2509a336   
   From: mailbox@dmitry-kazakov.de   
      
   On Mon, 20 Oct 2008 04:58:05 -0700 (PDT), 151 wrote:   
      
   > I have queried my tutors for further info but while awaiting a   
   > response I thought it would be worth asking the community here for a   
   > "for dummies" step by step explanation which will hopefully open up   
   > the whole concept of PROVING how fuzzy proposals extend classic set   
   > operations.   
      
   Step 1. Define a *injective* mapping from crisp values to fuzzy values.   
      
   Step 2. Take an operation and prove that for each combination of the   
   arguments, if you substituted crisp values of the arguments by their fuzzy   
   counterparts and applied the fuzzy equivalent operation, then the crisp   
   equivalent of the result would be the same value as if you applied the   
   crisp operation.   
      
   Step 3. Since the mapping was *injective* q.e.d.   
      
   Example:   
      
   1. F -> 0, T -> 1  (injection {F,T} -> [0,1])   
      
   2. /\ -> min   
      
   F /\ F = F, min (0, 0) = 0   
   F /\ T = F, min (0, 1) = 0   
   T /\ F = F, min (1, 0) = 0   
   T /\ T = T, min (1, 1) = 1   
      
   is a proof that min (Zadeh's "and") is an "extension" of /\.   
      
   For further details see category theory, which is basically all about such   
   stuff (relation between mathematical structures):   
      
      http://en.wikipedia.org/wiki/Category_theory   
      http://mathworld.wolfram.com/Category.html   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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