The main goal of the paper is to propose and analyze a generalization of the Hukuhara difference to the setting of interval and fuzzy arithmetic. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers. The two topics have been addressed by a consistent scientific activity and numerous papers appeared in the literature. Applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown. For the difference, the general setting of set-valued arithmetic is considered and some generalizations (the classical Hukuhara difference is a special case) are suggeste for compact sets, for compact convex sets (in particular compact intervals) and for fuzzy sets with compact and convex α-cuts (in particular fuzzy numbers). For the division, we consider the setting of interval-valued arithmetic and of fuzzy numbers. Both operations are related to two important problems in the interval and fuzzy arithmetic and analysis: (a) the solution of linear equations, and (b) the (generalized) differentiability of interval and fuzzy valued functions and the use of the corresponding derivative in the formulation of interval and fuzzy differential equations.

### A generalization of Hukuhara difference and division for interval and fuzzy arithmetic

#### Abstract

The main goal of the paper is to propose and analyze a generalization of the Hukuhara difference to the setting of interval and fuzzy arithmetic. First, the case of compact convex sets is examined; then, the results are applied to generalize the Hukuhara difference of fuzzy numbers, using their compact and convex level-cuts. Finally, a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers. The two topics have been addressed by a consistent scientific activity and numerous papers appeared in the literature. Applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown. For the difference, the general setting of set-valued arithmetic is considered and some generalizations (the classical Hukuhara difference is a special case) are suggeste for compact sets, for compact convex sets (in particular compact intervals) and for fuzzy sets with compact and convex α-cuts (in particular fuzzy numbers). For the division, we consider the setting of interval-valued arithmetic and of fuzzy numbers. Both operations are related to two important problems in the interval and fuzzy arithmetic and analysis: (a) the solution of linear equations, and (b) the (generalized) differentiability of interval and fuzzy valued functions and the use of the corresponding derivative in the formulation of interval and fuzzy differential equations.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11576/2502168`
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