## 4 Multidimensional, Epistemic Fuzzy Arithmetic of Type-2 Fuzzy Numbers
We then obtain the \(FOUSP_{\bar{C}=\bar{A}+\bar{B}}\) shown in Figure 19.  
It should be remembered that \(FOUSP_{\bar{C}=\bar{A}+\bar{B}}\) is not the algebraic and universal result of adding \(\bar{A}+\bar{B}\), i.e., that equation \(\bar{A}+\bar{B}=FOUSP_{\bar{C}=\bar{A}+\bar{B}}\) is not satisfied as suggest low dimensional fuzzy arithmetic types. Moreover, it can easily be checked that: \(\bar{A}\
eq FOUSP_{\bar{C}=\bar{A}+\bar{B}}-\bar{B}\) and \(\bar{B}\
eq FOUSP_{\bar{C}=\bar{A}+\bar{B}}-\bar{A}\). The algebraic and universal result of adding \(\bar{A}+\bar{B}\) is only the 5D number \(\bar{C}\) defined by the Formula (37). Only the number \(\bar{C}\) satisfies all possible, alternative equalities \(\bar{A}=\bar{C}-\bar{B}\), \(\bar{B}=\bar{C}-\bar{A}\), and \(\bar{A}+\bar{B}-\bar{C}=0\). **Negative type 2 fuzzy number \(-\bar{A}\)**  
T2 FN \(\bar{A}\) is given by the general Formula (28). The components of the number \(\bar{A}\) are given by the Formulas (29) and (30). Hence, the specific number \(\bar{A}\) is expressed by Formula (31). The negative fuzzy number \(-\bar{A}\) in its general form is given by the Formula (39), and its specific version by the Formula (40). \[-\bar{A}(c,\gamma_{\bar{A}},\mu):x_{-\bar{A}}(c,\gamma_{\bar{A}},\mu)=-x_{\bar {A}}(c=0,\gamma_{\bar{A}},\mu)-c[x_{\bar{A}}(c=1,\gamma_{\bar{A}},\mu)-x_{\bar {A}}(c=0,\gamma_{\bar{A}},\mu)],c,\gamma_{\bar{A}},\mu\in[0,1] \tag{39}\]  
\[-\bar{A}(c,\gamma_{\bar{A}},\mu):x_{-\bar{A}}(c,\gamma_{\bar{A}},\mu)=-(1+\mu +2c)-\gamma_{\bar{A}}(8-3\mu-3c),c,\gamma_{\bar{A}},\mu\in[0,1] \tag{40}\]  
Figure 20 shows T2 FN \(\bar{A}\) and its negative version in 3D space and Figure 21 shows cuts of \(\bar{A}\) and \(-\bar{A}\) in the form of \(FOU\bar{A}\) and \(FOU(-\bar{A})\). **Inverse type 2 fuzzy number**  
T2 FN \(\tilde{A}\) is determined by general Formula (41) and its specific triangular version is given by (42), \(0\
otin\tilde{A}\). \[\tilde{A}(c,\gamma_{A},\mu):x_{\tilde{A}}(c,\gamma_{A},\mu)=x_{\tilde{A}}(c=0, \gamma_{A},\mu)+c[x_{\tilde{A}}(c=1,\gamma_{\tilde{A}},\mu)-x_{\tilde{A}}(c=0, \gamma_{\tilde{A}},\mu)],c,\gamma_{\tilde{A}},\mu\in[0,1] \tag{41}\]  
\[\tilde{A}(c,\gamma_{\tilde{A}},\mu):x_{\tilde{A}}(c,\gamma_{\tilde{A}},\mu)=(1+ 3\mu+2c-2c\mu)+\gamma_{\tilde{A}}(5-5\mu-3c+3c\mu),c,\gamma_{\tilde{A}},\mu\in[ 0,1] \tag{42}\]  
Inverse T2 FN \(\tilde{A}^{-1}\) is determined by general Formula (43) and its specific version by (44). \[\tilde{A}^{-1}(c,\gamma_{\tilde{A}},\mu):x_{\tilde{A}^{-1}}(c,\gamma_{\tilde{A }},\mu)\in\tilde{A}^{-1}(c,\gamma_{\tilde{A}},\mu) \tag{43}\]  
\[x_{\tilde{A}^{-1}}(c,\gamma_{\tilde{A}},\mu)=(x_{\tilde{A}}(c=0,\gamma_{ \tilde{A}},\mu))^{-1}+c[(x_{\tilde{A}}(c=1,\gamma_{\tilde{A}},\mu))^{-1}-(x_{ \tilde{A}}(c=0,\gamma_{\tilde{A}},\mu))^{-1}],c,\gamma_{\tilde{A}},\mu\in[0,1] \tag{44}\]  
Figure 16:0  
Figure 16:1  
Figure 22 shows the triangular T2 FN \(\tilde{A}\) in 3D space and Figure 23 in 2D space in the form of its _FOU_. Figure 16:2 Cuts of T2 MF \(\tilde{A}\) and of \(-\tilde{A}\) made for various confidence \(c\)-values or \(FOU\tilde{A}\) and \(FOU(-\tilde{A})\). Figure 16:3 Type 2 fuzzy number \(\tilde{A}\) and its negative version \(-\tilde{A}\) (opposite number). Figure 24 shows the inverse T2 FN \(\bar{A}^{-1}\) in 3D space and Figure 25 in 2D space. If T2 FN contains zero, some knowledge about the inverse number \(1/\bar{A}\) can be obtained, see [13]. **Multiplication of T2 FNs without zero**  
Let us consider two T2 FNs \(\bar{A}\) and \(\bar{B}\) given by the general Formulas (28) and (32), where their specific versions are given by the Formulas (31) and (35), respectively. _FOU_ of these numbers are shown in Figures 15 and 17. The operation of the multiplication of T2 FNs \(\bar{A}\) and \(\bar{B}\) is performed according to general Formula (45). Figure 16:4  
For specific T2 FNS \(\bar{A}\) and \(\bar{B}\) given by Formulas (31) and (35), the result of the multiplication is given by the Formula (46).