The greatest fuzzy set and the maximum fuzzy set have been defined by using fuzzy logic.By using the relation between cut sets

it is proved that

under reflexive condition

the maximum fuzzy set contains the greatest fuzzy set

and that

under both reflexive and complete conditions

the two sets are equal to each other.In addition

counter examples show that

under S-complete or transitive conditions

the maximum fuzzy set does not equal to the greatest fuzzy set.