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Transcript
where Χ is the universe, µ A ( x ) represents the membership function and µ: Χ → [ 0,1] . For
example, one could define a membership function for the set of numbers much less than 100
as follows:
µ <<100 ( x ) =
1
x2
1+
100
Typically, the following definitions of fundamental logical operations (intersection, union
and complement) on sets are used:
µ
A∩ B
µ
( x ) = min( µ ( x ), µ ( x ))
A
B
(2)
( x ) = max( µ ( x ), µ ( x ))
A
B
(3)
µ ( x) = 1 − µ (x)
A
A
(4)
A∪ B
The above operators satisfy certain desired properties. For example, if set
containment is defined as A ⊆ B if ∀x ∈ X µ A ( x ) ≤ µ B ( x ), then the following always holds
A ⊆ A ∪ B and A ∩ B ⊆ A . Depending on the application, other operators from within the
triangular norm and co-norm classes may be more appropriate than the above minimum and
maximum functions [16]. For example, the framework of Dombi [17] is used in this work.
Specifically:
µ
A∩ B
1
( x) =
1
1 + ((
with λ ≥ 1.
(5)
1
1
− 1) λ + (
− 1)λ ) λ
µ (x)
µ ( x)
A
B
Increasing the parameter λ will increase the emphasis on the smaller
membership value. One can define the union operation by allowing λ ≤ -1. Notice as
λ → ∞ , (5) approaches either (2) or (3) and in practice, values of λ larger than two or three
renders this form essentially equivalent to minimum and maximum
8