Biomedical Engineering Reference
In-Depth Information
character recognition.
∗
In the following section, we describe the basic prop-
erties of fuzzy sets and their extension to fuzzy logic applications. Readers
interested in further details can consult many other resources on fuzzy theory
and applications (Zadeh, 1965, 1973; Zadeh et al., 2002; Brubaker, 1992).
3.5.1 Fuzzy Sets
A fuzzy set is defined as a group or collection of elements with membership
values between 0 (completely not in the set) and 1 (completely in the set).
The membership values give the property of fuzziness to elements of a fuzzy
set in that an element can partially belong to a set. The following definition
formalizes the fuzzy set.
DEFINITION 3.5.1
(Fuzzy Set [Zadeh, 1965])
A fuzzy set is character-
ized by a membership function, which maps the elements of a domain, space,
or an universe of discourse
X
to the unit interval
[0
,
1]
, written as
A
=
X
→
[0
,
1]
Hence a fuzzy set A is represented by a set of pairs consisting of the element
x
∈
X
and the value of its membership and written as
A
=
{
m
(
x
)
|
x
∈
X
}
A crisp set is our standard set where membership values take either 0 or 1.
Fuzzy sets can be defined on finite or infinite universes where when the uni-
verse is finite and discrete with cardinality
n
, the fuzzy set is an
n
-dimensional
vector with values corresponding to the membership grades and the position
in the vector corresponding to the set elements. When the universe is contin-
uous, the fuzzy set may be represented by the integral
A
=
x
m
(
x
)
/x
(3.60)
where the integral shows the union of elements not the standard summation
for integrals. The quantity
m
(
x
)
/x
denotes the element
x
with membership
grade
m
(
x
).
An example of a fuzzy set is the set of points on a real line (Pedrycz and
Vasilakos, 2001). Their membership functions can be described by a delta
function where for
x, p
∈
, the membership function is
p
)=
1 f
x
=
p
0 f
x
=
p
m
(
x
)=
δ
(
x
−
(3.61)
∗
http://austinlinks.com/Fuzzy/
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