1. Extension Principle — Concepts
To generalize crisp mathematical concepts to fuzzy sets.
Extension Principle

2. Extension Principle
Let X be a cartesian product of universes X=X1…Xr, and be r fuzzy sets in X1,…,Xr, respectively. f is a mapping from X to a universe Y, y=f(x1,…,xr), Then the extension principle allows us to define a fuzzy set in Y by
where
Extension Principle

3. Example 1
f(x)=x2
Extension Principle

4. Fuzzy Numbers
To qualify as a fuzzy number, a fuzzy set on R must possess at least the following three properties:
must be a normal fuzzy set
must be a closed interval for every α(0,1] (convex)
the support of , must be bounded
Extension Principle

5. Positive (negative) fuzzy number
A fuzzy number is called positive (negative) if its membership function is such that
Extension Principle

6. Increasing (Decreasing) Operation
A binary operation  in R is called increasing (decreasing) if
for x1>y1 and x2>y2
x1x2>y1y2 (x1x2<y1y2)
Extension Principle

7. Example 2 f(x,y)=x+y is an increasing operation
f(x,y)=x•y is an increasing operation on R+
f(x,y)=-(x+y) is an decreasing operation
Extension Principle

8. Notation of fuzzy numbers’ algebraic operations
If the normal algebraic operations +,-,*,/ are extended to operations on fuzzy numbers they shall be denoted by 
Extension Principle

9. Theorem 1
If and are fuzzy numbers whose membership functions are continuous and surjective from R to [0,1] and  is a continuous increasing (decreasing) binary operation, then is a fuzzy number whose membership function is continuous and surjective from R to [0,1]. 
Extension Principle

10. Theorem 2
If , F(R) (set of real fuzzy number) with and continuous membership functions, then by application of the extension principle for the binary operation : R  R→R the membership function of the fuzzy number is given by 
Extension Principle

11. Special Extended Operations
If f:X→Y, X=X1 the extension principle reduces for all F(R) to
Extension Principle

12. Example 31
For f(x)=-x the opposite of a fuzzy number is given with , where
If f(x)=1/x, then the inverse of a fuzzy number is given with
, where
Extension Principle

13. Example 32
For λR\{0} and f(x)=λx then the scalar multiplication o a fuzzy number is given by , where
Extension Principle

14. Extended Addition 
Since addition is an increasing operation → extended addition  of fuzzy numbers that
is a fuzzy number — that is
Extension Principle

15. Properties of  ( )( )  is commutative  is associative
( )( )
 is commutative
 is associative
0RF(R) is the neutral element for , that is , 0= , F(R)
For  there does not exist an inverse element, that is,
Extension Principle

16. Extended Product
Since multiplication is an increasing operation on R+ and a decreasing operation on R-, the product of positive fuzzy numbers or of negative fuzzy numbers results in a positive fuzzy number.
Let be a positive and a negative fuzzy number then is also negative and results in a negative fuzzy number.
Extension Principle

17. Properties of ( ) = is commutative is associative
, 1RF(R) is the neutral element for , that is , , F(R)
For there does not exist an inverse element, that is, 1= 1=
Extension Principle

18. Theorem 3
If is either a positive or a negative fuzzy number, and and are both either positive or negative fuzzy numbers then
Extension Principle

19. Extended Subtraction
Since subtraction is neither an increasing nor a decreasing operation,
is written as ( )
Extension Principle

20. Extended Division
Division is also neither an increasing nor a decreasing operation. If and are strictly positive fuzzy numbers then
The same is true if and are strictly negative.
Extension Principle

21. Note
Extended operations on the basis of min-max can’t directly applied to “fuzzy numbers” with discrete supports.
Example
Let ={(1,0.3),(2,1),(3,0.4)}, ={(2,0.7),(3,1),(4,0.2)} then
={(2,0.3),(3,0.3),(4,0.7),(6,1),(8,0.2),(9,0.4),(12,0.2)}
No longer be convex → not fuzzy number
Extension Principle

22. Extended Operations for LR-Representation of Fuzzy Sets
Extended operations with fuzzy numbers involve rather extensive computations as long as no restrictions are put on the type of membership functions allowed.
LR-representation of fuzzy sets increases computational efficiency without limiting the generality beyond acceptable limits.
Extension Principle

23. Definition of L (and R) type
Map R+→[0,1], decreasing, shape functions if
L(0)=1
L(x)<1, for x>0
L(x)>0 for x<1
L(1)=0 or [L(x)>0, x and L(+∞)=0]
Extension Principle

24. Definition of LR-type fuzzy number1
A fuzzy number is of LR-type if there exist reference functions L(for left). R(for right), and scalars α>0, β>0 with
Extension Principle

25. Definition of LR-type fuzzy number2
m; called the mean value of , is a real number
α,β called the left and right spreads, respectively.
is denoted by (m,α,β)LR
Extension Principle

26. Example 4 Let L(x)=1/(1+x2), R(x)=1/(1+2|x|), α=2, β=3, m=5 then
Extension Principle

27. Fuzzy Interval
A fuzzy interval is of LR-type if there exist shape functions L and R and four parameters , α, β and the membership function of is
The fuzzy interval is denoted by
Extension Principle

28. Different type of fuzzy interval
is a real crisp number for mR →
=(m,m,0,0)LR L, R
If is a crisp interval, →
=(a,b,0,0)LR L, R
If is a “trapezoidal fuzzy number” → L(x)=R(x)=max(0,1-x)
Extension Principle

29. Theorem 4
Let , be two fuzzy numbers of LR-type: =(m,α,β)LR, =(n,γ,δ)LR Then
(m, α, β)LR(n, γ,δ)LR=(m+n, α+γ, β+δ)LR
-(m, α, β)LR=(-m, β, α)LR
(m, α, β)LR (n, γ, δ)LR=(m-n, α+δ, β+γ)LR
Extension Principle

30. Example 5 L(x)=R(x)=1/(1+x2) =(1,0.5,0.8)LR =(2,0.6,0.2)LR
Extension Principle

31. Theorem 5 Let , be fuzzy numbers →
(m, α, β)LR (n, γ, δ)LR ≈(mn,mγ+nα,mδ+nβ)LR for , positive
(m, α, β)LR (n, γ, δ)LR ≈(mn,nα-mδ,nβ-mγ)LR for positive, negative
(m, α, β)LR (n, γ, δ)LR ≈(mn,-nβ-mδ,-nα-mγ)LR for , negative
Extension Principle