Triangle Type-2 Fuzzy Sets And Its Set-theoretic Operations

The concept of fuzzy sets (i.e. Type-1 fuzzy sets) was first proposed by Zedeh, the basic idea is to quantify the middle transition of fuzzy concept by means of giving elements the corresponding values of membership degree. Type-2 fuzzy set (T2 FS) is introduced as the extension of type-1 fuzzy sets. Such sets can effectively handle some uncertain information which is much fuzzier. In this paper, we simply introduce some relevant content of Type-1 fuzzy sets. The concept of T2 FSs, its basic operations and laws of operations were discussed in detail.There are just theoretical expressions of operations instead of easily operational formulas for performing union, intersection, and complement on T2 FSs. So it is difficult to apply those operations to type-2 fuzzy logical systems, say nothing of real-world applications. Therefore, many people begin to talk about some special type-2 fuzzy sets for example interval type-2 fuzzy sets (IT2 FSs), convex type-2 fuzzy sets, embedded type-2 fuzzy sets and so on. In the paper, the concept of these special type-2 fuzzy sets and their simplified operations were briefly introduced.A triangle type-2 fuzzy set (TT2 FSs) is also one of the special type-2 fuzzy sets. The calculation complexity of it is between interval type-2 fuzzy sets and general type-2 fuzzy sets. So, it can achieve balance between complexity-reduced and information-protected as much as possible. In this paper, on the one hand, we mainly talk about set-theoretic operations on TT2 FSs, find out some drawbacks of presented theory on TT2 FSs. On the other hand, we reconstruct the framework of set-theoretic operations on TT2 FSs using the method of simplified operations on convex type-2 fuzzy sets, and newly operational formulas for union, intersection and complement of TT2 FSs are derived. Because of only using parameters rather than extension principle, those formulas are computationally simple and manageable.The framework of set-theoretic operations on TT2 FSs is reasonable, but if there are so many parameters, it will be more complex when compute continuous, which restrict the application of triangle type-2 fuzzy sets. For further study, we try to transform polygon type-2 fuzzy set or an n-gon type 2 fuzzy set (n-gon T2 FS) to triangle type-2 fuzzy sets reasonably, and finally give more manageable formulas.