Chapter 2 The Operation of Fuzzy Set
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Transcript Chapter 2 The Operation of Fuzzy Set
Chapter 2 The Operations of
Fuzzy Set
Outline
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Standard operations of fuzzy set
Fuzzy complement
Fuzzy union
Fuzzy intersection
Other operations in fuzzy set
Disjunctive sum
Difference
Distance
Cartesian product
• T-norms and t-conorms
Standard operation of fuzzy set
• Complement
A ( x) 1 A ( x), x X
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Standard operation of fuzzy set
• Union
AB ( x) max( A ( x), B ( x)), x X
Standard operation of fuzzy set
• Intersection
AB ( x) min( A ( x), B ( x)), x X
Fuzzy complement
• C:[0,1][0,1]
Fuzzy complement
Fuzzy complement
• Axioms C1 and C2 called “axiomatic skeleton ”
are fundamental requisites to be a
complement function, i.e., for any function
C:[0,1][0,1] that satisfies axioms C1 and C2
is called a fuzzy complement.
• Additional requirements
Fuzzy complement
• Example 1 : Standard function
Axiom C1
Axiom C2
Axiom C3
Axiom C4
Fuzzy complement
• Example 2 :
Axiom C1
Axiom C2
X Axiom C3
X Axiom C4
Fuzzy complement
• Example 3:
Axiom C1
Axiom C2
Axiom C3
X Axiom C4
Fuzzy complement
• Example 4: Yager’s function
Axiom C1
Axiom C2
Axiom C3
Axiom C4
Fuzzy complement
• Fuzzy partition
If m subsets are defined in X, m-tuple (A1,
A2,…,Am) holding the following conditions is
called a fuzzy partition.
Fuzzy union
Fuzzy union
• Axioms U1 ,U2,U3 and U4 called “axiomatic
skeleton ” are fundamental requisites to be a
union function, i.e., for any function
U:[0,1]X[0,1][0,1] that satisfies axioms
U1,U2,U3 and U4 is called a fuzzy union.
• Additional requirements
Fuzzy union
• Example 1 : Standard function
Axiom U1
Axiom U2
Axiom U3
Axiom U4
Axiom U5
Axiom U6
Fuzzy union
• Example 2: Yager’s function
Axiom U1
Axiom U2
Axiom U3
Axiom U4
Axiom U5
X Axiom U6
Fuzzy union
Fuzzy union
• Some frequently used fuzzy unions
– Probabilistic sum (Algebraic Sum):
U as ( x, y) x y x y
– Bounded Sum (Bold union):
Ubs ( x, y) min{1, x y}
– Drastic Sum:
max{x, y}, if min{x, y} 0
U ds ( x, y)
1, x, y 0
– Hamacher’s Sum
x y (2 ) x y
U hs ( x, y)
, 0
1 (1 ) x y
Fuzzy union
Fuzzy intersection
Fuzzy intersection
• Axioms I1 ,I2,I3 and I4 called “axiomatic
skeleton ” are fundamental requisites to be a
intersection function, i.e., for any function
I:[0,1]X[0,1][0,1] that satisfies axioms
I1,I2,I3 and I4 is called a fuzzy intersection.
• Additional requirements
Fuzzy intersection
• Example 1 : Standard function
Axiom I1
Axiom I2
Axiom I3
Axiom I4
Axiom I5
Axiom I6
Fuzzy intersection
• Example 2: Yager’s function
Axiom I1
Axiom I2
Axiom I3
Axiom I4
Axiom I5
X Axiom I6
Fuzzy intersection
Fuzzy intersection
• Some frequently used fuzzy intersections
– Probabilistic product (Algebraic product):
I ap ( x, y) x y
– Bounded product (Bold intersection):
I bd ( x, y) max{0, x y 1}
– Drastic product :
min{x, y}, if max{x, y} 1
I dp ( x, y)
0, x, y 1
– Hamacher’s product
x y
I hp ( x, y )
, 0
(1 )(x y x y )
Fuzzy intersection
Other operations
• Disjunctive sum (exclusive OR)
Other operations
Other operations
Other operations
• Disjoint sum (elimination of common area)
Other operations
• Difference
Crisp set
Fuzzy set : Simple difference
By using standard complement and intersection
operations.
Fuzzy set : Bounded difference
Other operations
• Example
Simple difference
Other operations
• Example
Bounded difference
Other operations
• Distance and difference
Other operations
• Distance
Hamming distance
Relative Hamming distance
Other operations
Euclidean distance
Relative Euclidean distance
Minkowski distance
(w=1-> Hamming and w=2-> Euclidean)
Other operations
• Cartesian product
Power
Cartesian product
Other operations
• Example:
– A = { (x1, 0.2), (x2, 0.5), (x3, 1) }
– B = { (y1, 0.3), (y2, 0.9) }
t-norms and t-conorms (s-norms)
t-norms and t-conorms (s-norms)
t-norms and t-conorms (s-norms)
• Duality of t-norms and t-conorms
Applying complements
( x, y) 1 T (1 x,1 y) 1 T ( x, y) T ( x, y),
: t - conorms T : t - norms
DeMorgan’s law