On 01-09-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE>  WE> So far this is gab, where's the math?  
 
 WE>  JB> "Fuzzy Logic", he wrote wasn't a "math" book; as, he said 
 WE>  JB> there were other books that dealt with the math.  
 
 WE> Sigh. let's go for the gusto, what level of mathematics can you hack? 
    
  Sigh, in answer to your question, my BS degree (some 14 years ago) 
is in engineering with courses in -mathematics- such as calculus, 
differential equations, statistics and linear algebra.  Other 
courses requiring mathematics include Thermo dynmaics, physics, 
Statics, Dynamics, Fluid Mechanics, Mechanics of Materials, etc. 
  My doctorate rests in biological sciences with little use of such
mathematics over the past 14 years except possibility a minimum 
use of statistics. 
  Now, in answer to your question, "what level of mathematics can 
[I] hack?"  It depends upon -what level- you define -hack-?  I can 
still differentiate and integrate simple equations (such as 
polynomials, some natural log equations, some trigometric equations, 
etc) but doubt I would be able to -EASILY- preform harder equations.  
I still vaguely remember Escholen (? spelling) row reduction 
(a method of solving for unknowns in a system of linear equations) 
Eigen values and vectors, the kernal, etc.   
  
 WE>  JB> In Fuzzy logic set theroy, a "thing" may belong to a set 
 WE>  JB> (A) -AND- (not A) at the same time giving truth values   
 WE>  JB> between and inclusive of zero and one.    
 
 WE>  WE> What's the definitions? 
 
 WE>  JB> I am unsure. 
 
 WE> Have I cover this in my rock and river example? 
   
  When you ask for "the definitions?"  I translate this 
to mean "what are the definitions according to -some- 
standard reference book?"  According to that translated 
question, I don't have an answer for you.  
  
 WE>  WE> fuzzy 
 WE>  WE> set theory an element x belongs to a set A with degree d, 0   
 WE>  WE> <= d <= 1, x member A (d).  For example, A = {rocks in   
 
 WE>  JB> This is my understanding of it. 
 
 WE> Good, we're talking about the same thing.  x e A (d). 
 
  Yes we are, but the question is, "Is this the definition 
of fuzzy sets by standard books on the topic?" 
  
 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,  
 
  Yes it does. 
  
 WE> how the degree of fuzziness is notated I'm not sure, hence 
 WE> my guess x e A (d). 
   
  It is good as any; such notation would have been something I 
would have used.  
  
 WE>  WE> In general x member notA (d) when x member A (1-d), 
 WE>  WE> the definition of not A.  
 
 WE>  JB> Yes, it is, I am going to print it out to ponder it a bit. 
 
 WE> This is the definition of the fuzzy set not A, for the 
 
  The defintion would only fit assuming the limit is one, 
1, and I am not sure x epsilon not A is 1-d, assuming x epsilon 
A is d. 
  I wish I had the time to spend in study on this topic 
as well as Godel's Incompleteness Theorum.  
  
 WE> fuzzy set A & B and the fuzzy set A or B, maybe these are 
 WE> the definitions.  
 
    Perhaps, I don't know what the reference books say. 
  
 WE> x e A or B (max(a,b)) iff x e A (a) or x e B (b) 
 
  Seems reasonable to assume.  IOW, to put another 
way, the truth value of x epsilon belonging to the sets 
A or B is the maximam value of the individual truth 
values of such sets. 
 
 WE> x e A & B (min(a,b)) iff x e A (a) and x e B (b) 
 
  Seems reasonable to assume, IOW, to put another way, 
the truth value of x epsilon belonging to the sets A and 
B is the minimum value of x epsilon beling to the individual 
truth values of such sets. 
  
 WE> 'iff' means 'if and only if'. 
 
  I am sorry, I can't offer any -reference- material definitions. 
 
Take care, 
John 
 
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