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Implications Adriano Joaquim de O Cruz ©2002 NCE/UFRJ adriano@nce.ufrj.br

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@2002 Adriano Cruz NCE e IM - UFRJRelações 2 Implication n If x A then y B. n P is a proposition described by the set A n Q is a proposition described by the set B n P Q: If x A then y B n P implies Q

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@2002 Adriano Cruz NCE e IM - UFRJRelações 3 Implication n An implication is a logical construction that tell us if one condition is true, then another condition must also be true. n Implication is not if and only if. n Implication P -> Q is true even if only Q is true. n Elephants can fly, therefore it is hot today. n This statement is true if it is hot today.

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@2002 Adriano Cruz NCE e IM - UFRJRelações 4 Implication – Truth table ABA -> B FFT FTT TFF TTT

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@2002 Adriano Cruz NCE e IM - UFRJRelações 5 Implication n If I am elected I will decrease taxes n I was not elected, I did not decrease taxes n I was not elected, I did decrease taxes n I was elected, I did not decrease taxes n I was elected, I did decrease taxes

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@2002 Adriano Cruz NCE e IM - UFRJRelações 6 Implication Relation n Implication can be described as a relation

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@2002 Adriano Cruz NCE e IM - UFRJRelações 7 Interpretations of Implication n There are two ways to interpret the fuzzy rule if … then … else n First: A -> B as A coupled with B n Where T is a T-norm operator

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@2002 Adriano Cruz NCE e IM - UFRJRelações 8 A coupled with B n Commonly used T-norms are: n Mamdani: n Larson n Bounded Product

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@2002 Adriano Cruz NCE e IM - UFRJRelações 9 Interpretations of Implication n Second: A -> B as A entails B n Where entails can be represented in different ways

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@2002 Adriano Cruz NCE e IM - UFRJRelações 10 Families of Implication n Generalization of Material Implication

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@2002 Adriano Cruz NCE e IM - UFRJRelações 11 Families of Implication n Propositional Calculus

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@2002 Adriano Cruz NCE e IM - UFRJRelações 12 Families of Implication n Generalization of Modus Ponens n Goguen n Godel

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@2002 Adriano Cruz NCE e IM - UFRJRelações 13 Implications n If x is A then y is B n This rules is an implication R(x,y) n If x is A, we want to know whether y is B n B= A R(x,y) n B (y)= x [ A (x) R (x,y)]

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@2002 Adriano Cruz NCE e IM - UFRJRelações 14 Example 12345678910 u A (3) 12345678910 uB(y)uB(y) 123456789 u A (4)=1.0 0.5 1.0 0.33 0.67 1.00

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@2002 Adriano Cruz NCE e IM - UFRJRelações 15 Example

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@2002 Adriano Cruz NCE e IM - UFRJRelações 16 Example n We will use the Mamdani implication function

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@2002 Adriano Cruz NCE e IM - UFRJRelações 17 Example n B(y i )=A(x i )oR(x i,y i )

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@2002 Adriano Cruz NCE e IM - UFRJRelações 18 Example 12345678910 uB(y)uB(y) 0.33 0.67 1.00 12345678910 uB(y)uB(y) 0.33 0.50

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