Implications Adriano Joaquim de O Cruz ©2002 NCE/UFRJ

@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

@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.

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

@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

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

@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

@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

@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

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

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

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

@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)]

@2002 Adriano Cruz NCE e IM - UFRJRelações 14 Example u A (3) uB(y)uB(y) u A (4)=

@2002 Adriano Cruz NCE e IM - UFRJRelações 15 Example

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

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

@2002 Adriano Cruz NCE e IM - UFRJRelações 18 Example uB(y)uB(y) uB(y)uB(y)