Fuzzy subset

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The notion of fuzzy set

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 comprehension 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 a set S is the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_X : S\rightarrow \{0,1\} } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_X(x) = 1 } if x is an element in X and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_X(x) =0 } otherwise. Obviously, it is possible to identify X with its characteristic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_X} . 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 the complete lattice [0,1]. The following is a precise definition.


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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(S) } the class of all the fuzzy subsets of S. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1,...,S_n} are nonempty sets then a fuzzy subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_1\times . . . \times S_n} is called a fuzzy relation.


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.

We say that a fuzzy subset s is crisp if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(x)\in\{0,1\}} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in S} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} with the fuzzy subset constantly equal to 0 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} with the fuzzy subset constantly equal to 1.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vee, \wedge, \neg} . In order to define the same operations for fuzzy subsets, we have to fix a multi-valued logic and therefore suitable operations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oplus, \otimes} and ~ to interpret these connectives. Once this was done, we can set

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s\cup t)(x) = s(x)\oplus t(x)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s\cap t)(x) = s(x)\otimes t(x)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-s)(x) = ~s(x)} .

Also, the inclusion relation is defined by setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s\subseteq t \Leftrightarrow s(x)\leq t(x)} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in S} .


In such a way an algebraic structure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (F(S), \cup, \cap, -, \emptyset, S)} is defined and this structure is the direct power of the structure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ([0,1],\oplus, \otimes,} ~,0,1) with index set S. In Zadeh's original papers the operations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oplus, \otimes} , ~ are defined by setting for every x and y in [0,1]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\otimes y } = min(x, y)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\oplus y } = max(x,y)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~x } = 1-x.

In such a case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (F(S),\cup, \cap, -,\emptyset, S)} is a complete, completely distributive lattice with an involution. Several authors prefer to consider different operations, as an example to assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \otimes} is a triangular norm and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \oplus } 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (P(S), \cup, \cap, -, \emptyset, S)} into the algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (F(S), \cup, \cap, -, \emptyset, S)} .

An extension of these definitions to the general case in which instead of [0,1] we consider different algebraic structures L is obvious. In such a case one uses the expression L-subset.


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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is neither fully true nor fully false and we can express that by assigning to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} a number, as an example 0.8, as a subjective measure of our degree of belief in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} (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

Bibliography

  • Cox E., The Fuzzy Systems Handbook (1994), ISBN 0-12-194270-8
  • Elkan C.. The Paradoxical Success of Fuzzy Logic. November 1993. Available from Elkan's home page.
  • Gerla G., Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer, 2001.
  • 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.
  • 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
  • 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).
  • 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) 338­353.
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