[7] What are fuzzy numbers and fuzzy arithmetic?

Date: 15-APR-93

Fuzzy numbers are fuzzy subsets of the real line. They have a peak or
plateau with membership grade 1, over which the members of the
universe are completely in the set.  The membership function is
increasing towards the peak and decreasing away from it.

Fuzzy numbers are used very widely in fuzzy control applications. A typical
case is the triangular fuzzy number 

1.0 +                   +
    |                  / \
    |                 /   \
0.5 +                /     \
    |               /       \
    |              /         \
0.0 +-------------+-----+-----+--------------
                  |     |     |
                 5.0   7.0   9.0

which is one form of the fuzzy number 7. Slope and trapezoidal functions
are also used, as are exponential curves similar to Gaussian probability
densities.

For more information, see:

   Dubois, Didier, and Prade, Henri, "Fuzzy Numbers: An Overview", in
   Analysis of Fuzzy Information 1:3-39, CRC Press, Boca Raton, 1987.

   Dubois, Didier, and Prade, Henri, "Mean Value of a Fuzzy Number", 
   Fuzzy Sets and Systems 24(3):279-300, 1987.

   Kaufmann, A., and Gupta, M.M., "Introduction to Fuzzy Arithmetic",
   Reinhold, New York, 1985.

================================================================