Level sets and the representation theorem for intuitionistic fuzzy sets

We discuss the concept of a level set of a fuzzy set and the related ideas of the representation theorem and Zadeh’s extension principle. We then describe the extension of these ideas to the case of interval valued fuzzy sets (IVFS). We then recall the formal equivalence between IVFS and intuitionistic fuzzy sets (IFS). This equivalence allows us to naturally extend the concepts of level sets, representation theorem and extension principle from the domain of IVFS to the domain of IFS. What is important to note here is that in the case of these non-standard fuzzy sets, interval valued and intuitionistic, the number of distinct level sets can be greater then the number of distinct membership grades of the fuzzy set being represented. This is a result of the fact that the distinct level sets are generated by the power set of the membership grades. In particular, the minimum of each subset of membership grades provides a level set. In the case of the standard fuzzy sets the minimum of a subset of membership grades results in one of the elements in the subset. In the case of the non-standard fuzzy sets, the membership grades are not linearly ordered and hence taking the minimum of a subset of these can result in a value that was not one of the members of the subset.