Re: Algorithm for fuzzy "OR" calculations?

Fred A Watkins (fwatkins@hyperlogic.com)
Mon, 2 Mar 1998 18:18:17 +0100 (MET)

"Alfred Kellner" <alfkellner@magnet.at> wrote:
>...
>
>I doubt the Algorithm X OR Y = x+y-xy
>since
> X OR Y <= X
>and X OR Y <= Y
>must hold true.
>When i.e. Y=X=0.7 then x+y-xy == 0.7+0.7-0.7*0.7 == 0.91
>So ( 0.91 <= 0.7 ) == false !!
>
>Am I fuzzy missing something ?

Afraid so: The OR of two objects is at least as "big" as either
thing OR'ed. An easy way to write it is

(X OR Y) >= MAX(X, Y)

The above recalls the fact that the MAX operator is the smallest
of all "t-conorms", the usual choices for disjunction (OR). Dually,
it is the case that

(X AND Y) <= MIN(X, Y)

which says that among all "t-norms" (AND-operators), MIN is the largest.

Thus MIN and MAX are a distinguished t-norm/t-conorm pair: they are the
*extremal* objects. (Between them lie the *weighted averages*, usually
represented as integrals.)

(In the above, the inequalities are *pointwise*; the fuzzy sets X and Y
are *functions* from some universe into the unit interval [0, 1].)

Noise about t-norms/t-conorms aside, you simply had the key relationship
backwards.

Exercise: Show that 0 <= x + y - xy <= 1 for all x,y in [0, 1].

Fred A Watkins, Ph.D.
HyperLogic Corporation
PO Box 300010
Escondido, CA 92030-0010 USA
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email: fwatkins@hyperlogic.com