Type-reduction of the discretised interval type-2 fuzzy set: What happens as discretisation becomes finer?

The defuzzification of a type-2 fuzzy set is a two stage process consisting of firstly type-reduction, and secondly defuzzification of the resultant type-1 set. This paper considers three approaches to discrete interval type-reduction: 1) The exhaustive method which produces the Type-Reduced Set (TRS), 2) the Greenfield-Chiclana Collapsing Defuzzifier which gives rise to the Representative Embedded Set Approximation (RESA), and 3) the Nie-Tan Method from which the Nie-Tan Set (NTS) is derived. In the discrete case these three type-1 sets are distinct. This paper investigates the relationships between the three sets under fine discretisation, in order to shed light on the relationships between the continuous versions of these type-1 sets.