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Chapter 3 FUZZY RELATION AND COMPOSITION G.Anuradha
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!
