Fuzzy subset: Difference between revisions

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imported>Giangiacomo Gerla
imported>Giangiacomo Gerla
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== The notion of [[fuzzy set|fuzzy subset]] ==
Given a well defined property ''P'' and a set ''S'', the axiom of abstraction reads that there exists a set ''B'' whose members are precisely those objects in ''S'' that satisfy ''P''. Such a set is called ''the extension of P''. For example if ''S'' is the set of natural numbers and ''P'' is the property "to be prime", then the set ''B'' of prime numbers is defined. Assume that ''P'' is a vague property as ''"to be big"'', ''"to be young"'': is there a way to define the extension of ''P'' ? For example: is there a precise definition of the notion of ''set of big numbers'' ? 
The definition of ''fuzzy subset'' was proposed in 1965 by Lotfi Zadeh as an attempt to give an answer to such a question.  At the same time, the same definition was proposed by Dieter Klaua in the framework of multi-valued logic. Now recall that the ''[[characteristic function]]'' of a classical subset ''X'' of ''S'' is the map ''c''<sub>X</sub> : → {0,1} such that ''c<sub>X</sub>''(''x'') = 1 if ''x'' is an element in ''X'' and ''c<sub>X</sub>''(''x'') =0 otherwise. Obviously, it is possible to identify every subset ''X'' with its characteristic function ''c<sub>X</sub>'' and therefore the extension of a property with a suitable characteristic function. This suggests that we can define the subset of big elements by a generalized characteristic function in which instead of the Boolean algebra {0,1} we can consider, for example, a bounded [[lattice (order)|lattice]] ''L''. The following is a precise definition.
'''Definition.''' Let ''S'' be a nonempty set and ''L'' be a bounded lattice. Then an ''L''-subset or ''fuzzy subset'' of ''S'' is a map ''s'' from ''S'' into ''L''. We denote by ''L''<sup>S</sup> the class of all the fuzzy subsets of ''S''. If S<sub>1</sub>,...S<sub>n</sub> are nonempty sets then a fuzzy subset of S<sub>1</sub>×. . .×S<sub>n</sub> is called an ''n-ary L-relation''.
The elements in ''L'' are interpreted as truth values and, in accordance, for every ''x'' in ''S'', the value ''s(x)'' is interpreted as the membership degree of ''x'' to ''s''. Usually, ''L'' coincides with the lattice [0,1]. We say that a fuzzy subset ''s'' is ''crisp'' if ''s''(''x'') is in {0,1} for every ''x'' in S. By associating every classical subsets of ''S'' with its characteristic function, we can identify the subsets of ''S'' with the crisp fuzzy subsets. In particular we call ''"empty subset"'' of ''S'' the fuzzy subset of ''S'' constantly equal to 0. Notice that in such a way there is not a unique empty subsets.
== Some set-theoretical notions for fuzzy subsets ==
== 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 <math>\vee, \wedge, \neg</math>. In order to define the same operations for fuzzy subsets, we have to fix suitable operations <math> \oplus, \otimes</math> and ~ in ''L'' to interpret these connectives. Once this was done, we can set
In classical mathematics the definitions of union, intersection and complement are related with the interpretation of the basic logical connectives <math>\vee, \wedge, \neg</math>. In order to define the same operations for fuzzy subsets, we have to fix suitable operations <math> \oplus, \otimes</math> and ~ in ''L'' to interpret these connectives. Once this was done, we can set
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:<math> x\otimes y </math> = min(''x'', ''y'') ; <math> x\oplus y </math> = max(''x'',''y'') ; <math> ~x </math> = 1-''x''.
:<math> x\otimes y </math> = min(''x'', ''y'') ; <math> x\oplus y </math> = max(''x'',''y'') ; <math> ~x </math> = 1-''x''.


In such a case <math>([0,1]^F, \cup, \cap, -, \emptyset, S)</math> is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that <math>\otimes</math> is a triangular norm in [0,1] and that <math>\oplus </math> is the corresponding triangular co-norm.
In such a case <math>([0,1]^F, \cup, \cap, -, \emptyset, S)</math> is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that <math>\otimes</math> is a [[triangular norm]] in [0,1] and that <math>\oplus </math> 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 <math>(\{0,1\}^S, \cup, \cap, -, \emptyset, S)</math> into the algebra <math>(L^S, \cup, \cap, -, \emptyset, S)</math>.
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 <math>(\{0,1\}^S, \cup, \cap, -, \emptyset, S)</math> into the algebra <math>(L^S, \cup, \cap, -, \emptyset, S)</math>.

Revision as of 04:56, 4 January 2009

The notion of fuzzy subset

Given a well defined property P and a set S, the axiom of abstraction reads that there exists a set B whose members are precisely those objects in S that satisfy P. Such a set is called the extension of P. For example if S is the set of natural numbers and P is the property "to be prime", then the set B of prime numbers is defined. Assume that P is a vague property as "to be big", "to be young": is there a way to define the extension of P ? For example: is there a precise definition of the notion of set of big numbers ?

The definition of fuzzy subset was proposed in 1965 by Lotfi Zadeh as an attempt to give an answer to such a question. At the same time, the same definition was proposed by Dieter Klaua in the framework of multi-valued logic. Now recall that the characteristic function of a classical subset X of S is the map cX : → {0,1} such that cX(x) = 1 if x is an element in X and cX(x) =0 otherwise. Obviously, it is possible to identify every subset X with its characteristic function cX and therefore the extension of a property with a suitable characteristic function. This suggests that we can define the subset of big elements by a generalized characteristic function in which instead of the Boolean algebra {0,1} we can consider, for example, a bounded lattice L. The following is a precise definition.

Definition. Let S be a nonempty set and L be a bounded lattice. Then an L-subset or fuzzy subset of S is a map s from S into L. We denote by LS the class of all the fuzzy subsets of S. If S1,...Sn are nonempty sets then a fuzzy subset of S1×. . .×Sn is called an n-ary L-relation.

The elements in L are interpreted as truth values and, in accordance, for every x in S, the value s(x) is interpreted as the membership degree of x to s. Usually, L coincides with the lattice [0,1]. We say that a fuzzy subset s is crisp if s(x) is in {0,1} for every x in S. By associating every classical subsets of S with its characteristic function, we can identify the subsets of S with the crisp fuzzy subsets. In particular we call "empty subset" of S the fuzzy subset of S constantly equal to 0. Notice that in such a way there is not a unique empty subsets.

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 .