From: mailbox@dmitry-kazakov.de   
      
   Hi Nico!   
      
   On Wed, 29 Sep 2004 00:28:18 +0200, Nico du Bois wrote:   
      
   > Can you please give some more explenation? Now its like your saying that all   
   > fuzzy numbers "x" are the same, because the singleton "x" is the same. I'm   
   > sure I must be missing something.   
      
   I meant the following. Let's consider for simplicity intervals (they are   
   examples of fuzzy numbers with a rectangular membership function):   
      
   Let   
      
   a = [1,2]   
      
   then   
      
   a/a = [1,2] * [0.5,1] = [min(0.5,1,1,2), max(0.5,1,1,2)] = [0.5,2]   
      
   So a/a is not the singleton [1,1], but [1,1] is a true subset of [0.5,2].   
      
   This result can be deduced from Zadeh's extension principle, I think.   
   Further, I believe it should hold for all alpha cuts, not just for ones at   
   the level of truth = 1.   
      
   BTW, the result above could be improved, should we take into account   
   dependence of arguments. Compare:   
      
   [1,2] + [1,2] = [1,4]   
      
   but   
      
   [1,2] * 2 = [2,4]   
      
   The second is better because it "knows" that [1,2] is the same thing. We   
   could have the same result for [1,2]+[1,2] if the joint membership function   
   of arguments were diagonal and not just min, as it is for independent   
   arguments.   
      
   If we do this thing with a/a considering them dependent, then the result   
   will be:   
      
   (a/a)(x) =   
      
   = Sup a(y,z) =   
   y,z|y/z=x   
      
   = Sup 1 =   
   y,z|y/z=x & y=z   
      
   = 1 iff x=1   
      
   i.e. a/a will be exact 1.   
      
   Unfortunately, in general case we never know...   
      
   With matrices the situation is exactly same. We can be sure that thank to   
   Zadeh's extension principle the result is always correct, but it may get   
   blurred.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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