1 chapter 2 fuzzy sets versus crisp sets part one: theory
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Chapter 2Fuzzy Sets Versus
Crisp Sets
Part one: Theory
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2.1 Additional properties of alpha-cuts
The standard fuzzy intersection and fuzzy union are both
cutworthy when applied to two fuzzy sets.
The standard fuzzy intersection and fuzzy union are both strong cutworthy when applied to two
fuzzy sets.
The standard fuzzy complement is neither cutworthy nor strong cutworthy.
Alpha-cuts and strong alpha-cuts are always monotonic decreasing
with respect to alpha
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2.1 Additional properties of alpha-cuts
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2.1 Additional properties of alpha-cuts
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2.1 Additional properties of alpha-cuts
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2.1 Additional properties of alpha-cuts
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2.1 Additional properties of alpha-cuts
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2.1 Additional properties of alpha-cuts
An example: (vi) (a)
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1)( ,(
, However,
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1)1
1(sup)(sup))(( ,
,1
1)(Let
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ixAXx
ANi
XxA
ixAxAXx
Nii
xA
iii
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2.1 Additional properties of alpha-cuts
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2.1 Additional properties of alpha-cuts
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2.2 Representations of fuzzy sets
In this section, we show that each fuzzy set can uniquely be represented by either the family of all its -cuts or the family of all its strong -cuts.
Representations of fuzzy sets by crisp sets (the first one): An example: Considering the fuzzy set
this can be represented by its -cuts:
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2.2 Representations of fuzzy sets
Define a fuzzy set
we obtain
Now, it is easy to see that
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2.2 Representations of fuzzy sets
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2.2 Representations of fuzzy sets
For example:
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2.2 Representations of fuzzy sets
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2.2 Representations of fuzzy sets
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2.2 Representations of fuzzy sets
For example: The level set of A:
and
Xx x
AA 0 0 0
0
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2.3 Extension principle for fuzzy set
A crisp function:
f : X Y A fuzzified function
Its inverse function
An extension principle:a principle for fuzzifying crisp functions
Now, we first discuss the extended functions which are restricted to crisp power sets.
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2.3 Extension principle for fuzzy set
X Yf
x y
P(X) P(Y)f
x yA B
B(y) =
A(x) =
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2.3 Extension principle for fuzzy set
An example: Let X={a, b, c} and Y={1,2}
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2.3 Extension principle for fuzzy set
B(y) =
A(x) =
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2.3 Extension principle for fuzzy set
0.2
0.4
0.4
0.70.8
0.8
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2.3 Extension principle for fuzzy set
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2.3 Extension principle for fuzzy set
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2.3 Extension principle for fuzzy set
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2.3 Extension principle for fuzzy set
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2.3 Extension principle for fuzzy set