# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 5 Type-2 Fuzzy Sets vs. Rough-Fuzzy Sets for T-FRI
### Conceptual Comparison  
The uncertain information captured in the primary memberships of a general type-2 fuzzy set consists of a region FOU that is bounded by the lower and upper membership functions. However,  
\begin{table}
\begin{tabular}{c c} \hline \hline Rule 1 & \(\tilde{A}_{1}=<(0,5,6;1),(0,5,6;1)>\) \\  & \(\tilde{B}_{1}=<(0,2,4;1),(0,2,4;1)>\) \\ \hline Rule 2 & \(\tilde{A}_{2}=<(11,13,14;1),(11,13,14;1)>\) \\  & \(\tilde{B}_{2}=<(10,11,13;1),(10,11,13;1)>\) \\ \hline Observation & \(\tilde{A}^{*}=<(7,8,9;1),(7,8,9;1)>\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Involved interval type-2 fuzzy sets for Case 4.  
Figure 7: Interpolation for type-1 fuzzy sets case.  
for an interval type-2 fuzzy set, because the secondary membership grades convey no further information, the FOU is a complete description of all that is embedded in such a fuzzy set [32]. Thus, this single region is sufficiently used to describe uncertainty in interval type-2 fuzzy sets.  
The concept of rough-fuzzy sets is derived from the definition of rough sets [41; 42]. In particular, a rough-fuzzy set is constructed by the lower and upper approximation membership functions, where the lower approximation (LA) is defined by the intersection of all of the given fuzzy sets (certainty) and the upper approximation (UA) by the union of the given sets. That is, the LA captures the minimal level of certainty regarding the information embedded in the original fuzzy sets, and the UA reflects the maximal level. The boundary region, bounded by the two approximations, indicates the region that can possibly, but not always certainly, be partitioned with the information contained within the original fuzzy sets.  
Comparing these two approaches to capturing higher-order uncertainty in describing given information content, it can be revealed that the UA in rough-fuzzy sets plays the same role as the interval in interval-valued fuzzy sets. In fact, the FOU in interval type-2 fuzzy sets is the union of all primary memberships (intervals) [32]; therefore, it is also represented as an interval, covering all uncertain information involved. Both FOU and UA address all uncertain information. As the third-dimension of type-2 fuzzy sets becomes fixed with a membership value of 1, the FOU in such an interval type-2 fuzzy set captures all uncertain information with full certainty, whilst such information can also be captured by means of the LA of a rough-fuzzy set. As a consequence, type-2 fuzzy sets and rough-fuzzy sets are two different extensions of conventional fuzzy sets; the former is fuzzy-fuzzy sets, while the latter is rough-fuzzy sets. Nevertheless, the mathematical foundations of them are different, but they both can be utilised to represent knowledge involving uncertainty information, leading to the development of similar extensions to the conventional transformation-based fuzzy rule interpolation. From this point of view, the approach taken in this work is general, being minor in comparison to the methodology as reported in [23], where rough-fuzzy sets are utilised as the mathematical underpinning.