Operators for oversets, undersets and offsets (and for all types of fuzzy and fuzzy extensions sets) in single-valued neutrosophic logic

Abstract

We defined for the first time the concepts of Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset, along with their corresponding logics, in 1995, and published them in 2007. Between 1995 and 2016, we presented these notions at various national and international conferences and seminars. These new constructs are fundamentally different from other existing sets, logics, or probabilities.

We extended the neutrosophic set as follows:

• to a Neutrosophic Overset, when some neutrosophic component is greater than 1;


• to a Neutrosophic Underset, when some neutrosophic component is less than 0;


• to a Neutrosophic Offset, when some components are outside the [0, 1] interval, i.e., one component > 1 and another < 0.

This is not surprising, as the real world provides numerous examples and applications of over-, under-, and off-neutrosophic components.