Fuzzy subset

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Revision as of 04:57, 2 January 2009 by imported>Giangiacomo Gerla (→‎Some set-theoretical notions for fuzzy subsets)
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Some set-theoretical notions for fuzzy subsets

In classical mathematics the definitions of union, intersection and complement are related with the interpretation of the basic logical connectives . In order to define the same operations for fuzzy subsets, we have to fix suitable operations and ~ in L to interpret these connectives. Once this was done, we can set

,
,
.


In such a way an algebraic structure is defined and this structure is the direct power of the structure ~,0 ,1) with index set S. In Zadeh's original papers the operations , ~ are defined by setting for every x and y in [0,1]:

= min(x, y) ; = max(x,y) ; = 1-x.

In such a case is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that is a triangular norm in [0,1] and that is the corresponding triangular co-norm.

In all the cases the interpretation of a logical connective is conservative in the sense that its restriction to {0,1} coincides with the classical one. This entails that the map associating any subset X of a set S with the related characteristic function is an embedding of the Boolean algebra into the algebra .