Chapter 2 The Operations of
Fuzzy Set
Outline
• 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
A x x X
3
Standard operation of fuzzy set
•
Union
x
max(
A x
B x x X
Standard operation of fuzzy set
•
Intersection
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 (A
1
,
A
2
,…,A m
) 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
– Bounded Sum (Bold union):
U bs
( x , y )
min{ 1 , x
y y }
– Drastic Sum:
U ds
( x , y )
– Hamacher’s Sum
max{ x , y },
1 if
, min{ x , y
x ,
0 y }
0
U hs
( x , y )
x
y
1
( 2
( 1
) x
) x y
y
,
0
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 :
I dp
( x , y )
– Hamacher’s product
min{ x , y }, if max{ x , y }
1
0 , x , y
1
I hp
( x , y )
x
( 1
)( x y
y
x
y )
,
0
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 )
: t conorms T : t norms
1
T ( x , y )
T ( x , y ) ,
DeMorgan’s law