## 3 Border and Body Definition of T2 FS  
The concept of a type 1 fuzzy set was introduced by [PERSON] in 1965 in [27]. A T1 FS \(A\) on the universe of discourse \(X\) is characterized by its membership function \(\mu_{A}(x)\) and represented as follows (11), [PERSON] [43]. \[A=\{(x,\mu_{A}(x))|\forall x\in X,u_{A}(x)\in[0,1]\} \tag{11}\]  
where \(X\)--universe of discourse, \(\mu_{A}\)--membership function. As the left and right border of MF \(\mu_{A}(x)\) are sometimes difficult to identify accurately, there is some uncertainty about their position. This is why [PERSON], in 1975, in [2], introduced the concept of T2 FS. However, this original definition of T2 FS was a little vague, which prompted attempts to refine it, as well as giving different interpretations of T2 FS. Some scientific confusion has arisen. Therefore, [PERSON], [PERSON] and [PERSON] proposed the ordering of this definition in [43; 44]. In this ordered definition, \(X\) is the universe of discourse for primary variable \(x\) and \(U\) for the secondary variable \(u\in[0,1]\). A T2 FS denoted \(\tilde{A}\) is determined by a T2 MF \(\mu_{\tilde{A}}(x,u)\), where \(x\) belongs to \(X\) and \(u\) belongs to \(U\) (12). \[\tilde{A}=\{((x,u),\mu_{\tilde{A}}(x,u))|x\in X,u\in[0,1]\} \tag{12}\]  
In the current definition (12) of T2 FS, the primary membership \(J_{x}\) does not occur, Equation (13). \[J_{x}=\{(x,u)|u\in[0,1],\mu_{\tilde{A}}(x,u)>0\} \tag{13}\]  
The set \(\tilde{A}\) can also be expressed [44] in a special fuzzy notation (14). \[\tilde{A}=\int\limits_{x\in X}\int\limits_{u\in[0,1]}\mu_{\tilde{A}}(x,u)/(x,u) \tag{14}\]  
The symbol \(\int\) denotes here union over all admissible values of \(x\) and \(u\). In definition of \(\tilde{A}\), the concept of secondary MF [43] is used (15). \[\mu_{\tilde{A}(x)}(u)=\int\limits_{u\in[0,1]}\mu_{\tilde{A}}(x,u)/u=\int \limits_{u\in[0,1]}f_{x}(u)/u \tag{15}\]  
\(\tilde{A}(x)\) is a T1 FS which is referred to as a secondary set. It can be represented by its \(\alpha\)-cut decomposition. \(\{\mu_{\tilde{A}}(u)|u\in U\}\) can be called vertical slice of \(\mu_{\tilde{A}}(x,u)\). In the updated border definition of T2 FS, the concept of footprint of uncertainty (_FOU_) can be used further (16). \[FOU(\tilde{A})=\{(x,u)|x\in X\text{ and }u\in[\underline{\mu}_{\tilde{A}}(x), \overline{\mu}_{\tilde{A}}(x)]\} \tag{16}\]  
In Formula (16), \(\underline{\mu}_{\tilde{A}}(x)\) means lower MF (LMF) and \(\overline{\mu}_{\tilde{A}}(x)\) means upper MF (UMF) [44]. The MFs are given by (17) and (18). \[\underline{\mu}_{\tilde{A}}(x)=\inf\{u\,|\,u\in[0,1],\mu_{\tilde{A}}(x,u)>0\} \tag{17}\]  
\[\overline{\mu}_{\tilde{A}}(x)=\sup\{u|u\in[0,1],\mu_{\tilde{A}}(x,u)>0\} \tag{18}\]A special type of T2 FS is the interval T2 FS (IT2 FS) defined by (19), where all secondary grades \(\mu_{\hat{A}}(x,u)=1\). \[\bar{A}=\{((x,u),\mu_{\hat{A}}(x,u)=1)|\forall x\in X,\forall u\in J_{x}\subseteq[ 0,1]\} \tag{19}\]  
Visualization of a triangular T2 FS is shown in Figure 6.  
Figure 6 clearly shows the border nature of the current definition of T2 FS, which defines only the left and right border of this set. The first definition of T1 FS by [PERSON] from [27] has the same boundary character. Figure 7 shows an example of a triangular MF defining a T1 FS. In addition to the border definition of T1 FS, [PERSON] also published a second, less known definition of this set [2] based on \(\alpha\)-cuts called \(\alpha\)-level sets by him. This definition is readily used in FA. If \(A\) is a fuzzy set of \(X\), then \(\alpha\)-level set of \(A\) is a nonfuzzy set denoted by \(A_{\alpha}\), which comprises all elements of \(X\) whose grade \(\mu\) of membership in \(A\) is greater or equal to \(\alpha\). In symbols  
\[\tilde{A}=\{((x,u),\mu_{\tilde{A}}(x,u))|x\in X,u\in[0,1]\} \tag{12}\]0  
A fuzzy set \(A\) may be decomposed into level sets through the resolution identity (21) or (22).