Type 2 Fuzzy Sets and Examples of FOU
The member function of the unqualified type 2 fuzzy set A is three- dimensional, where the third dimension is the value of the member function at each point in the two-dimensional domain tabbed in the Footprint of Uncertainty (FOU) (Gray, 2011).
The membership function of the unqualified type 2 fuzzy set is three- dimensional (3D). The cross section and others are in the FOU.
For an interval type 2 fuzzy set, the 3D values are the same everywhere, so no new information is included in the three dimensions of the interval type 2 fuzzy set. So, for such a set, the third dimension is ignored and only the FOU is used to describe it. For this reason, spacing type 2 fuzzy sets are sometimes represented as tabs in the first-order uncertainty fuzzy set model, while unqualified type 2 fuzzy sets (with useful three dimensions) are also referred to as secondary.
FOU stands for the ambiguity of the type 1 membership function. It is fully described as two boundary functions: LMF (lower membership function) and UMF (upper membership function), both of which are type 1 fuzzy sets! As a result, type 1 fuzzy set math can be used to type and work with interval type 2 fuzzy sets. In this way, engineers and scientists who once knew type 1 fuzzy sets would not have to spend a lot of time learning almost unqualified type 2 fuzzy set math to understand and use interval type 2 fuzzy sets.
Work on type 2 fuzzy sets withered during the early 1980s and mid-1990s, despite the fact that a handful of products were almost published. Since people were still contemplating what to do with a type 1 fuzzy set, the plane was proposed by Zadeh in 1976 as a type 2 fuzzy set, but it wasn't time to give up what the researchers did with the type 1 fuzzy set. This came back in the late 1990s as a result of research by Professor Jerry Mendel and his students on type 2 fuzzy sets and systems. Since then, more and more researchers have been creating products close to type 2 fuzzy sets and systems almost worldwide.
Upper and Lower Membership Functions
I assume that the level of information is not irreversible in specifying the member functions exactly. For example, we can only know' the upper and low'er premises of the membership class for each element of the universe for a fuzzy set. These fuzzy sets are described as interval value membership functions. For a particular factor x=z, membership is fuzzy set A. That is, pA (z) is expressed as the membership interval [a,, aj. A fuzzy set with an interval value can be generalized further by allowing that interval to fuzzy wither. Then, each membership interval becomes a regular fuzzy set.
A Type 1 Purge Set Represented by a Type 2 Fuzzy Set
A type 1 fuzzy set can be interpreted as a type 2 fuzzy set with all of the second ranks being single (i.e. all flags being 1). In fact, a type 1 fuzzy set is an instance of a type 2 fuzzy set. This is a clear version of the type 2 fuzzy set. Given this case, it makes sense to consider using the type 1 definitions for joins, intersections, and complements as a starting point and generalizing them to a type 2 fuzzy set, which is a type 1 fuzzy set.
and 1 Membership of Type 2 Fuzzy Set
The fuzzy set's membership function is a generalization of the classic set's indicator function (Chatzis et al., 2000). In fuzzy logic, it represents the layer of truth as an extension of valuation. Ambiguous truth is conceptually distinguished when considering that it is a member of an ambiguously specified set rather than the likelihood of some event or condition, but the degree of truth often falls with the probability. The membership feature was introduced by Zadeh in his first paper on fuzzy sets (1965). Zadeh suggested in fuzzy set theory to use a membership function (with range tent (0,1)) that works in the domain of all possible values.