A Survey on Fuzzy Differences <s> Introduction <s> In this paper the Radstrom embedding theorem (Proc. Amer. Math. Soc. 3 (1952), 165) is generalized and is used to define the concept of the differential of a fuzzy function. <s> BIB001 </s> A Survey on Fuzzy Differences <s> Introduction <s> In this paper we consider additions of interactive fuzzy numbers. The interactivity relation between fuzzy numbers are defined by their joint possibility distribution. We show that Nguyen's theorem remains valid in this environment. We give explicit formulas for the /spl gamma/-level sets of the extended sum of two completely correlated fuzzy numbers. We show that the interactive and the non-interactive sums have the same membership function for any pair of completely positively correlated fuzzy numbers. Finally, we prove that (i) the interactive sum of two completely negatively correlated fuzzy numbers A and B with A(x) = B(-x) for all x /spl isin/ /spl Ropf/, is (crisp) zero; (ii) the interactive difference of two completely positively correlated fuzzy numbers A and B with identical membership function, that is, a(x) - b(x) for all x /spl isin/ /spl Ropf/, is (crisp) zero. <s> BIB002 </s> A Survey on Fuzzy Differences <s> Introduction <s> There are two approaches that restore the algebraic properties of real arithmetic to fuzzy arithmetic - constraint interval arithmetic on alpha-cuts and arithmetic on intervals of gradual numbers. In this study, the relationships between fuzzy interval analysis from a constraint interval approach and from a gradual number approach will be discussed. <s> BIB003 </s> A Survey on Fuzzy Differences <s> Introduction <s> In the present paper we introduce and study a generalization of the Hukuhara differ- ence and also generalizations of the Hukuhara differentiability to the case of interval valued functions. We consider several possible definitions for the derivative of an interval valued function and we study connections between them and their proper- ties. Using these concepts we study interval differential equations. Local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ODE systems and by differential algebraic equations. We also show some connection with differential inclusions. The thoretical results are turned into practical algorithms to solve interval differential equations. <s> BIB004
The standard difference between two fuzzy numbers A and B is based on the difference between intervals  . For each α in the unit interval, it takes into account all the possible combinations between two elements, one from the α-level of A and the other from the α-level of B. Consequently, the result is always greater (in diameter) than any of the sets involved in the operation. Thus, the difference between two non-crisp numbers is always a non-crisp number and subtracting a non-crisp number from itself is never the crisp number zero. Using the Hukuhara difference BIB001 , the result of subtracting a non-crisp number A from itself (A ⊖ H A) is, in fact, zero. However, for this case, a necessary condition for the subtraction between two different fuzzy numbers A and B to exist is the first term to have bigger diameter than the second one. Generalizing the idea of this difference, BIB004  proposed the generalized Hukuhara difference, which also satisfies A − gH A = 0 and is defined for a bigger class of fuzzy numbers than the Hukuhara difference. An extension of generalized Hukuhara difference is the generalized difference   , which has the same results of the generalized Hukuhara operator (when it exists), but is defined for more pairs of fuzzy numbers. Another possibility is CIA (Constraint Interval Arithmetic) BIB003 . In this case, the diameter of the difference between two fuzzy numbers using CIA is smaller than using standard difference. All differences mentioned above makes use of the interval arithmetic on α-levels. Extensions to fuzzy numbers are drawn up via Stacking Theorem  over the resulting α-levels. Another way is similar to arithmetic for random variables, that is, the subtractions between fuzzy numbers are obtained using the joint possibility (or membership) distribution between the involved fuzzy numbers BIB002 . The comparison between the results obtained from two approaches is made via a kind of Nguyen's theorem extension BIB002 . In this study we present the definitions of all cited differences as well as its derivatives and we analyze its relations and exhibit an illustrative example.