probability
theory, degrees of membership do not have to add up to 1 across all
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. 14.1 Membership function for the young set.
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