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

P. 23

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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)

))((

)11

1)( ,(

, However,

))(( ,1Let

1)1

1(sup)(sup))(( ,

,1

1)(Let

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1

1

xAXA

ixAXx

ANi

XxA

ixAxAXx

Nii

xA

iii

ii

i

i

ii

ii

ii

i

i

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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

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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