On 01-12-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
 
        [snip] 
  
 WE>  JB> Now, in answer to your question, "what level of mathematics can 
 WE>  JB> [I] hack?"  It depends upon -what level- you define -hack-?   
 
 WE> Just wanted to know how fuzzy to get.  It would be nice to get a text. 
 WE>  The  
 WE> mathematical one's I've see did include applications which  
 WE> are the best way to get a philosophical grasp of the  
 WE> subject.  I prefer the mathematical text to the engineering  
 WE> text as I lack engineering background, the mathematical  
 WE> text is easier. 
   
  Ah, I am a "practical" person, so, I like to see how such 
and such can be used, typical engineer, eh . 
  
        [snip] 
  
 WE>  JB> When you ask for "the definitions?"  I translate this 
 WE>  JB> to mean "what are the definitions according to -some-  
 WE>  JB> standard reference book?"  According to that translated  
 WE>  JB> question, I don't have an answer for you.   
 
 WE> Sigh, I was hoping that the book you're reading was a text 
 WE> book instead of a philosophical description of fuzz.   
   
  No, it was an introduction into Fuzzy Logic.  I do have 
a textbook, "Fuzzy Logic for the Management of Uncertaintiy;" 
however, I have not had time to get into it; as, I have 
been waiting to finish Bart Kosko's book (I have been 
on page 30 for several years) due to other items that 
take my effort as in work, time here writing, reading 
my required study material (I have to read about 20 to 30 
biological (mostly medical) magazines per month), etc. 
  
 WE>  JB> Yes we are, but the question is, "Is this the definition 
 WE>  JB> of fuzzy sets by standard books on the topic?"  
 
 WE> It's my recollection of a text book definition.  A fuzzy 
 WE> set A is an assignment d, 0<=d<=1, to each element x for  
 WE> the universe of elements.  Notation x e A (d), x belongs to  
 WE> A  with the degree d.  A rock half in a river:  rock e  
 WE> River (1/2). 
   
  In Fuzzy logic, it is my recollection from the book, the 
"truth values" of -an item- ranges from 0 to 1. 
  
 WE>  WE> e is epsilon, Ascii 238, an open e like looking character. 
 WE>  WE> Does it come thru as such?  x e A is x is a member of A,   
 
 WE>  JB> Yes it does. 
 
 WE> It doesn't come back to me that way, it comes as an 'e'. 
  
 Ah, my ability to add Ascii characters, into the editor I 
use, "EDIT", for Offline is limited; so, it would have come 
back at you as "e." 
  Or, perhaps, I should rephrase that, I lack the knowledge 
to add characters that require more than -simple-, 
depending how one defines "simple," typing. 
  
 WE> Not A is the fuzzy set assigning to x the degree 1-d where x e A (d). 
   
  I don't remember the book mentioning this.  The reason for 
my uncertainity in making this definitional point, the set 
not A is quite broad and would include classes (items not part 
A) that would be also part of A, due to their fuzzy nature.   
  
 WE>  JB> The defintion would only fit assuming the limit is one, 
 WE>  JB> 1, and I am not sure x epsilon not A is 1-d, assuming x epsilon  
 WE>  JB> A is d.  
 
 WE> Indeed, refer to the original definition of a fuzzy set 
 WE> where d is in the closed interval [0,1]. 
 
  I don't remember the book mentioning this.  The reason for 
my uncertainity in making this definitional point, the set 
not A is quite broad and would include sets (items not part 
of A) but yet be part of A, due to their fuzzy nature. 
  Assuming the set not A would include sets B, C, D, E, F, 
G, H, I, ....., translating the set Not A, then becomes the 
set {B, C, D, E. AND ....} 
  Using the defintion of x e A and B (min(a,b)) iff x e A(a) 
and x e B(b) (where the little letters stand for x elements 
degree of inclusion into each set), and extending it to the 
total set {B, C, D, E, F,....}, we are to assume -all- these 
elements degrees taken together -somehow- become 1-a.  At 
the present, I am unsure how this is done. 
  
 WE>  WE> x e A or B (max(a,b)) iff x e A (a) or x e B (b) 
 
 WE>  JB> Seems reasonable to assume.  IOW, to put another 
 WE>  JB> way, the truth value of x epsilon belonging to the sets  
 WE>  JB> A or B is the maximam value of the individual truth  
 WE>  JB> values of such sets.  
 
 WE> A or B is the fuzzy set assigning to x the degree max(a,b) 
 WE> where x e A (a) and x e B (b). 
   
  Seems reasonable. 
  
 WE>  WE> x e A & B (min(a,b)) iff x e A (a) and x e B (b) 
 
 WE>  JB> Seems reasonable to assume, IOW, to put another way, 
 WE>  JB> the truth value of x epsilon belonging to the sets A and  
 WE>  JB> B is the minimum value of x epsilon beling to the individual  
 WE>  JB> truth values of such sets.  
 
 WE> A and B is the fuzzy set assigning to x the degree min(a,b) 
 WE> where x e A (a) and x e B (b). 
   
  Seems reasonable. 
  
 WE>  JB> I am sorry, I can't offer any -reference- material definitions. 
 
 WE> I would be nice to have a text to study.  My definitions 
 
   Yes, it would.  I do have at least one mentioned above.  I 
do, however, consider, "Fuzzy Logic" by Bart Kosko, a "textbook" 
to begin with.   
 
 WE> could be checked.  Fuzziness was a subject that I wanted to 
 WE> cover but just didn't.  It took a second to topology.   
 
  Ah, topology would have been fun to take. 
  
 WE> Anyway, were you to pick up a mathematical text, perhaps by 
 WE> the same author as I seem to recollect his name, we could  
 WE> dig into it. 
 
   It would be ok, but my time is limited; so, it would have 
to be in piece meal.  
 
Take care, 
John 
 
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