Chapter 2 The Operations of Fuzzy Set. Outline Standard operations of fuzzy set Fuzzy complement...
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Transcript of Chapter 2 The Operations of Fuzzy Set. Outline Standard operations of fuzzy set Fuzzy complement...
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