Membership Functions Representing a Number vs. Representing a Set: Proof of Unique Reconstruction

Authors

Hung T. Nguyen, New Mexico State University - Main CampusFollow
Vladik Kreinovich, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow

Publication Date

4-2016

Comments

Technical Report: UTEP-CS-15-96a

To appear in Proceedings of the 2016 IEEE International Conference on Fuzzy Systems FUZZ-IEEE'2016, Vancouver, Canada, July 24-29, 2016.

Abstract

In some cases, a membership function m(x) represents an unknown number, but in many other cases, it represents an unknown crisp set. In this case, for each crisp set S, we can estimate the degree m(S) to which this set S is the desired one. A natural question is: once we know the values m(S) corresponding to all possible crisp sets S, can we reconstruct the original membership function? In this paper, we show that the original membership function m(x) can indeed be uniquely reconstructed from the values m(S).