# Fuzzy subset

To be Completed !!

## 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 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.

**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 *S*_{1},...*S*_{n} are nonempty sets, then a fuzzy subset of *S*_{1}×. . .×*S*_{n} is called an *n-ary fuzzy relation*.

The elements in [0,1] are interpreted as truth degree 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 as in set theory.

## Some set-theoretical notions

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

- <math>(s\cup t)(x) = s(x)\oplus t(x)</math>,

- <math>(s\cap t)(x) = s(x)\otimes t(x)</math>,

- <math>(-s)(x) = \backsim s(x)</math>.

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

- <math> x\otimes y = min(x, y)</math> ; <math> x\oplus y = max(x,y)</math> ; <math> \backsim x = 1-x</math>.

In such a case <math>([0,1]^S, \cup, \cap, -, \emptyset, S)</math> is a complete, completely distributive lattice with an involution. Usually one assumes that <math>\otimes</math> is a triangular norm in [0,1] and that <math>\oplus </math> is the corresponding triangular co-norm defined by setting <math> x\oplus y = \backsim ((\backsim x)\otimes (\backsim y))</math>. 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 <math>(\{0,1\}^S, \cup, \cap, -, \emptyset, S)</math> into the algebra <math>(L^S, \cup, \cap, -, \emptyset, S)</math>.

## L-subsets

The notion of fuzzy subset can be extended by substituting the interval [0,1] by any bounded lattice *L*. Indeed, we define an *L-subset*as a map *s* from a set *S* into the lattice *L*. Again one assumes that in *L* suitable operations are defined to interpret the logical connectives and therefore to extend the set theoretical operations. 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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