Foundations of Fuzzy Sets Theory
2.1.
Basic Concepts and Properties of Fuzzy Sets
2.1.1.
Basic types of fuzzy sets
Definition. Fuzzy sets
Let X be a classical set of objects, called the universe, whose generic elements are denoted x.
Membership in a classical subset A of X is often viewed as a characteristic function
μ
A
from X to {0,1}
such that
μ
A
(x) =
⎩
⎨
⎧
∉
∈
A
x
if
0
A
x
if
1
where {0,1} is called a valuation set; 1 indicates membership while 0 - non-membership.
If the valuation set is allowed to be in the real interval [0,1], then A is called a fuzzy set [1-6].
μ
A
(x) is
the grade of
membership of x in A
]
1
,
0
[
X
:
A
→
μ
.
As closer the value of
μ
A
(x) is to 1, so much x belongs to A.
A is completely characterized by the set of pair.
}
X
x
)),
x
(
,
x
{(
A
A
∈
μ
=
.
Example
A=0.1/1+0.3/2+0.5/3+0.7/4+0.8/5+0.9/6+0.95/7+1.0/8+0.97/9+
+0.9/10+0.8/11+0.7/12+0.5/13+0.3/14+0.1/15.
A is a fuzzy set. A classical version of this fuzzy set is
A={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
or
A=1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12+1/13+
+1/14+1/15.
For graphical representation see Figure 2.1.
Figure 2.1. Fuzzy set
0.5
1.0
5
10
15
20
x
)
x
(
A
μ