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== Introduction ==
{{Image|Fuzzy1.JPG|left|150px|The fuzzy subset of ''"small numbers"'' and the fuzzy subset of ''"numbers close to 6"''.}}
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]].


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== The notion of [[fuzzy set|fuzzy subset]] ==
Fuzzy set theory is an attempt to represent the extension of vague properties. 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 youngh"'': 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'' ? 
In order to give an aswer to this question recall that the ''caracteristic function'' of a classical subset ''X'' of ''S'' is the map <math> c_X : S\rightarrow \{0,1\} </math> such that <math> c_X(x) = 1 </math> if ''x'' is an element in ''X'' and <math>c_X(x) =0 </math> otherwise. Obviously, it is possible to identify ''X'' with its characteristic function <math>c_X</math>. This suggests that we can define the subset of big elements by a generalized caracteristic function in which instead of the Boolean algebra {0,1} we can consider, for example, the complete lattice [0,1]. The following is a precise definition.


'''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''<sub>1</sub>,...''S''<sub>''n''</sub> are nonempty sets, then a fuzzy subset of ''S''<sub>1</sub>×. . .×''S''<sub>''n''</sub> is called an ''n-ary fuzzy relation''.


'''Definition.''' Given a nonempty set ''S'', a ''fuzzy subset'' of ''S'' is a map ''s'' from ''S'' into the interval [0,1]. We denote by [0,1]<sup>S</sup> the class of all the fuzzy subsets of ''S''. If S<sub>1</sub>,...S<sub>n</sub> are nonempty sets then a fuzzy subset of S<sub>1</sub><math>\times . . . \times</math> S<sub>n</sub> is called a ''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 ==
{{Image|Fuzzy2.JPG|right|150px|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).}}


The elements in [0,1] 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''. More in general, if ''L'' is a bounded lattice, we obtain the notion of
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
''L-subset'' by considering ''L'' instead of [0,1].
We say that a fuzzy subset ''s'' is ''crisp'' if <math>s(x)\in\{0,1\}</math> for every <math>x\in S</math>. By associating every classical subsets of ''S'' with its caracteristic function, we can identify the subsets of ''S'' with the crisp fuzzy subsets. In particular we identify <math>\emptyset</math> with the fuzzy subset constantly equal to 0 and <math>S</math> with the fuzzy subset constantly equal to 1.


== Some set-theoretical notions for fuzzy subsets ==
:<math>(s\cup t)(x) = s(x)\oplus t(x)</math>,  
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 fuzzy subsets, we have to fix suitable operations <math> \oplus, \otimes</math> and ~ to interpret these connectives. Once this was done, we can set


<math>(s\cup t)(x) = s(x)\oplus t(x)</math>,  
:<math>(s\cap t)(x) = s(x)\otimes t(x)</math>,  


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


<math>(-s)(x) = ~s(x)</math>.
If we denote by [0,1]<sup>S</sup> 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]:


Also, the ''inclusion relation'' is defined by setting
:<math> x\otimes y = min(x, y)</math>  ;  <math> x\oplus y  = max(x,y)</math>  ;  <math> \backsim x = 1-x</math>.


<math>s\subseteq t \Leftrightarrow s(x)\leq t(x)</math> for every <math>x\in S</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]].


In such a way an algebraic structure <math>([0,1]^S, \cup, \cap, -, \emptyset, S)</math> is defined and this structure is the direct power of the structure <math>([0,1],\oplus, \otimes,</math>~,0,1) with index set ''S''.
==See also==  
In Zadeh's original papers the operations <math> \oplus, \otimes</math>, '''~''' are defined by setting for every ''x'' and ''y'' in [0,1]:
* [[Fuzzy logic]]  
 
* [[Fuzzy control system]]  
<math> x\otimes y </math> = min(''x'', ''y'') ; <math> x\oplus y </math> = max(''x'',''y'') ; <math> ~x </math> = 1-''x''.
 
In such a case <math>([0,1]^F, \cup, \cap, -, \emptyset, S)</math> is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that <math>\otimes</math> is a triangular norm and that <math>\oplus </math> is the corresponding triangular co-norm.
 
In all the cases the interpretation of a logical connectives 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 caracteristic function is an embedding of the Boolean algebra <math>([0,1]^S, \cup, \cap, -, \emptyset, S)</math> into the algebra <math>([0,1]^S, \cup, \cap, -, \emptyset, S)</math>.
 
An extension of these definitions to the general case in which instead of [0,1] we consider a any lattice ''L'' equipped by suitable operations is obvious.
 
== 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 <math>\alpha</math> 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 <math>\alpha</math> is neither fully true nor fully false and we can express that by assigning to <math>\alpha</math> 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 siuation of complete information.
Now, imagine a world in which all the roses are either clearly red or clearly yellow. In such a world <math>\alpha</math> 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 <math>\alpha</math> a number, as an example 0.8, as a subjective measure of our degree of belief in <math>\alpha</math> (probability). In such a case this number depends strongly from 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.
 
== See also ==
* [[Formal fuzzy logic]]
* [[Fuzzy logic]]
* [[Fuzzy control system]]
* [[Neuro-fuzzy]]
* [[Neuro-fuzzy]]
* [[Fuzzy subalgebra]]
* [[Fuzzy subalgebra]]  
* [[Fuzzy associative matrix]]
* [[Fuzzy associative matrix]]  
* [[FuzzyCLIPS]] expert system
* [[FuzzyCLIPS expert system]]
* [[Paradox of the heap]]
* [[Paradox of the heap]]
* [[Pattern recognition]]
* [[Pattern recognition]]
* [[Rough set]]
* [[Rough set]]


== Bibliography ==
==Bibliography==  
* Cox E., ''The Fuzzy Systems Handbook'' (1994), ISBN 0-12-194270-8
* Cox E., The Fuzzy Systems Handbook (1994), ISBN 0-12-194270-8  
* Elkan C.. ''The Paradoxical Success of Fuzzy Logic''. November 1993. Available from [http://www.cse.ucsd.edu/users/elkan/ Elkan's home page].
* Gerla G., Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer, 2001.
* 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.
* Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.  
* Höppner F., Klawonn F., Kruse R. and Runkler T., ''Fuzzy Cluster Analysis'' (1999), ISBN 0-471-98864-2.
* 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 Folger T., Fuzzy Sets, Uncertainty, and Information (1988), ISBN 0-13-345984-5.  
* Klir G. , UTE H. St. Clair and Bo Yuan ''Fuzzy Set Theory Foundations and Applications'',1997.
* Klir G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 0-13-101171-5  
* 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  
* [[Bart Kosko]], ''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).  
* 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  
* 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.  
* 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.[[Category:Suggestion Bot Tag]]
* Zadeh L.A., Fuzzy Sets, ‘’Information and Control’’, 8 (1965) 338­353.

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Introduction

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

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 S1,...Sn are nonempty sets, then a fuzzy subset of S1×. . .×Sn 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

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 . 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-subsetas 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

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.