Chapter 3 FUZZY RELATION AND COMPOSITION

Chapter 3

FUZZY RELATION AND COMPOSITION

G.Anuradha

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Outline

• Product set

• Crisp / fuzzy relations

• Composition / decomposition

• Projection / cylindrical extension

• Extension of fuzzy set / fuzzy relation

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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

• Bipartigraph

(Crisp) (Fuzzy)

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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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Domain and range of fuzzy relation

• Domain:

• Range :( ) ( ) max ( , )dom R R

y Bx x y

( ) ( ) max ( , )ran R Rx A

y x y

domain range

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Domain and range of fuzzy relation

• Fuzzy matrix

( ) 1

( ) 2

( ) 3

( ) 4

( ) 5

( ) 6

( ) 1.0

( ) 0.4

( ) 1.0

( ) 1.0

( ) 0.5

( ) 0.2

dom R

dom R

dom R

dom R

dom R

dom R

x

x

x

x

x

x

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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 RR

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 RR

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

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α-cut of fuzzy relation

•

• Example

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α-cut of fuzzy relation

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Decomposition of relation

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Decomposition of relation

0.9 0.4 0.0

0.0 1.0 0.4

0.0 0.7 1.0

0.4 0.0 0.0

RM

0

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Decomposition of relation

0.9 0.4 0.0

0.0 1.0 0.4

0.0 0.7 1.0

0.4 0.0 0.0

RM

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Projection / cylindrical extension

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Projection / cylindrical extension

( ) ( ) max ( , )dom R Ry B

x x y

( ) ( ) max ( , )ran R Rx A

y x y

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Projection in n dimension

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Projection

Projection

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Projection

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max(0.4,0.5) 0.5

max(0.2,0.1) 0.2

Projection

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Projection / cylindrical extension

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Cylindrical extension

Functions with Fuzzy Arguments

• A crisp function maps its crisp input argument to its image.

• Fuzzy arguments have membership degrees.• When computing a fuzzy mapping it is

necessary to compute the image and its membership value.

Crisp Mappings

Other operations on fuzzy sets

• Cartesian product• Mth power• Algebraic sum• Bounded sum• Bounded difference • Algebraic product

Cartesian product

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Thanks for your attention!