Understanding the Extension Principle and Fuzzy Numbers in Fuzzy Set Theory

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 surjectivefromR 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 • 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. ( ) ( ) = 1= 1= 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, 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. ={(2,0.3),(3,0.3),(4,0.7),(6,1),(8,0.2),(9,0.4),(12,0.2)} 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 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 •  =(3,1.1,1)LR • =(-1,0.7,1.4)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