8.6.4. Transformations Between Fuzzy Sets and Ordinary Sets

 

Definition 7. Given  Œ F(W), for any l Œ [0, 1]

Al = { w‘m ³ l } is called the l-level cut of  .

If any w whose membership value exceeds l is considered as a member of  , then fuzzy set  becomes ordinary set Al .

For instance, from  = (1, 0.8, 0.4, 0) we have

C1 = {0} , C0.5 = C0.8 = {a, b} .

If  Œ F(W) , it can be proven that

m =  ,

where XAl is the characteristic function for Al . This theorem and the level cut concept are the linkages for conversions between fuzzy sets and ordinary sets.