# Chapter 2 The Operation of Fuzzy Set

```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)  1   A ( x), x  X
3
Standard operation of fuzzy set
• Union
AB ( x)  max( A ( x), B ( x)), x  X
Standard operation of fuzzy set
• Intersection
AB ( 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.
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.
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.
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
```