fuzzy maths
DESCRIPTION
fuzzy mathsTRANSCRIPT
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