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