# Re: Fuzzy Set Intersection

ca314159 (ca314159@bestweb.net)
Sun, 7 Jun 1998 00:50:57 +0200 (MET DST)

WSiler wrote:
>
> 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
>
>
> 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

Thanks for that perspective. I was not aware of the Lukasiewicz distinction.
I'll look into that.

The measures where taken to be in the range 0-1, percentages or probabilities or
anything that maps isomorphically into this range.
(Specifically, I'm looking at quantum mechanical inner products, both real
symmetrical 0-1 and complex -1 to 1)

I was confused as to why A ZAND B is taken as min(A(x),B(x)) since
the ranges defined above are the correct possibilities for "A \cap B".

For instance in the case A=.1 and B=.2 the possible range of overlap
can be expressed graphically as:

Minimum overlap
1234567890
A =
B ==
-0-

Maximum overlap
1234567890
A =
B ==
- .1

Minimum overlap
1234567890
A ======
B ========
---- .4

for A=.6 and B=.8
Maximum overlap
1234567890
A ======
B ========
------ .6

(or use overlapping pie charts)

If A = percentage of students failing question A and,
B = percentage of students failing question B,
then if
C = percentage of students failing both questions A and B,
C will lie in these ranges, and C < = A ZAND B,
what is the rational for A ZAND B = min(A(x),B(x)) then ?

```--

http://www.bestweb.net/~ca314159/

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