>>> John Boone on Fuzzy 
 JB> No, it was an introduction into Fuzzy Logic.  I do have 
 JB> a textbook, "Fuzzy Logic for the Management of Uncertaintiy;" 
Is the intro just descriptive, without any theory or serious material  I'd 
suggest skipping it.  How about the other book, what's introductory chapters 
like?  With substance?
 JB> my required study material (I have to read about 20 to 30 
 JB> biological (mostly medical) magazines per month), etc. 
Are there biological applications for fuzziness?  How about the text book, 
can you find any biological examples?
 JB> In Fuzzy logic, it is my recollection from the book, the 
 JB> "truth values" of -an item- ranges from 0 to 1. 
Is this like probability that assigns a probability to statements.  How is it 
different?
 WE> Not A is the fuzzy set assigning to x the degree 1-d where x e A (d). 
 JB> 
 JB> I don't remember the book mentioning this.  The reason for 
 JB> my uncertainity in making this definitional point, the set 
 JB> not A is quite broad and would include classes (items not part 
 JB> A) that would be also part of A, due to their fuzzy nature.   
No, the notion of set is a stratified notion.  First there are elements, then 
there are sets of elements.  Third level is sets of the first two levels.  
Etc.  So there is a universe U of elements, distinct from sets.  A fuzzy set 
is an assignment of degrees of each element in U to a real number d in the 
closed unit interval.  So the fuzzy set not A is the assignment of 1 - d to x 
when x is assigned d by A.  For simplicity, I happy to stop at the first 
level.  Now if you want to extend this to higher strata you may do so.  
Beware these strata ascend unto infinities upon infinities culminating in 
paradoxes.  In aggregate, it is called the constructed universe as it is 
constructed from a bunch of elements and sets generated therefrom.
 JB> I don't remember the book mentioning this.  The reason for 
Fuzzy sets and fuzzy logic are different.  Perhaps that is why.
 JB> Assuming the set not A would include sets B, C, D, E, F, 
 JB> G, H, I, ....., translating the set Not A, then becomes the 
 JB> set {B, C, D, E. AND ....} 
No, I didn't assume that.  In fact I assumed the contrary.
 JB> Using the defintion of x e A and B (min(a,b)) iff x e A(a) 
 JB> and x e B(b) (where the little letters stand for x elements 
 JB> degree of inclusion into each set), and extending it to the 
 JB> total set {B, C, D, E, F,....}, we are to assume -all- these 
 JB> elements degrees taken together -somehow- become 1-a.  At 
 JB> the present, I am unsure how this is done. 
Me neither as I'm not up on extensions of fuzzy set theory beyond the first 
strata.
 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). 
 JB> Seems reasonable. 
 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). 
 JB> Seems reasonable. 
A is included in B, A <= B when for each element x in U the degree a of x in 
A is <= the degree b of x in B.  A <= B if for all elements x in U, x e A (a) 
& x e B (b) implies a <= b.  Note that A included in B and B included in A 
implies that A = B, that is for all elements x in U, x e A (a) & x e B (b) 
implies a = b.  The assignments A and B of degrees to the elements are the 
same.
 JB> Yes, it would.  I do have at least one mentioned above.  I 
 JB> do, however, consider, "Fuzzy Logic" by Bart Kosko, a "textbook" 
 JB> to begin with.   
Does it have any theory in it?  If so what?
 
 JB> Ah, topology would have been fun to take. 
It was.  Took a class in set theoretic topology which was interesting but 
discovered that the really interesting topology, the stuff that imagined 
different types of surfaces or tied you into knots was more advanced.
 JB> It would be ok, but my time is limited; so, it would have 
 JB> to be in piece meal.  
Well let's piece it together. -)  What mathematical stuff is in the books you 
already have?  I'd like to get a sharper fix on the difference between fuzzy 
logic and fuzzy sets.
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