50
B. Uzun et al.
Intersection:
μ ˜
A∩ ˜
B (x) = μ ˜
A ∧ μ ˜
B , ∀x ∈ U
(5)
∧ represents the ‘min’ operation.
Complement:
μ
( ˜
A)
(x) = 1 − μ ˜
A (x)
(6)
There may be some cases there;
˜
A ∩ ˜
A
= 0
8.2.3 Fuzzy Sets Features
Fuzzy sets are described as sets that contain variables with similar values or
membership. The fuzzy sets are characterized by different properties, which are
as follows:
(a) Commutativity:
The Commutativity property involves two variables and relating them together by
analyzing them. Many systems contain the commutativity property; a clear example
is the case of two or more inputs of a similar character. This involves fuzzy set ˜
A
and ˜
B, and states that:
˜
A ∪ ˜
B = ˜
B ∪ ˜
A
(7)
˜
A ∩ ˜
B = ˜
B ∩ ˜
A
(8)
(b) Associativity
This involves a particular mathematical property of a derived binary operation which
cannot have effect on the given result. This involves fuzzy sets ˜
A, ˜
B and ˜
C, and
states that:
˜
A ∪
˜
B ∪ ˜
C
=
˜
A ∪ ˜
B
∪ ˜
C
(9)
˜
A ∩
˜
B ∩ ˜
C
=
˜
A ∩ ˜
B
∩ ˜
C
(10)