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# 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 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 . In order to define the same operations for the fuzzy subsets of a given set, we have to fix suitable operations and in [0,1] to interpret these connectives. Once this was done, we can define these operations by the equations

- ,

- ,

- .

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

- ; ; .

In such a case is a complete, completely distributive lattice with an involution. Usually one assumes that is a triangular norm in [0,1] and that is the corresponding triangular co-norm defined by setting . 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 into the algebra .

## 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

## Bibliography

- Cox E., The Fuzzy Systems Handbook (1994), ISBN 0-12-194270-8
- Gerla G., Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer, 2001.
- Gottwald S., A treatase on Multi-Valued Logics, Research Studies Press LTD, Baldock 2001.
- Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.
- Klaua D., Über einen Ansatz zur mehrwertigen Mengenlehre, Monatsberichte der Deutschen Akademie der Wissenschaften Berlin, vol 7 (1965), pp 859-867.
- Klir G. and Folger T., Fuzzy Sets, Uncertainty, and Information (1988), ISBN 0-13-345984-5.
- Klir G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 0-13-101171-5
- Kosko B., Fuzzy Thinking: The New Science of Fuzzy Logic (1993), Hyperion. ISBN 0-7868-8021-X
- Novák V., Perfilieva I, Mockor J., Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, (1999).
- Yager R. and Filev D., Essentials of Fuzzy Modeling and Control (1994), ISBN 0-471-01761-2
- Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 0-7923-7435-5.
- Zadeh L.A., Fuzzy Sets, Information and Control, 8 (1965) 338-353.