Fuzzy subset

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 c_X : → {0,1} such that c_X(x) = 1 if x is an element in X and c_X(x) =0 otherwise. Obviously, it is possible to identify every subset X with its characteristic function c_X 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 L^S the class of all the fuzzy subsets of S. If S₁,...S_n are nonempty sets then a fuzzy subset of S₁×. . .×S_n 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 $\vee ,\wedge ,\neg$. In order to define the same operations for fuzzy subsets, we have to fix suitable operations $\oplus ,\otimes$ and ~ in L to interpret these connectives. Once this was done, we can set

$(s\cup t)(x)=s(x)\oplus t(x)$,

$(s\cap t)(x)=s(x)\otimes t(x)$,

$(-s)(x)=~s(x)$.

In such a way an algebraic structure $(L^{S},\cup ,\cap ,-,\emptyset ,S)$ is defined and this structure is the direct power of the structure $(L,\oplus ,\otimes ,$ ~,0 ,1) with index set S. In Zadeh's original papers the operations $\oplus ,\otimes$, ~ are defined by setting for every x and y in [0,1]:

$x\otimes y$ = min(x, y) ; $x\oplus y$ = max(x,y) ; $~x$ = 1-x.

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