Membership modification and level sets

We introduce the concept of a membership modification function and describe its role in transforming one fuzzy set into another. We discuss the related idea of a membership modification program consisting of a collection of related membership modification functions and show how the concept of level sets is an example of a membership modification program. Using this idea of membership modification allows us to consider other transformations of fuzzy sets that soften the idea of level sets. Using these ideas we provide an extension of the Jaccard similarity index.