>>> John Boone on Fuzzy 
 JB> Sigh, in answer to your question, my BS degree (some 14 years ago) 
 JB> is in engineering with courses in -mathematics- such as calculus, 
Good, so we can get into the fuzz.
 JB> Now, in answer to your question, "what level of mathematics can 
 JB> [I] hack?"  It depends upon -what level- you define -hack-?  
Just wanted to know how fuzzy to get.  It would be nice to get a text.  The 
mathematical one's I've see did include applications which are the best way 
to get a philosophical grasp of the subject.  I prefer the mathematical text 
to the engineering text as I lack engineering background, the mathematical 
text is easier.
 JB> I still vaguely remember Escholen (? spelling) row reduction 
 JB> (a method of solving for unknowns in a system of linear equations) 
 JB> Eigen values and vectors, the kernal, etc.   
Never got a clear notion of Eigen values myself.
 JB> When you ask for "the definitions?"  I translate this 
 JB> to mean "what are the definitions according to -some- 
 JB> standard reference book?"  According to that translated 
 JB> question, I don't have an answer for you.  
Sigh, I was hoping that the book you're reading was a text book instead of a 
philosophical description of fuzz.  
 JB> Yes we are, but the question is, "Is this the definition 
 JB> of fuzzy sets by standard books on the topic?" 
It's my recollection of a text book definition.  A fuzzy set A is an 
assignment d, 0<=d<=1, to each element x for the universe of elements.  
Notation x e A (d), x belongs to A  with the degree d.  A rock half in a 
river:  rock e River (1/2).
 WE> e is epsilon, Ascii 238, an open e like looking character. 
 WE> Does it come thru as such?  x e A is x is a member of A,  
 JB> Yes it does. 
It doesn't come back to me that way, it comes as an 'e'.
Not A is the fuzzy set assigning to x the degree 1-d where x e A (d).
 JB> The defintion would only fit assuming the limit is one, 
 JB> 1, and I am not sure x epsilon not A is 1-d, assuming x epsilon 
 JB> A is d. 
Indeed, refer to the original definition of a fuzzy set where d is in the 
closed interval [0,1].
 WE> x e A or B (max(a,b)) iff x e A (a) or x e B (b) 
 JB> Seems reasonable to assume.  IOW, to put another 
 JB> way, the truth value of x epsilon belonging to the sets 
 JB> A or B is the maximam value of the individual truth 
 JB> values of such sets. 
A or B is the fuzzy set assigning to x the degree max(a,b) where x e A (a) 
and x e B (b).
 WE> x e A & B (min(a,b)) iff x e A (a) and x e B (b) 
 JB> Seems reasonable to assume, IOW, to put another way, 
 JB> the truth value of x epsilon belonging to the sets A and 
 JB> B is the minimum value of x epsilon beling to the individual 
 JB> truth values of such sets. 
A and B is the fuzzy set assigning to x the degree min(a,b) where x e A (a) 
and x e B (b).
 JB> I am sorry, I can't offer any -reference- material definitions. 
I would be nice to have a text to study.  My definitions could be checked.  
Fuzziness was a subject that I wanted to cover but just didn't.  It took a 
second to topology.  Anyway, were you to pick up a mathematical text, perhaps 
by the same author as I seem to recollect his name, we could dig into it.
---
---------------