**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.