From: mailbox@dmitry-kazakov.de   
      
   On 9 Feb 2004 08:29:16 -0800, joehv@web.de (joe) wrote:   
      
   >The term level-2 fuzzy set indicates fuzzy sets whose elements are   
   >fuzzy sets.   
   >Am I right? If yes, is it possible to calculate the grades of   
   >membership of a level-2 fuzzy set subject to the grades of membership   
   >of the underlying level-1 fuzzy sets?   
   >   
   >Example:   
   >Universal set:   
   >X = { 1, 2, 3}   
   >Level-1 fuzzy sets:   
   >A1 = { 1/1, 0.7/2, 0.5/3}   
   >A2 = { 0.1/1, 0.4/2, 0.8/3}   
   >Level-2 fuzzy set:   
   >B = { 0.3/A1, 0.9/A2} -> B = { ?/1, ?/2, ?/3} !?   
      
   The second is of course not B, but some other object B*.   
      
   Formally:   
      
   X={1,2,3}   
      
   A1:X->[0,1]   
   A2:X->[0,1]   
   A={A1, A2}   
      
   B:A->[0,1]   
      
   Now you want to build some   
      
   B*:X->[0,1]   
      
   Based on the assumption:   
      
   " if x is in Ai and Ai is in B then x is also in B* "   
      
   Provided that truth values are possibilities. You can estimate the   
   possibility that some x is in B*   
      
   Pos (B*| {x}) <=   
   = Sup min ( Pos (Ai | {x}), Pos (B | Ai) ) =   
   = Max min ( Ai (x), B (Ai) )   
      
   This is the upper estimation of B*: { 0.3/1, 0.4/2, 0.8/3 }   
      
   The lower estimation is:   
      
   Nec (B*| {x}) >= 1 - Pos (not B* | {x}) =   
   = Min max ( 1- Ai (x), B (Ai) )   
      
   It gives: { 0.3/1, 0.3/2, 0.5/3 }   
      
   As you see they are not equal. This is why B* is not fuzzy, but   
   intuitionistic. At best we can say that an imaginary fuzzy B* lies   
   between the upper and the lower boundary sets:   
      
   { 0.3/1, 0.3/2, 0.5/3 } <= B* <= { 0.3/1, 0.4/2, 0.8/3 }   
      
   Wellcome to intuitionistic fuzzy sets!   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   www.dmitry-kazakov.de   
      
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