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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
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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
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Example of fuzzy set
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{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
