FUZZEKS [to Index]
Overview: Membership function window

The task of the membership functions is to transform the values of the input parameters to a common scale (as preparation for the aggregation).

The membership windows are opened by clicking the desired row in the second column of the management window.

Membership functions return values in the range [0,1], as can be seen on the membership-axis of the graph in the membership function window (there also is an example).

### Theoretical considerations

The membership function (m) supplies a value in the interval [0,1] for every input value (the input values' range is R; R is the set of real numbers) and can be denoted
m: R -> [0,1]

This function m already defines a fuzzy set.
Therefore (and because this fuzzy set can be viewed as value of a linguistic variable) in the membership function window it is possible to give this fuzzy set a name (refer to the input field "Name of fuzzy set").

In order to find out about possible membership functions, refer to "Types of membership functions".

Because FUZZEKS allows fuzzy numbers as input the function m is extended by the so-called Extension Principle (often used in fuzzy theory, see also references at bottom of Overview/References, especially [Zadeh, 1965]) to a function
m_fuzzy : F(R) -> F([0,1])
(where F(X) denotes the set of all fuzzy subsets of a set X).
This means that the function takes fuzzy numbers as input and delivers fuzzy values as well. The use of the so-called Extension Principle assures that m_fuzzy is a function equal with m if used on crisp real numbers only (crisp real numbers are a special case of fuzzy numbers).

Remark: Do not confuse the transformation function above (the named fuzzy set, defined by m) with the fuzziness of the input data (fuzzy numbers, which make necessary to define m_fuzzy). There are actually two concepts here which use fuzzy sets.