Re: Why alpha-cut?

Marco A. Vera (m-vera@uniandes.edu.co)
Tue, 20 Apr 1999 17:01:21 +0200 (MET DST)

Hi Ulf and Fuzzy people,

After reading very carefully Ulf's message -Why Alpha-cut?, and revised in
the fuzzy-mail's index the discussion about it, I just want to add that
alpha-cuts are useful in "fuzzy arithmetic" of fuzzy numbers (FN). (See Klir
and Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Apps", Chapter 4).

Suppose A, B, are continuous FNs, defined on R (the real line), the four
basic arithmetic operations:
C=A+B , C=A-B, C=A*B, C=A / B, define a continuous C FN (on R), that can be
evaluated with the help of the alpha-cut definition.

The FAQ of these kind of problems is: How to calculate the membership
function (MF) of C?.
As it's already known, Fuzzy arithmetic is based on two important
properties:
1- Each FN can fully and uniquely be represented by its alpha-cuts, and
2- The alpha-cuts of each FN are a family of closed intervals (classical
(crisp) sets) of real numbers for all alpha within (0 1].

These two properties enable us to define fuzzy ops on FN in terms of
arithmetic ops on their alpha-cuts. Then the fuzzy arithmetic ops are just
solved with classic arithmetic on intervals.

For instance the addition of two triangular-shape FNs, A and B, is equal to
the sum of the limits of the closed intervals for each alpha cut set, i.e.:

1 | /\ A / \ B
/\ C=A+B
| / \ / \
/ \
alpha|______ /__\_______/__ \_________ /__ \_____
:for an specific alpha
0|_____ /___\______/_ \____ __/____\______________
a1 a2 b1 b2 a1+b1 c2=a2+b2
R

Now you can begin to add FNs as you learn to add (crisp) numbers at school.
Just a simple reason to justify why many authors view alpha-cuts as a bridge
between classical and fuzzy sets (logic), in the sense of the "extension
principle". I think that's worth enough to bother with this topic.

Marco

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
\_ Marco A. Vera \_/ Department of Electrical and _/
\_ m-vera@uniandes.edu.co \_/ Electronic Engineering _/
\_ macu@hotmail.com \_/ Los Andes University _/
\_ ICQ #: 27566863 \_/ Bogota, D.C., Colombia _/
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