1. Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ

2. 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

3. 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

4. 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

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

6. 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

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

8. 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

9. 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

10. 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

11. Monotonic Continuous Functionsy y x
u(y)
u(x) x
@2002 Adriano Cruz
NCE e IM - UFRJ

12. 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

13. 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

14. 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

15. Nonmonotonic Continuous Functionsy y x
u(y)
u(x) x
@2002 Adriano Cruz
NCE e IM - UFRJ

16. 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

17. Nonmonotonic Continuous Functionsy 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

18. Function Example 1 Consider Consider fuzzy set Result
@2002 Adriano Cruz
NCE e IM - UFRJ

19. Function Example 2 Result according to the principle
@2002 Adriano Cruz
NCE e IM - UFRJ

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

21. 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

22. 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

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

24. 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

25. 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

26. 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

27. Fuzzy addition B, y=11 A, x=3 C, x=14 0.6 0.5 0.3 @2002 Adriano Cruz
NCE e IM - UFRJ

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