# Re: Algebraic sum

N.G. du Bois (*n.dubois@wxs.nl*)

*Sun, 14 Jun 1998 22:26:56 +0200 (MET DST)*

ca314159 wrote:

*> *

*> WSiler wrote:*

*> >*

*> > CA13159 wrote:*

*> >*

*> > > 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 (A \cap B) will lie in the range: max(A+B-1,0)<= C <= min(A(x),B(x))*

*> What is the fuzzy theory rationale for A ZAND B = min(A(x),B(x)) ?*

*> *

*> > >*

*> >*

*> > It is probably improper to use the min-max operator here. If we are using*

*> > probabilities, then (if we want to preserve the laws of excluded middle and*

*> > contradiction)the proper operator to use depends on the prior association (or*

*> > correlation) between A and B. In your case, the probability of failing question*

*> > A and of failing question B are almost certainly somewhat imperfectly*

*> > correlated.*

*> *

*> I tried to choose a very bland case, making no assumptions about*

*> priors or correlations. Just two percentages given and no other knowledge.*

*> In this case I chose to use test questions which was misleading, since*

*> one can make some empirically derived subjective constraints based on*

*> that senario.*

*> *

*> But in general, given membership percentages for two sets, and no*

*> other information, the min-max would seem fundementally appropriate for*

*> defining this set intersection completely. We have no reason to conclude*

*> that it will be a fixed probability, but must instead assume a*

*> distribution. (this is in a sense why von Neuman said: "there are*

*> no dispersion-free states")*

*> *

*> The violation of the law of contradiction, is in fact*

*> quite an interesting aspect of fuzzy theory, because we see something*

*> like this taking place in quantum mechanics. We see that when a particle*

*> can travel between two possible paths, the degree of our knowledge of*

*> which path it took determines the degree to which that particle obeys*

*> classical probability theory.*

*> *

*> Let A, and (not A) represent the two possible paths respectively.*

*> Then the degree of knowledge of whether the particle took the path*

*> A is given as a probability 0-1. If we do not know anything about*

*> which path it took, then A=.5 and (not A)=.5 and in fuzzy theory*

*> the overlap (A zand not A) > 0, which is actually OK since we observe*

*> an interference pattern in this instance which cannot be explained*

*> under a strict particle theory without this idea of (orthogonality)*

*> contradiction violation. So for quantum physics at least, the violation*

*> fits the bill nicely. I'm not sure, but I wouldn't be surprised if*

*> "vacuum fluctuations" (the supposed something from nothing violations*

*> that are theoretically allowed to happen) can be described similarly*

*> *

*> > Let me review three of the inifinity of possible operators.*

*> *

*> <really nice review snipped>*

*> *

*> --*

*> *

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

I like the discussion and the answers, but I still have questions

I also encountered the term "bounded sum". Is'nt that a better term for

sums like A + B - AB; or A + B + C - AB - AC - BC; etcetera.

And should the term algebraic sum not only be used for sums like A + B;

or A + B + C and so on?

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