Re: Fuzzy AND and OR

Stephen Paul King (spking1@mindspring.com)
Wed, 26 Aug 1998 22:54:51 +0200 (MET DST)

Here is a forwarded post on the subject:

Date: 01 Jun 1998 20:21:01 GMT
Newsgroups: comp.ai.fuzzy
To: ca314159@bestweb.net
From: wsiler@aol.com (WSiler)
Organization: AOL http://www.aol.com
References: <6kmaq3$q1f$1@nnrp1.dejanews.com>
Subject: Re: Fuzzy Set Intersection
X-UIDL: 0dcee935437219ddc5162bfa3224f976
X-Mozilla-Status: 0011

Your posting is a little confusing, since you have not stated what
your
measures correspond to in terms of the AND and OR operators.

To use a little more compact notation, let ZAND = the Zadeh AND, ZOR
= the
Zadeh OR, LAND = the Lukasiewicz AND and LOR = the Lukasiewicz OR,
with A LAND
B = max(A+B-1, 0) and A LOR B = min(A+B, 1).

You propose the following measures:

P = min(A(x), B(x)) = A ZAND B
Q = max(A(x), B(x)) = A ZOR B

You continue with

min(A \cap B) = min(P+Q, 1) = P LOR Q
max(A \cap B) = min(A, B) = A ZAND B

We can then state your final result as:

min(A \cap B) = P LAND Q = (A ZAND B) LAND (A ZOR B)
max(A \cap B) = A ZOR B

Using your examples:

A = .1
B = .2
P = A ZAND B = .1
Q = A ZOR B = .2
min(A \cap B) = .1 LAND .2 = 0
max(A \cap B) = .1 ZAND .3 = .1

A = .6
B = .8
P = A ZAND B = .6
Q = A ZOR B = .8
min(A \cap B) = .6 LAND .8 = .4
max(A \cap B) = .6 ZAND .8 = .6

yielding the same answers as you obtained.

However, the reasoning with which you achieved your operators is quite
unclear
to me. Some clarification would be appreciated.

William Siler

Carlos Gershenson <carlos@jlagunez.iquimica.unam.mx> wrote:

>Hi all.
>
>Well, I know that Zadeh defined the AND as the minimum and the OR as the
>maximum. But I have the idea that I read somewhere that there were
>other(s) definitions for the fuzzy AND and OR. Does someone knows these
>definitions?
>
>thanks
>
>"There is no Truth but that of Eternal struggle..."
> -Orunlu the Keeper
>
> Carlos Gershenson
> http://132.248.11.4/~carlos/
>
>
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