fuzzy subset

fuzzy subset

Fuzzy set

theory is based on the idea that vague notions as “big”, “near”, “hold” can be modelled by “fuzzy subsets”. The idea of a fuzzy subset

T

of a set

S

is the following: each

element

x

∈

S

, there is a number

p

∈

[

0

,

1

]

such that

p

x

is the “probability” that

x

is in

T

.

To formally define a fuzzy set, let us first recall a well-known fact about subsets: a subset

T

of a set

S

corresponds uniquely to the

characteristic function

c

T

:

S

→

{

0

,

1

}

, such that

c

T

⁢

(

x

)

=

1

iff

x

∈

T

. So if one were to replace

{

0

,

1

}

with the the closed unit interval

[

0

,

1

]

, one obtains a fuzzy subset:

A

fuzzy subset

of a set

S

is a map

s

:

S

→

[

0

,

1

]

from

S

into the interval

[

0

,

1

]

.

More precisely, the interval

[

0

,

1

]

is considered as a

complete lattice

with an involution

1

-

x

.
We call

fuzzy subset of

S

any element of the

direct power

[

0

,

1

]

S

. Whereas there are

2

|

S

|

subsets of

S

, there are

ℵ

1

|

S

|

fuzzy subsets of

S

.

The join and meet

operations

in the complete lattice

[

0

,

1

]

S

are named

union

and

intersection

, respectively. The operation

induced

by the involution is called

complement

. This means that if

s

and

t

are two fuzzy subsets, then the fuzzy subsets

s

∪

t

,

s

∩

t

,

-

s

, are defined by the equations

(

s

∪

t

)

⁢

(

x

)

=

max

⁡

{

s

⁢

(

x

)

,

t

⁢

(

x

)

}

;

(

s

∩

t

)

⁢

(

x

)

=

min

⁡

{

s

⁢

(

x

)

,

t

⁢

(

x

)

}

;

-

s

⁢

(

x

)

=

1

-

s

⁢

(

x

)

.

It is also possible to consider any

lattice

L

instead of

[

0

,

1

]

. In such a case we call

L

-subset of

S

any element of the direct power

L

S

and the union and the intersection are defined by setting

(

s

∪

t

)

⁢

(

x

)

=

s

⁢

(

x

)

∨

t

⁢

(

x

)

;

(

s

∩

t

)

⁢

(

x

)

=

s

⁢

(

x

)

∧

t

⁢

(

x

)

where

∨

and

∧

denote the join and the meet operations in

L

, respectively. In the case an order reversing

function

¬

:

L

→

L

is defined in

L

, the

complement

-

s

of

s

is defined by setting

-

s

⁢

(

x

)

=

¬

⁢

s

⁢

(

x

)

.

Fuzzy set theory is devoted mainly to applications. The main success is perhaps

fuzzy control

.

References

1

Cignoli R., D Ottaviano I. M. L. and Mundici D.,

Algebraic

Foundations

of Many-Valued Reasoning

. Kluwer, Dordrecht, (1999).

2

Elkan C.,

The

Paradoxical

Success of

Fuzzy Logic

. (November 1993). Available from

http://www.cse.ucsd.edu/users/elkan/

http://www.cse.ucsd.edu/users/elkan/ Elkan’s home page.

3

Gerla G.,

Fuzzy logic: Mathematical tools for approximate reasoning

, Kluwer Academic Publishers, Dordrecht, (2001).

4

Goguen J., The logic of inexact

concepts

,

Synthese

, vol. 19 (1968/69)

5

Gottwald S.,

A treatise on

many-valued logics

, Research Studies Press, Baldock (2000).

6

HÃÂ¡jek P.,

Metamathematics

of fuzzy logic

. Kluwer (1998).

7

Klir G. , UTE H. St.Clair and Bo Yuan,

Fuzzy Set Theory Foundations and Applications

, (1997).

8

Zimmermann H.,

Fuzzy Set Theory and its Applications

(2001), ISBN 0-7923-7435-5.

9

Zadeh L.A., Fuzzy Sets,

Information and Control

, 8 (1965) 338ÃÂ­-353.

10

Zadeh L. A., The concept of a linguistic

variable

and its application to approximate reasoning I, II, III,

Information Sciences

, vol. 8, 9(1975), pp. 199-275, 301-357, 43-80.

Title

fuzzy subset

Canonical name

FuzzySubset

Date of creation

2013-03-22 16:34:54

Last modified on

2013-03-22 16:34:54

Owner

ggerla (15808)

Last modified by

ggerla (15808)

Numerical id

11

Author

ggerla (15808)

Entry type

Definition

Classification

msc 03E72

Classification

msc 03G20

Synonym

fuzzy set

Synonym

L-subset

Related topic

Logic

Related topic

FuzzyLogic2

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