On 01-17-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE>  WE> other book, what's introductory chapters like?  With substance?  
 
 WE>  JB> It depends upon the definiton of "substance."  I have, 
 WE>  JB> however, found it usefull as I learned from it, only  
 WE>  JB> on page 30 for the past 2 years.   
 
 WE> Substance is explicit definitions, theorems, proofs, formulas. 
    
  So far some defintions via the logo, Greek meaning word, 
method with out any mathematical treatment, so far no  
theoriums, proofs, formulas. 
  
        [snip] 
  
 WE> JB> Not unexpected, this definitional point is key for the rest 
 WE> JB> of our discussion.  Let me try to read and share some of  
 WE> JB> what I read at a future date, but I need to start to taper  
 WE> JB> my other disussions on this echo.   
 
 WE> A fuzzy set A is a function from the universe U of elements into the 
 WE> closed  
 WE> unit interval, x e A (d) could be written A(x) = d.  As a  
 WE> function is a set of ordered pairs, the answer is yes.   
 
 WE> Note that every fuzzy set contains all of the elements x e 
 WE> U to varying degrees.  The fuzzy set U contains all the  
 
  Yep, it would appear that way, if we have the definition 
correct. 
 
 WE> elements with degree 1.  The fuzzy null set not U contains 
 WE> all the elements with degree 0. 
   
 WE> Note the fuzzy set U is different than the set U.  U is the  
 WE> set of elements, for example {x,y,z}, while the fuzzy set U  
 WE> is the function U(x) = 1, U(y) = 1, U(z) = 1 thus assuring  
 WE> that x e U (1), y e U (1), z e U (1) for all the elements  
 WE> in the universe {x,y,z} of my example. 
   
 WE>  JB> I remember from Bart Kosko's  
 WE>  JB> book, somewhere between page 0 and 30, I believe around page  
 WE>  JB> 6 or so, Bart introduced the concept of fuzzy sets, e.g.,  
 WE>  JB> the set of {apples.}  He asked at which point, in consumption  
 WE>  JB> as in eating (at least we are having fun , does the  
 WE>  JB> apple become {not apple?}  
 
 WE> This sounds like descriptive stuff.  Does he give any explicit 
 
   Yes, but I think attempts to explain the point. 
 
 WE> definitions 
 WE> anywhere in the book?  The apple is not an apple when it's  
 
  Ah, but in the world of fuzzy, a half eaten apple belongs to 
the set apple.  Take a apple, take a bite, does it belong to 
the set apples?  
 
 WE> an apple core.  A half eaten apple is half apple.  A slice 
 WE> of an apple is 1/8 apple.  Simple enuf.  Now when a apple  
 WE> is rotten, how included in Apples is it?  How about apple  
 WE> sauce, cider?  Oh how language can out strip mathematical  
 WE> formalism. 
 
  You got it, this is what Fuzzy logis is supposed to help with. 
 
Take care, 
John 
 
___ 
 * OFFLINE 1.54 
--- Maximus 3.01
---------------