Fuzzy subset

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Introduction

The notion of fuzzy subset extends the one of subset from set theory. Indeed, 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. B 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 as "to be small", "to be close to 6", then the following question arises: is there a way to define the extension of P ? For example:

is there a precise definition of notions as "the subset of small numbers", "the subset of numbers close to 6 ?

The notion of fuzzy subset

The fuzzy subset of small numbers and the fuzzy 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, for every 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 there is not a unique empty subsets.

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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