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Fuzzy Logic  
    • Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth -- truth values between "completely true" and "completely false". It was introduced by Dr. Lotfi Zadeh of UC/Berkeley in the 1960's as a means to model the uncertainty of natural language.
    • It is basically a multi-valued logic that allows intermediate values to be defined between conventional boolean logic like true/false, yes/no, black/white.
 
   
Fuzzy set theory  
Just as there is a strong relationship between Boolean logic and the concept of a subset, there is a similar strong relationship between fuzzy logic and fuzzy subset theory.

In classical set theory, a subset U of a set S can be defined as a mapping from the elements of S to the elements of the set {0, 1},

U: S --> {0, 1}

This mapping may be represented as a set of ordered pairs, with exactly one ordered pair present for each element of S. The first element of the ordered pair is an element of the set S, and the second element is an element of the set {0, 1}. The value zero is used to represent non-membership, and the value one is used to represent membership. The truth or falsity of the statement

x is in U

is determined by finding the ordered pair whose first element is x. The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0.

Similarly, a fuzzy subset F of a set S can be defined as a set of ordered pairs, each with the first element from S, and the second element from the interval [0,1], with exactly one ordered pair present for each element of S. This defines a mapping between elements of the set S and values in the interval [0,1]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate DEGREES OF MEMBERSHIP. The set S is referred to as the UNIVERSE OF DISCOURSE for the fuzzy subset F. Frequently, the mapping is described as a function, the MEMBERSHIP FUNCTION of F. The degree to which the statement

x is in F

is true is determined by finding the ordered pair whose first element is x. The DEGREE OF TRUTH of the statement is the second element of the ordered pair.

 
   
References  
[1] Shyue-Liang Wang, Jenn-Shing Tsai and Tzung-Pei Hong, Mining Functional Dependencies from Fuzzy Relational Databases, ACM SAC 2000, Como, Italy, March 2000.
[2] B.P. Buckles and F.E. Petry, “A fuzzy representation of data for relational databases”, Fuzzy Sets and Systems, 7, 1982, 213-226.
[3] Chan Man Kuok, Ada Fu and Man Hon Wong, “Mining Fuzzy Association Rules in Databases”, SIGMOD vol.27 I, March 1998. 41-46.
[4] M. Anvari, and G.F. Rose. “Fuzzy relational databases”, in Bezdek, Ed., Analysis of Fuzzy Information. Vol II (CRC Press, Boca Raton, FL, 1987).
[5] J.M. Medina, M.A. Vila, J.C. Cubero, and O. Pons, “ Towards the implementation of a generalized fuzzy relational database model”, Fuzzy Sets and Systems, 75. 1995. 273-289.
[6] Tzung-Pei Hong, Chan-Sheng Kuo, Sheng-Chai Chi and Shyue-Liang Wang, “Mining Fuzzy Rules from Quantitative Data Based on the AprioriTid Algorithm”, ACM SAC 2000 Como, Italy. March 2000.490-493.
[7] K. V. S. V. N. Raju and A. K. Majumdar: "Fuzzy Functional Dependencies and Lossless Join Decomposition of Fuzzy Relational Database Systems", ACM Transactions on Database Systems, Vol. 13, NO. 2, June 1988, 129-166.
[8] L.A. Zadeh and J. Kacprzyk: "Fuzzy Logic for the Management of Uncertainty". John Wiley & Sons Inc., 1992.
[9] von Altrock, "Fuzzy Logic and NeuroFuzzy Applications Explained", ISBN 0-1336-8465-2, Prentice Hall 1995
[10] von Altrock, C. and Krause, B., "On-Line-Development Tools for Fuzzy Knowledge-Base Systems of Higher Order", 2nd Int'l Conference on Fuzzy Logic and Neural Networks Proceedings, IIZUKA, Japan (1992), ISBN 4-938717-01-8.
[11] R. Belohlavek. Lattices of fixed points of fuzzy Galois connections. Mathematical Logic Quarterly, to appear.
[12] R. Belohlavek. Lattices generated by binary fuzzy relations. Tatra Mountains Mathematical Publications, 16, 11–19, 1999 (special issue on fuzzy sets).
[13] R. Belohlavek. Fuzzy Galois connections. Mathematical Logic Quarterly, 45, 497–504, 1999.
[14] P. Hajek. Metamathematics of Fuzzy Logic. Kluwer, 1998.
[15] G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice Hall, 1995.
[16] V. Novak, I. Perfilieva and J. Mockor. Mathematical Principles of Fuzzy Logic. Kluwer, 1999.
 
   
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