Algebraic sum

N.G. du Bois (n.dubois@wxs.nl)
Tue, 13 Oct 1998 23:46:52 +0200 (MET DST)

A few months ago I asked some questions about the algebraic sum. i got
some usefulanswers, but I was not completely satisfied.
The crux for me was the question: what is the algebraic sum of two sets
A,B : 1. A + B (this is stated by Zadeh in his original article
on fuzzy sets, or
2. A + B - AB ( this is used now by many fuzzy experts).

I think that Zadeh was right. By using the term algebraic he ment simply
arithmetic operations, e.g. addition and multiplication.
So A + B and AB are the algebraic operations on sets.

But A + B is NOT the dual of AB! That is wat Zadeh also stated in his
original article.

When you call AB the probabilistic AND (P AND), then the dual is the
probabilistic OR: A + B - AB (P OR).
So what many fuzzy scientists call the algebraic sum is to my opinion
better called the "probabilistic sum".

Let me illustrate this with some formula's.

A + B = algebraic sum
Suppose A = {1,2,3} and B = {3,4,5}.
Then A + B = { 1,2,3,3,4,5}
AND is {3}, and OR is {1,2,3,4,5}, and XOR is {1,2,4,5}.

So in general:

Algebraic sum - AND = OR [A + B - A ^ B = A v B = OR]
Algebraic sum - 2(AND) = XOR [A + B - 2(A ^ B)= XOR]
Algebraic sum - P AND = P OR [A + B - AB = P OR]
Algebraic sum - 2(P AND) =P XOR [A + B - 2(AB)= P XOR]

This way the terminology makes sense to me.
I look out for some reactions.

Nico du Bois.

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