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== Fuzzy Set ==  
{{subpages}}
To be Completed !!
{{TOC|right}}
== 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]].


Given a nonempty set ''S'', a ''fuzzy subset'' of ''S'' is a map ''s'' from ''S'' into the interval [0,1]. Then an element in [0,1] is 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''. In other words, a fuzzy subset is a characteristic function in which graded truth values are admitted.
Such a notion enables us to represent the extension of predicates and relations as "big","slow", "near" "similar", which are vague in nature.
Observe that there are two possible interpretations of the word "fuzzy logic".
The first one is related with an informal utilization of the notion of fuzzy set and it is devoted to the applications. In such a case should be better expressions as "[[fuzzy set theory]]" or "fuzzy logic in board sense".
Another interpretation is given in considering fuzzy logic as a chapter of formal logic. In such a case one uses the expression "fuzzy logic in narrow sense" or "[[formal fuzzy logic]]".


== Fuzzy logic and probability ==
'''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''.


any 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:
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.
Let <math>\alpha</math> be the claim "the rose on the table is red" and imagine we can freely examine the rose (complete knowledge) 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 value, as an example 0.8, different from 0 and 1 (fuzziness). This truth value does not depend on the information we have since this information is complete.


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.
== 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).}}


== Some set-theoretical notions for fuzzy subsets ==
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
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>. Likewise, the same operations for fuzzy subsets are defined once in a multi-valued logic these connectives are interpreted by suitable operations <math> \oplus, \otimes</math>, '''-'''. In fac, the union, intersection and complement are defined by setting


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


In Zadeh's original papers the operations <math> \oplus, \otimes</math>, '''-''' are defined by setting for every ''x'' and ''y'' in [0,1]:
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]:


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


<math> x\oplus y </math> = max(''x'',''y'')
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>.


<math> - x </math> = 1-''x''.
==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]].


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.
==See also==  
 
* [[Fuzzy logic]]
An extension of these definitions to the general case in which instead of [0,1] we consider different algebraic structures is obvious.
* [[Fuzzy control system]]
 
== See also ==
* [[Neuro-fuzzy]]
* [[Neuro-fuzzy]]
* [[Fuzzy subalgebra]]
* [[Fuzzy subalgebra]]  
* [[Fuzzy associative matrix]]
* [[Fuzzy associative matrix]]  
* [[FuzzyCLIPS]] expert system
* [[FuzzyCLIPS expert system]]
* [[Fuzzy control system]]
* [[Fuzzy set]]
* [[Paradox of the heap]]
* [[Paradox of the heap]]
* [[Pattern recognition]]
* [[Pattern recognition]]
* [[Rough set]]
* [[Rough set]]


== Bibliography ==
==Bibliography==  
* Chang C. C.,Keisler H. J., ''Continuous Model Theory'', Princeton University Press, Princeton, 1996.
* Cox E., The Fuzzy Systems Handbook (1994), ISBN 0-12-194270-8  
* Cignoli R., D’Ottaviano I. M. L. , Mundici D. , ‘’Algebraic Foundations of Many-Valued Reasoning’’. Kluwer, Dordrecht, 1999.
* Gerla G., Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer, 2001.  
* Cox E., ''The Fuzzy Systems Handbook'' (1994), ISBN 0-12-194270-8
* Gottwald S., A treatase on Multi-Valued Logics, Research Studies Press LTD, Baldock 2001.  
* Elkan C.. ''The Paradoxical Success of Fuzzy Logic''. November 1993. Available from [http://www.cse.ucsd.edu/users/elkan/ Elkan's home page].
* Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.  
* 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.  
* Hájek P., Fuzzy logic and arithmetical hierarchy, ''Fuzzy Sets and Systems'', 3, (1995), 359-363.
* Klir G. and Folger T., Fuzzy Sets, Uncertainty, and Information (1988), ISBN 0-13-345984-5.  
* Höppner F., Klawonn F., Kruse R. and Runkler T., ''Fuzzy Cluster Analysis'' (1999), ISBN 0-471-98864-2.
* Klir G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995) ISBN 0-13-101171-5  
* Klir G. and Folger T., ''Fuzzy Sets, Uncertainty, and Information'' (1988), ISBN 0-13-345984-5.
* Kosko B., Fuzzy Thinking: The New Science of Fuzzy Logic (1993), Hyperion. ISBN 0-7868-8021-X  
* Klir G. , UTE H. St. Clair and Bo Yuan ''Fuzzy Set Theory Foundations and Applications'',1997.
* Novák V., Perfilieva I, Mockor J., Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, (1999).  
* Klir G. and Bo Yuan, ''Fuzzy Sets and Fuzzy Logic'' (1995) ISBN 0-13-101171-5
* Yager R. and Filev D., Essentials of Fuzzy Modeling and Control (1994), ISBN 0-471-01761-2  
* [[Bart Kosko]], ''Fuzzy Thinking: The New Science of Fuzzy Logic'' (1993), Hyperion. ISBN 0-7868-8021-X  
* Zimmermann H., Fuzzy Set Theory and its Applications (2001), ISBN 0-7923-7435-5.  
* Montagna F., Three complexity problems in quantified fuzzy logic. ''Studia Logica'', 68,(2001), 143-152.
* Zadeh L.A., Fuzzy Sets, Information and Control, 8 (1965) 338-353.[[Category:Suggestion Bot Tag]]
* 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.
* Kevin M. Passino and Stephen Yurkovich, ''Fuzzy Control'', Addison Wesley Longman, Menlo Park, CA, 1998.
* Wiedermann J. , Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines, ''Theor. Comput. Sci.'' 317, (2004), 61-69.
* Zadeh L.A., Fuzzy algorithms, ''Information and Control'', 5,(1968), 94-102.
* Zadeh L.A., Fuzzy Sets, ‘’Information and Control’’, 8 (1965) 338­353.
* Zemankova-Leech, M., ''Fuzzy Relational Data Bases'' (1983), Ph. D. Dissertation, Florida State University.
 
[[category:CZ Live]]
[[category:Computers Workgroup]]
[[category:Mathematics Workgroup]]
[[category:Philosophy Workgroup]]

Latest revision as of 17:00, 19 August 2024

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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.