Transcript Fuzzy Set and Opertion

Fuzzy Set and Opertion
Outline
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Fuzzy Set and Crisp Set
Expanding concepts
Standard operation of fuzzy set
Fuzzy relations
Operations on fuzzy relations
Set
• Membership Function
– 𝜇𝐴 𝑥
• Universal Set
– user specify (crisp set)
• 𝐴 = (𝑥, 𝜇𝐴 𝑥 |𝑥 ∈ 𝑋}
Crisp set and Fuzzy set
• Crisp set
– membership function
1, if and only if x  A
 A ( x)  
0, if and only if x  A
– membership degree: {0,1}
• Fuzzy set
– membership function: user specify
– membership degree: [0,1]
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Crisp set and Fuzzy set
• Example
– Universal set 𝑋 = {1,2,3,4}
• Crisp Set
–𝐴=
0, 𝑥 < 2
𝑥, 𝜇𝐴 𝑥 𝑥 ∈ 𝑋
𝜇𝐴 𝑥 =
1, 𝑥 ≥ 2
𝐴 = {2,3,4}
• Fuzzy Set
– 𝐵 = {(𝑥, 𝜇𝐵 (𝑥)|𝑥 ∈ 𝑋)} 𝜇𝐵 𝑥 = 0.1𝑥
𝐵 = { 1,0.1 , 2,0.2 , 3,0.3 , (4,0.4)}
Expression of fuzzy set
A  {( x,  A ( x))}
n
A    A ( xi ) / xi
i 1
A  {(2,1.0),(3,0.5)}
1.0 0.5
A

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3
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Example of fuzzy set
X  {5,15, 25,35, 45,55,65,75,85}
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Support
• Support of A
support( A)  {x  X | A ( x)  0}
• example
support(young)  {15, 25,35, 45,55}
support(adult)  {15, 25,35, 45,55,65,75,85}
support(infant)  
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Height
• The maximum value of the membership degree
height (infant )  0
height ( young )  1
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Normalized fuzzy set
• Normalized fuzzy set
– height is 1
– young, adult, and senior are normalized fuzzy sets
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 -cut
•  -cut set
A  {x  X |  A ( x)  }
• Example
senior0.6  {65,75,85}
5 25,35, 45}
young0.2  {12,
young0.8  {25,35}
• If  '   , A  A '
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 -cut
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Level set
• Level set
 A  { | x  X ,  A ( x)   ,   0}
• Example
young  {0,0.1,0.2,0.4,0.8,1}
senior  {0,0.1,0.2,0.6,1}
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Convex fuzzy set
•  A (t )  min( A (r ),  A (s))
where t   r  (1   )s, r , s  R,  [0,1]
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Non-convex fuzzy set
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Relation of fuzzy sets
• A and B are equivalent
A  B iff  A ( x)  B ( x)
• A is a subset of B
A  B iff  A ( x)  B ( x)
• A is a proper subset of B
 A ( x)   B ( x), x  X
A  B iff
A  B and A  B
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Subset of fuzzy set
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Standard operation of fuzzy set
• Complement
• Union
• Intersection
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Complement
•  A ( x)  1   A ( x), x  X
• Example
A  {(5,0),(15,0.1),(25,0.9),(35,1),(45,1),...,,(85,1)}
A  {(5,1),(15,0.9),(25,0.1)}
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Union
• AB ( x)  max( A ( x), B ( x)), x  X
• Example
" young "  " adult "
 {(15,0.2),(25,1),(35,1),(45,1),(55,1),(65,1),(75,1),(85,1)}
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Intersection
• AB ( x)  min( A ( x), B ( x)), x  X
• Example
" young "  " adult "
 {(15, 0.1), (25, 0.9), (35, 0.8), (45, 0.4), (55, 0.1)}
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Other operations
• Disjunctive sum (exclusive OR)
Other operations
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Other operations
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Product set
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Product set
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Product set
• A={a1,a2} B={b1,b2} C={c1,c2}
• AxBxC =
{(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a
2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)}
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Crisp relation
• A relation among crisp sets A1 , A2 , , An is a
subset of the Cartesian product. It is denoted
by R .
R  A1  A2 
 An
• Using the membership function defines the
crisp relation R :
1 iff (x1 , x2 , ..., xn )  R,
 R ( x1 , x2 , ,xn )  
0 otherwise
where x1  A1 , x2  A2 ,..., xn  An
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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.
R : A1  A2  ...  An  [0,1]
R  (( x1 , x2 ,..., xn ), R ) | R ( x1 , x2 ,..., xn )  0, x1  A1 , x2  A2 ,..., xn  An 
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Representation methods
• Matrix
B
y1
y2
y3
B
y4
x1
x1
x2
x2
x3
x3
x4
x4
(Crisp)
y1
y2
y3
y4
(Fuzzy)
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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: A  B  max[aij , bij ]
• Example
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Operations on fuzzy matrices
• Max product: C = A・B=AB=
• Example
C12  ?
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Max product
• Example
C12  0.1
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Max product
• Example
C13  0.5
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Max product
• Example
C
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Operations on fuzzy matrices
• Scalar product:  A where 0    1
• Example
a
0.5 A 
b
c
a
b
0.1 0.25 0.0
0.2 0.5 0.05
c
0.0
0.5
0.0
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Operations on fuzzy relations
• Union relation
( x, y)  A  B
RS ( x, y)  max( R ( x, y), s ( x, y))
 R ( x, y)  s ( x, y)
• For n relations
( x, y )  A  B
 R  R ... R ( x, y )    R ( x, y )
1
2
n
Ri
i
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Union relation
• Example
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Operations on fuzzy relations
• Intersection relation
( x, y)  A  B
 RS ( x, y)  min(  R ( x, y), s ( x, y))
  R ( x, y)  s ( x, y)
• For n relations
( x, y )  A  B
 R  R ... R ( x, y )    R ( x, y )
1
2
n
Ri
i
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Intersection relation
• Example
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Operations on fuzzy relations
• Complement relation:
( x, y)  A  B
R ( x, y)  1  R ( x, y)
• Example
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Composition of fuzzy relations
• Max-min composition
( x, y)  A  B, ( y, z )  B  C
S R ( x, z)  max[min(R ( x, y), S ( y, z))]
y
 [R ( x, y)  S ( y, z)]
• Example
y
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Composition of fuzzy relations
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Composition of fuzzy relations
• Example
S R (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]  0.4
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Composition of fuzzy relations
• Example
S R (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]  0.2
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Composition of fuzzy relations
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