Fuzzy subset

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Introduction

In the everyday activity it is usual to adopt vague properties as "to be small", "to be close to 6" and so on. Now, in set theory given a set S and a "well defined" property P, the axiom of comprehension reads that a subset B of S exists whose members are precisely those objects in S satisfying P. Such a set is called "the extension of P in S". For example if S is the set of natural numbers and P is the property "to be prime", then the subset B of prime numbers is defined. Assume that P is a vague property, then the question arises whether there is a way to define notions as "the subset of small numbers", "the subset of numbers close to 6 ?

An attempt to give an answer to such a question was proposed in 1965 by Lotfi Zadeh and, at the same time, by Dieter Klaua in the framework of multi-valued logic. The idea is to extend the notion of characteristic function. 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.

Definition. Let S be a nonempty set, then a fuzzy subset of S is a map s from S into the real interval [0,1]. 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 [0,1] are interpreted as truth values and, in accordance, given x in S, the number s(x) is interpreted as the membership degree of x to s. 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 different sets have different empty subsets and therefore there is not a unique empty subset.

The fuzzy subset of "small numbers" and the fuzzy subset of "numbers close to 6".

Some set-theoretical notions

The intersection of the "fuzzy subset of small numbers" with the "fuzzy subset of numbers close to 6" (obtained by the minimum and the product).

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 the fuzzy subsets of a given set, we have to fix suitable operations $\oplus ,\otimes$ and $\backsim$ in [0,1] to interpret these connectives. Once this was done, we can set

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

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

$(-s)(x)=\backsim s(x)$.

If we denote by [0,1]^S the class of all the fuzzy subsets of S, then an algebraic structure $([0,1]^{S},\cup ,\cap ,-,\emptyset ,S)$ is defined. This structure is the direct power of the structure $([0,1],\oplus ,\otimes ,\backsim ,0,1)$ with index set S. In Zadeh's original papers the operations $\oplus ,\otimes ,\backsim$ are defined by setting for every x and y in [0,1]:

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

In such a case $([0,1]^{S},\cup ,\cap ,-,\emptyset ,S)$ is a complete, completely distributive lattice with an involution. More in general several authors admit different operations, as an example thay assume that $\otimes$ is any triangular norm in [0,1] and that $\oplus$ is the corresponding triangular co-norm defined by setting $x\oplus y=\backsim ((\backsim x)\otimes (\backsim y))$. For example, the picture represents the intersection of the fuzzy subset of small number with the fuzzy subset of numbers close to 6 obtained by the minimum and the product. 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 $(L^{S},\cup ,\cap ,-,\emptyset ,S)$.

L-subsets

The notion of fuzzy subset can be extended by substituting the interval [0,1] by any bounded lattice L. Again one assumes that in L suitable operations are defined to interpret the logical connectives. This extension was done mainly in the framework of fuzzy logic.

See also

• Fuzzy logic
• Fuzzy control system
• Neuro-fuzzy
• Fuzzy subalgebra
• Fuzzy associative matrix
• FuzzyCLIPS expert system
• Paradox of the heap
• Pattern recognition
• Rough set

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