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**Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ**

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Fuzzy Numbers A fuzzy number is fuzzy subset of the universe of a numerical number. A fuzzy real number is a fuzzy subset of the domain of real numbers. A fuzzy integer number is a fuzzy subset of the domain of integers. @2002 Adriano Cruz NCE e IM - UFRJ

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**Fuzzy Numbers - Example**

u(x) Fuzzy real number 10 5 10 15 x u(x) Fuzzy integer number 10 5 10 15 x @2002 Adriano Cruz NCE e IM - UFRJ

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**Functions with Fuzzy Arguments**

A crisp function maps its crisp input argument to its image. Fuzzy arguments have membership degrees. When computing a fuzzy mapping it is necessary to compute the image and its membership value. @2002 Adriano Cruz NCE e IM - UFRJ

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Crisp Mappings Y f(X) X @2002 Adriano Cruz NCE e IM - UFRJ

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**Functions applied to intervals**

Compute the image of the interval. An interval is a crisp set. y y=f(I) I x @2002 Adriano Cruz NCE e IM - UFRJ

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Mappings f(X) Y X Fuzzy argument? @2002 Adriano Cruz NCE e IM - UFRJ

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Extension Principle Suppose that f is a function from X to Y and A is a fuzzy set on X defined as A = µA(x1)/x1 + µA(x2)/x µA(xn)/xn The extension principle states that the image of fuzzy set A under the mapping f(.) can be expressed as a fuzzy set B. B = f(A) = µA(x1)/y1 + µA(x2)/y µA(xn)/yn where yi=f(xi) @2002 Adriano Cruz NCE e IM - UFRJ

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Extension Principle If f(.) is a many-to-one mapping, then there exist x1, x2 X, x1 x2, such that f(x1)=f(x2)=y*, y*Y. The membership grade at y=y* is the maximum of the membership grades at x1 and x2 more generally, we have @2002 Adriano Cruz NCE e IM - UFRJ

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**Monotonic Continuous Functions**

For each point in the interval Compute the image of the interval. The membership degrees are carried through. I @2002 Adriano Cruz NCE e IM - UFRJ

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**Monotonic Continuous Functions**

y y x u(y) u(x) x @2002 Adriano Cruz NCE e IM - UFRJ

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**Monotonic Continuous Ex.**

Function: y=f(x)=0.6*x+4 Input: Fuzzy number - around-5 Around-5 = 0.3 / / / 7 f(around-5) = 0.3/f(3) + 1/f(5) + 0.3/f(7) f(around-5) = 0.3/0.6* / 0.6* / 0.6*7+4 f(around-5) = 0.3/ / /8.2 I @2002 Adriano Cruz NCE e IM - UFRJ

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**Monotonic Continuous Ex.**

f(x) 8.2 10 5.8 4 x 5 10 1 0.3 u(x) 1 0.3 x 3 5 7 @2002 Adriano Cruz NCE e IM - UFRJ

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**Nonmonotonic Continuous Functions**

For each point in the interval Compute the image of the interval. The membership degrees are carried through. When different inputs map to the same value, combine the membership degrees. @2002 Adriano Cruz NCE e IM - UFRJ

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**Nonmonotonic Continuous Functions**

y y x u(y) u(x) x @2002 Adriano Cruz NCE e IM - UFRJ

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**Nonmonotonic Continuous Ex.**

Function: y=f(x)=x2-6x+11 Input: Fuzzy number - around-4 Around-4 = 0.3/2+0.6/3+1/4+0.6/5+0.3/6 y = 0.3/f(2)+0.6/f(3)+1/f(4)+0.6/f(5)+0.3/f(6) y = 0.3/3+0.6/2+1/3+0.6/6+0.3/11 y = 0.6/2+(0.3 v 1)/3+0.6/6+0.3/11 y = 0.6/2 + 1/ / /11 I @2002 Adriano Cruz NCE e IM - UFRJ

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**Nonmonotonic Continuous Functions**

y y 1 v 0.3 x 1 0.6 0.3 u(y) u(x) 1 0.6 0.3 x 2 3 4 5 6 @2002 Adriano Cruz NCE e IM - UFRJ

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**Function Example 1 Consider Consider fuzzy set Result**

@2002 Adriano Cruz NCE e IM - UFRJ

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**Function Example 2 Result according to the principle**

@2002 Adriano Cruz NCE e IM - UFRJ

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Function Example 3 @2002 Adriano Cruz NCE e IM - UFRJ

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Extension Principle Let f be a function with n arguments that maps a point in X1xX2x...xXn to a point in Y such that y=f(x1,…,xn). Let A1x…xAn be fuzzy subsets of X1xX2x...xXn The image of A under f is a subset of Y defined by @2002 Adriano Cruz NCE e IM - UFRJ

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**Arithmetic Operations**

Applying the extension principle to arithmetic operations it is possible to define fuzzy arithmetic operations Let x and y be the operands, z the result. Let A and B denote the fuzzy sets that represent the operands x and y respectively. @2002 Adriano Cruz NCE e IM - UFRJ

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Fuzzy addition Using the extension principle fuzzy addition is defined as @2002 Adriano Cruz NCE e IM - UFRJ

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**Fuzzy addition - Examples**

B =(11~)= 0.5/10 + 1/ /12 A+B=(0.3^0.5)/(1+10) + (0.6^0.5)/(2+10) + (1^0.5)/(3+10) + (0.6^0.5)/(4+10) + (0.3^0.5)/(5+10) + (0.3^1)/(1+11) + (0.6^1)/(2+11) + (1^1)/(3+11) + (0.6^1)/(4+11) + (0.3^1)/(5+11) +( 0.3^0.5)/(1+12) + (0.6^0.5)/(2+12) + (1^0.5)/(3+12) + (0.6^0.5)/(4+12) + (0.3^0.5)/(5+12) @2002 Adriano Cruz NCE e IM - UFRJ

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**Fuzzy addition - Examples**

B =(11~)= 0.5/10 + 1/ /12 Getting the minimum of the membership values A+B=0.3/ / / / / / /13 + 1/ / / / / / / /17 @2002 Adriano Cruz NCE e IM - UFRJ

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**Fuzzy addition - Examples**

B =(11~)= 0.5/10 + 1/ /12 Getting the maximum of the duplicated values A+B=0.3/11 + (0.5 V 0.3)/12 + (0.5 V 0.6 V 0.3)/13 + (0.5 V 1 V 0.5)/14 + (0.3 V 0.6 V 0.5)/15 + (0.3 V 0.5)/ /17 A+B=0.3 / / / / / / / 17 @2002 Adriano Cruz NCE e IM - UFRJ

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**Fuzzy addition B, y=11 A, x=3 C, x=14 0.6 0.5 0.3 @2002 Adriano Cruz**

NCE e IM - UFRJ

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Fuzzy Arithmetic Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as: @2002 Adriano Cruz NCE e IM - UFRJ

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