By Mittelmann H.D., Weber H. (eds.)

ISBN-10: 3764312041

ISBN-13: 9783764312046

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**Extra info for Bifurcation problems and their numerical solution. Proc. Dortmund**

**Example text**

The intersection of the extensions of all these fuzzy sets to the domain X 1 × X 2 × × X n . Once computed, the chosen definition for implication can be applied to the rule: IF < U1 ,U 2 , ,U n > is A1 × A2 × × An THEN V is B. , a fuzzy subset of X 1 × X 2 × × X n × Y . Finally, the fuzzy conclusion can be drawn with the compositional rule of inference as B ' ( y ) = A1′ × × An′ ( x1 ,…, xn ) R ( x1 ,…, xn , y ) The Compositional Rule of Inference with several rules takes the following form: Rule 1: IF U1 is A11 and …and U n is A1n THEN V is B1 Rule 2: IF U1 is A21 and …and U n is A2 n THEN V is B2 : : Rule k: IF U1 is Ak1 and …and U n is Akn THEN V is Bk Fact: U1 is A'1 and U 2 is A' 2 and … and U n is An′ Conclusion: V is B' Each rule is translated as above to form Ri ( x1 ,…, xn , y ) and then the compositional rule of inference is applied to that rule with the fact proposition to obtain Bi ' ( y ) = Ai′1 × × Ain′ ( x1 ,…, xn ) Ri ( x1 ,…, xn , y ) .

The goal is to provide sufficient pain medication without over dosing. This is a continuous membership. 3(b). 2 Basic fuzzy set operators Once fuzzy subsets of a universal set X are defined, definitions for the complement of a set, the union of two sets and the intersection of two sets are required to actually generate a “set theory”. In 1965, Zadeh proposed the following. Suppose A : X → [0,1] is a fuzzy subset of X. The complement Ac of A is defined by Ac ( x) = 1 − A( x) . 28 Applications of Fuzzy Logic in Bioinformatics Additionally, if B : X → [0,1] is another fuzzy subset of X, Zadeh defined ( A ∪ B)( x) = max{A( x), B( x)} = A( x) ∨ B( x) and ( A ∩ B)( x) = min{A( x), B( x)} = A( x) ∧ B( x) .

Three common definitions used in many fuzzy rule systems are: The Lukasiewicz implication (Zadeh’s original implication operator): Rz ( x, y ) = min(1,1 − A( x) + B( y )) Correlation min implication: Rcm ( x, y ) = min( A( x), B( y )) Correlation product implication: Rcp ( x, y ) = A( x) * B( y ) Note that a fuzzy implication proposition is just a (fuzzy) rule. The Compositional Rule of Inference or Generalized Modus Ponens can now be described to combine a fuzzy rule and a linguistic proposition.

### Bifurcation problems and their numerical solution. Proc. Dortmund by Mittelmann H.D., Weber H. (eds.)

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