Fuzzy subset

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

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 $\alpha$ 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 $\alpha$ is neither fully true nor fully false and we can express that by assigning to $\alpha$ 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 $\alpha$ 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 $\alpha$ a number, as an example 0.8, as a subjective measure of our degree of belief in $\alpha$ (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.

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 $\vee ,\wedge ,\neg$. Likewise, the same operations for fuzzy subsets are defined once in a multi-valued logic these connectives are interpreted by suitable operations $\oplus ,\otimes$, -. In fac, the union, intersection and complement are defined by setting

$(s\cup t)(x)=s(x)\oplus t(x)$,

$(s\cap t)(x)=s(x)\otimes t(x)$,

$(-s)(x)=-s(x)$.

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

$x\otimes y$ = min(x, y)

$x\oplus y$ = max(x,y)

$-x$ = 1-x.

Several authors prefer to consider different operations, as an example to assume that $\otimes$ is a triangular norm and that $\oplus$ is the corresponding triangular co-norm.

An extension of these definitions to the general case in which instead of [0,1] we consider different algebraic structures is obvious.

See also

• Neuro-fuzzy
• Fuzzy subalgebra
• Fuzzy associative matrix
• FuzzyCLIPS expert system
• Fuzzy control system
• Fuzzy set
• Paradox of the heap
• Pattern recognition
• Rough set

Bibliography

• Chang C. C.,Keisler H. J., Continuous Model Theory, Princeton University Press, Princeton, 1996.
• Cignoli R., D’Ottaviano I. M. L. , Mundici D. , ‘’Algebraic Foundations of Many-Valued Reasoning’’. Kluwer, Dordrecht, 1999.
• 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.
• Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.
• Hájek P., Fuzzy logic and arithmetical hierarchy, Fuzzy Sets and Systems, 3, (1995), 359-363.
• 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
• Montagna F., Three complexity problems in quantified fuzzy logic. Studia Logica, 68,(2001), 143-152.
• 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.