(OPP,INV,SUM,PROD,EVAL)

Maurice Clerc (maurice.clerc@writeme.com)
Thu, 18 Mar 1999 00:36:25 +0100 (MET)

Let X be a "trapezoidal" fuzzy set (x1,0),(x2,1),(x3,1),(x4,0), or, in short
X=(x1,x2,x3,x4).
In particular the null value is defined by Z=(0,0,0,0) and the unit
value by U=(1,1,1,1).

Let A and B be two suchl fuzzy sets. We define OPP(A),INV(A), SUM(A,B),
PROD(A,B) and EVAL(A) (a defuzzification operator).

I am looking for functions (OPP,INV,SUM,PROD,EVAL) so that
SUM(A,OPP(A))=Z
PROD(A,INV(A))=U
and, if possible,
EVAL(SUM(A,B))=EVAL(A)+EVAL(B)
EVAL(PROD(A,B))=EVAL(A)*EVAL(B)

Of course, I already have some ideas, but what I would like is something
"reasonable", not too far from the "meaning" of fuzzy sets, for example
OPP(A)=(-a4,-a3,-a2,-a1)
SUM(A,B)=(a1+b1,a2+b2,a3+b3,a4+b4)
..
EVAL((2*SQRT(a2*a3) + SQRT(a1*a4))/3) (which is far from perfect).

Thanks in advance, and, as usually, I will post a summary of all you
brilliant replies ;-)

Maurice Clerc

-----------------------------------------------------------
http://www.netword.com/ Maurice Clerc (or http://perso.wanadoo.fr/maurice.clerc/)
Kiam kien ni iru ?
"La fin est dans les moyens", "The aim is in the means", "Das Ziel ist
der Weg"
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