fuzzy maths

Chapter 2 The Operations of Fuzzy Set

Outline

• Standard operations of fuzzy set• Fuzzy complement• Fuzzy union• Fuzzy intersection• 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 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

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{),(

yxyxyx

yxIdp

}1,0max{),( yxyxIbd

yxyxIap ),(

0,))(1(

),(

yxyxyxyxIhp

Fuzzy intersection

Other operations

• Disjunctive sum (exclusive OR)

Other operations

Other operations

Other operations

• Disjoint sum (elimination of common area)

Other operations

• DifferenceCrisp setFuzzy set : Simple difference By using standard complement and intersection

operations.

Fuzzy set : Bounded difference

Other operations

• ExampleSimple difference

Other operations

• Example Bounded difference

Other operations

• Distance and difference

Other operations

• DistanceHamming distance

Relative Hamming distance

Other operationsEuclidean distance

Relative Euclidean distance

Minkowski distance (w=1-> Hamming and w=2-> Euclidean)

Other operations

• Cartesian productPower

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-conormsApplying complements

DeMorgan’s law

norms- t:T conorms-t: ,),(),(1)1,1(1),(

yxTyxTyxTyx