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 sumDifferenceDistanceCartesian product

• T-norms and t-conorms

Standard operation of fuzzy set• Complement

3

( ) 1 ( ), AA x x x X

Standard operation of fuzzy set

• Union( ) max( ( ), ( )), A B A Bx x x x X

Standard operation of fuzzy set

• Intersection( ) min( ( ), ( )), A B A Bx x 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 C1Axiom C2Axiom C3Axiom C4

Fuzzy complement

• Example 2 :

Axiom C1Axiom C2X Axiom C3X Axiom C4

Fuzzy complement

• Example 3:

Axiom C1Axiom C2Axiom C3X Axiom C4

Fuzzy complement

• Example 4: Yager’s function

Axiom C1Axiom C2Axiom C3Axiom 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 U1Axiom U2Axiom U3Axiom U4Axiom U5Axiom U6

Fuzzy union• Example 2: Yager’s function

Axiom U1Axiom U2Axiom U3Axiom U4Axiom U5X Axiom U6

Fuzzy union

Fuzzy union• Some frequently used fuzzy unions– Probabilistic sum (Algebraic Sum):

– Bounded Sum (Bold union):

– Drastic Sum:

– Hamacher’s Sum

0, ,1

0},min{ if },,max{),(

yx

yxyxyxU ds

},1min{),( yxyxUbs

yxyxyxU as ),(

0,)1(1

)2(),(

yx

yxyxyxU hs

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 I1Axiom I2Axiom I3Axiom I4Axiom I5Axiom I6

Fuzzy intersection• Example 2: Yager’s function

Axiom I1Axiom I2Axiom I3Axiom I4Axiom I5X Axiom I6

Fuzzy intersection

Fuzzy intersection• Some frequently used fuzzy intersections– Probabilistic product (Algebraic product):

– Bounded product (Bold intersection):

– Drastic product :

– Hamacher’s product

1, ,0

1},max{ if },,min{),(

yx

yxyxyxIdp

}1,0max{),( yxyxIbd

yxyxIap ),(

0,))(1(

),(

yxyx

yxyxIhp

Fuzzy intersection