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On a functional equation connected to the distributivity of fuzzy implications over triangular norms and conorms

Michał Baczyński, Tomasz Szostok, Wanda Niemyska (2014)

Kybernetika

Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see [9, 15] and [4]). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see [5]) f ( min ( x + y , a ) ) = min ( f ( x ) + f ( y ) , b ) , where a , b > 0 and f : [ 0 , a ] [ 0 , b ] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation f ( m 1 ( x + y ) ) = m 2 ( f ( x ) + f ( y ) ) , where m 1 , m 2 are functions...

On a new class of distances between fuzzy numbers.

Carlo Bertoluzza, Norberto Corral, Antonia Salas (1995)

Mathware and Soft Computing

In the course of the studies on fuzzy regression analysis, we encountered the problem of introducing a distance between fuzzy numbers, which replaces the classical (x - y)2 on the real line. Our proposal is to compute such a function as a suitable weighted mean of the distances between the α-cuts of the fuzzy numbers. The main difficulty is concerned with the definition of the distance between intervals, since the current definitions present some disadvantages which are undesirable in our context....

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