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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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DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

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