Cuts of intuitionistic fuzzy sets respecting fuzzy connectives

Intuitionistic fuzzy sets (IF-sets) create a suitable tool to describe cases where it is useful to account not only the grade of membership to a certain collection, but also the grade of its non-membership. We consider the αα-cuts of an IF-set A as crisp sets consisting of those elements x for which the truth value (in fuzzy logic) of the statement “x belongs to A and it is not true that x does not belong to A  ” is at least αα. We describe properties of such cuts depending on the chosen type of conjunction and negation.