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Fuzzy Logic (1) Intelligent System Course

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Apples, oranges or in between? A O Group of appleGroup of orange A A AA A A O O O O O O

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Lotfi Zadeh

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Thinking in Fuzzy A O A A A A A A O O O OO O Fuzzy group of applesFuzzy group of oranges

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Crisp vs Fuzzy Crisp sets handle only 0s and 1s Fuzzy sets handle all values between 0 and 1 Crisp NoYes Fuzzy NoSlightlyMostlyYes

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Fuzzy Set Notation Continuous Discrete

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Continuous Fuzzy Membership Function 0 1 2 3 x B (x) The set, B, of numbers near to two is

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Discrete Fuzzy Membership Function The set, B, of numbers near to two is 0 1 2 3 x B (x)

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Example Example Fuzzy Sets to Aggregate... A = { x | x is near an integer} B = { x | x is close to 2} 0 1 2 3 x A (x) 0 1 2 3 x B (x) 1

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Fuzzy Union Fuzzy Union (logic or) 4Meets crisp boundary conditions 4Commutative 4Associative 4Idempotent

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Fuzzy Union - example A OR B = A+B ={ x | (x is near an integer) OR (x is close to 2)} = MAX [ A (x), B (x) ] 0 1 2 3 x A+B (x)

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Fuzzy Intersection Fuzzy Intersection (logic and) 4Meets crisp boundary conditions 4Commutative 4Associative 4Idempotent

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Fuzzy Intersection - example A AND B = A·B ={ x | (x is near an integer) AND (x is close to 2)} = MIN [ A (x), B (x) ] 0 1 2 3 x A B (x)

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Fuzzy Complement The complement of a fuzzy set has a membership function... 4Meets crisp boundary conditions 4Two Negatives Should Make a Positive 4Law of Excluded Middle: NO! 4Law of Contradiction: NO!

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Fuzzy Complement - example 0 1 2 3 x 1 complement of A ={ x | x is not near an integer}

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Fuzzy Associativity (1) Min-Max fuzzy logic has intersection distributive over union... since min[ A,max(B,C) ]=min[ max(A,B), max(A,C) ]

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Fuzzy Associativity (2) Min-Max fuzzy logic has union distributive over intersection... since max[ A,min(B,C) ]= max[ min(A,B), min(A,C) ]

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Fuzzy and DeMorgans Law (1) Min-Max fuzzy logic obeys DeMorgans Law #1... since 1 - min(B,C)= max[ (1-B), (1-C)]

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Fuzzy and DeMorgans Law (2) Min-Max fuzzy logic obeys DeMorgans Law #2... since 1 - max(B,C)= min[(1-B), (1-C)]

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Fuzzy and Excluded Middle Min-Max fuzzy logic fails The Law of Excluded Middle. since min( A,1- A ) 0 Thus, (the set of numbers close to 2) AND (the set of numbers not close to 2) null set

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Fuzzy and the Law of Contradiction Min-Max fuzzy logic fails the The Law of Contradiction. since max( A,1- A ) 1 Thus, (the set of numbers close to 2) OR (the set of numbers not close to 2) universal set

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Cartesian Product The intersection and union operations can also be used to assign memberships on the Cartesian product of two sets. Consider, as an example, the fuzzy membership of a set, G, of liquids that taste good and the set, LA, of cities close to Los Angeles

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Cartesian Product We form the set... E = G · LA = liquids that taste good AND cities that are close to LA The following table results... LosAngeles(0.0) Chicago (0.5) New York (0.8) London(0.9) Swamp Water (0.0) 0.00 0.00 0.000.00 Radish Juice (0.5) 0.00 0.50 0.500.50 Grape Juice (0.9) 0.00 0.50 0.800.90

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