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": there is a way to define the extension of P ? For example: is there a precise definition of the notion of set of big objects ? 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 L be a bounded lattice. Then, given a nonempty set S, an L-subset of 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$$\times. . . \times$$ 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, one considers the lattice [0,1]. We say that a fuzzy subset s is crisp if $$s(x)\in\{0,1\}$$ 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 identify $$\emptyset$$ with the fuzzy subset constantly equal to 0 and $$S$$ with the fuzzy subset constantly equal to 1.

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

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

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

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

Also, the inclusion relation is defined by setting

$$s\subseteq t \Leftrightarrow s(x)\leq t(x)$$ for every $$x\in S$$.

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 $$(L^S, \cup, \cap, -, \emptyset, S)$$.

Truth degree and belief degree: fuzzy logic and probability
Many peoples compare fuzzy logic with probability theory since both refer to the interval [0,1]. However, they are conceptually distinct since we have not confuse a degree of truth with a probability measure. To illustrate the difference, consider the following example: Let $$\alpha$$ be the claim "the rose on the table is red" and imagine we can freely examine such a rose (complete information) but, as a matter of fact, the color looks not exactly red. Then $$\alpha$$ is neither fully true nor fully false and we can express that by assigning to $$\alpha$$ a truth degree, as an example 0.8, different from 0 and 1 (fuzziness). This truth degree does not depend on the information we have since it is assigned in a situation of complete information. Now, imagine a world in which all the roses are either clearly red or clearly yellow. In such a world $$\alpha$$ is either true or false but, inasmuch as we cannot examine the rose on the table, we are not able to know what is the case. Nevertheless, we have an opinion about the possible color of that rose and we could assign to $$\alpha$$ a number, as an example 0.8, as a subjective measure of our degree of belief in $$\alpha$$ (probability). In such a case this number depends on the information we have and, for example, it can vary if we have some new information on the taste of the possessor of the rose.