Fuzzy Set and Opertion

Outline

• Fuzzy Set and Crisp Set• Expanding concepts• Standard operation of fuzzy set• Fuzzy relations• Operations on fuzzy relations

Set

• Membership Function

• Universal Set– user specify (crisp set)

Crisp set and Fuzzy set

• Crisp set– membership function

– membership degree: {0,1}

• Fuzzy set– membership function: user specify– membership degree: [0,1]

1, if and only if ( )

0, if and only if A

x Ax

x A

4

Crisp set and Fuzzy set

• Example– Universal set

• Crisp Set–

• Fuzzy Set–

Expression of fuzzy set

{( , ( ))}AA x x

1

( ) /n

A i ii

A x x

{(2,1.0), (3,0.5)}A

1.0 0.5

2 3A

6

Example of fuzzy set

7

{5,15,25,35,45,55,65,75,85}X

Support

• Support of A

• example

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support( ) { | ( ) 0}AA x X x

support(young) {15,25,35,45,55}

support(adult) {15,25,35,45,55,65,75,85}support(infant)

Height

• The maximum value of the membership degree

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( ) 1height young

( ) 0height infant

Normalized fuzzy set

• Normalized fuzzy set– height is 1– young, adult, and senior are normalized fuzzy sets

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-cut

• -cut set

• Example

• If , 11

{ | ( ) }AA x X x

0.2young {12,25,35,45}

0.8young {25,35}

0.6senior {65,75,85}

' 'A A

5

-cut

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Level set

• Level set

• Example

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{ | , ( ) , 0}A Ax X x

young {0,0.1,0.2,0.4,0.8,1}

senior {0,0.1,0.2,0.6,1}

Convex fuzzy set

•

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( ) min( ( ), ( ))

where (1 ) , , , [0,1]A A At r s

t r s r s R

Non-convex fuzzy set

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Relation of fuzzy sets

• A and B are equivalent

• A is a subset of B

• A is a proper subset of B

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( ) ( )A BA B iff x x

( ) ( )A BA B iff x x

Xxxx BA ),()( BA and iff BABA

Subset of fuzzy set

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Standard operation of fuzzy set

• Complement• Union• Intersection

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• • Example

Complement

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( ) 1 ( ), AA x x x X

{(5,0), (15,0.1), (25,0.9), (35,1), (45,1),..., , (85,1)}A

{(5,1), (15,0.9), (25,0.1)}A

Union

• • Example

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( ) max( ( ), ( )), A B A Bx x x x X

" " " "

{(15,0.2), (25,1), (35,1), (45,1), (55,1), (65,1), (75,1), (85,1)}

young adult

Intersection

• • Example

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( ) min( ( ), ( )), A B A Bx x x x X

" " " "

{(15,0.1), (25,0.9), (35,0.8), (45,0.4), (55,0.1)}

young adult

Other operations

• Disjunctive sum (exclusive OR)

Other operations

•

Other operations

•

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Product set

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Product set

Product set

• A={a1,a2} B={b1,b2} C={c1,c2}

• AxBxC = {(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)}

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Crisp relation

• A relation among crisp sets is a subset of the Cartesian product. It is denoted by .

• Using the membership function defines the crisp relation R :

1 2 nR A A A

1 21 2

1 1 2 2

1 iff ( , , ..., ) ,( , , , )

0 otherwise

where , ,...,

nR n

n n

x x x Rx x x

x A x A x A

1 2, , , nA A A

R

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Fuzzy relation

• A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets A1, A2, ..., An where tuples (x1, x2, ..., xn) may have varying degrees of membership within the relation.

• The membership grade indicates the strength of the relation present between the elements of the tuple.

1 2

1 2 1 2 1 1 2 2

: ... [0,1]

(( , ,..., ), ) | ( , ,..., ) 0, , ,..., R n

n R R n n n

A A A

R x x x x x x x A x A x A

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Representation methods

• Matrix

(Crisp) (Fuzzy)

1 2 3 4 y y y y

1

2

3

4

x

x

x

x

1

2

3

4

x

x

x

x

1 2 3 4 y y y yB B

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Representation methods

• Digraph

(Crisp) (Fuzzy)

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α-cut of fuzzy relation

•

• Example

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α-cut of fuzzy relation

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Operations on fuzzy matrices

• Sum:

• Example

max[ , ]ij ijA B a b

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Operations on fuzzy matrices

• Max product: C = A・ B=AB=

• Example

12 ?C

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Max product

• Example

12 0.1C

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Max product

• Example

13 0.5C

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Max product

• Example

C

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Operations on fuzzy matrices

• Scalar product:

• Example

where 0 1A

0.1 0.25 0.0

0.5 0.2 0.5 0.05

0.0 0.5 0.0

a b c

a

A b

c

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Operations on fuzzy relations

• Union relation

• For n relations

( , )

( , ) max( ( , ), ( , ))

( , ) ( , )R S R s

R s

x y A B

x y x y x y

x y x y

1 2 ...

( , )

( , ) ( , )n i

iR R R R

R

x y A B

x y x y

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Union relation

• Example

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Operations on fuzzy relations

• Intersection relation

• For n relations

( , )

( , ) min( ( , ), ( , ))

( , ) ( , )R S R s

R s

x y A B

x y x y x y

x y x y

1 2 ...

( , )

( , ) ( , )n i

iR R R R

R

x y A B

x y x y

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Intersection relation

• Example

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Operations on fuzzy relations

• Complement relation:

• Example

( , )

( , ) 1 ( , )RR

x y A B

x y x y

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Composition of fuzzy relations

• Max-min composition

• Example

( , ) max[min( ( , ), ( , ))]

[ ( , ) ( , )]

S R R Sy

R Sy

x z x y y z

x y y z

( , ) , ( , )x y A B y z B C

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Composition of fuzzy relations

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Composition of fuzzy relations

• Example

(1, ) max[min(0.1,0.9), min(0.2,0.2), min(0.0,0.8), min(1.0,0.4)]

max[0.1,0.2,0.0,0.4] 0.4S R

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Composition of fuzzy relations

• Example

(1, ) max[min(0.1,0.0), min(0.2,1.0), min(0.0,0.0), min(1.0,0.2)]

max[0.0,0.2,0.0,0.2] 0.2S R

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Composition of fuzzy relations