## 4 Multidimensional, Epistemic Fuzzy Arithmetic of Type-2 Fuzzy Numbers
It is easy to see that there is no special need to represent the T2 FNs in the form of 3-dimensional figures. Presenting these numbers in the form of 2D-FOUs is sufficient. If the support-T1 MF transition in core-T1 MF is linear, it is enough to show only 2 cuts. If the transition is non-linear, 3 or more cuts of 3D-T2 MF can be shown. The number \(\tilde{B}\) with Definition 4 is given by the general Formula (32) and its constituent elements by the Formulas (33) and (34). \[\tilde{B}(c,\gamma_{\tilde{B}},\mu):x_{\tilde{B}}(c,\gamma_{\tilde{B}},\mu)=x_ {\tilde{B}}(c=0,\gamma_{\tilde{B}},\mu)+c[x_{\tilde{B}}(c=1,\gamma_{\tilde{B}},\mu)-x_{\tilde{B}}(c=0,\gamma_{\tilde{B}},\mu)],c,\gamma_{\tilde{B}},\mu\in[0,1] \tag{32}\]  
\[\tilde{B}(c=0,\gamma_{\tilde{B}},\mu):x_{\tilde{B}}(c=0,\gamma_{\tilde{B}},\mu )=[b_{1}+\mu(b_{2}-b_{1})]+\gamma_{\tilde{B}}\{[b_{4}-\mu(b_{4}-b_{3})]-[b_{1}+ \mu(b_{2}-b_{1})]\},\gamma_{\tilde{B}},\mu\in[0,1] \tag{33}\]  
\[\tilde{B}(c=1,\gamma_{\tilde{B}},\mu):x_{\tilde{B}}(c=1,\gamma_{\tilde{B}},\mu )=[b_{5}+\mu(b_{6}-b_{5})]+\gamma_{\tilde{B}}\{[b_{8}-\mu(b_{8}-b_{7})]-[b_{5} +\mu(b_{6}-b_{5})]\},\gamma_{\tilde{B}},\mu\in[0,1] \tag{34}\]  
\[\tilde{B}(c,\gamma_{\tilde{B}},\mu):x_{\tilde{B}}(c,\gamma_{\tilde{B}},\mu)=(1 0+3\mu+3c-c\mu)+\gamma_{\tilde{B}}(11-7\mu-7c+3c\mu),c,\gamma_{\tilde{B}},\mu \in[0,1] \tag{35}\]  
**Adding T2 FNs**  
For the assumed numbers \(\tilde{A}\), Formulas (28) and (31) and \(\tilde{B}\), Formulas (32) and (35), the result \(\tilde{C}\) of addition is given by the Formula (36). \[ \begin{array}{l}\tilde{A}(c,\gamma_{\tilde{A}},\mu)+\tilde{B}(c,\gamma_{ \tilde{B}},\mu)=\tilde{C}(c,\gamma_{\tilde{A}},\gamma_{\tilde{B}},\mu):x_{ \tilde{A}}(c,\gamma_{\tilde{A}},\mu)+x_{\tilde{B}}(c,\gamma_{\tilde{B}},\mu) =x_{\tilde{C}}(c,\gamma_{\tilde{A}},\gamma_{\tilde{B}},\mu),c,\gamma_{\tilde{A }},\gamma_{\tilde{B}},\mu\in[0,1]\\ x_{\tilde{C}}(c,\gamma_{\tilde{A}},\gamma_{\tilde{B}},\mu)=[(1+\mu+2c)+ \gamma_{\tilde{A}}(8-3\mu-3c)]+[(10+3\mu+3c-c\mu)+\gamma_{\tilde{B}}(11-7\mu -7c+3c\mu)]\end{array}  \tag{36}\]  
As can be seen from (36), the result \(x_{\tilde{C}}(c,\gamma_{\tilde{A}},\gamma_{\tilde{B}},\mu)\) exists in 5D space, \(\gamma_{\tilde{A}}\) corresponds to \(x_{\tilde{A}}\) here, \(\gamma_{\tilde{B}}\) corresponds to \(x_{\tilde{B}}\). The 5D result cannot be seen. However, you can see simplified information about it in a 2D or 3D space, such as its span, cardinality distribution, or gravity center. However, fuzzy systems specialists are most often interested in the horizontal uncertainty of the calculated results in the form of span. In the case of a 5D-result \(\tilde{C}=\tilde{A}+\tilde{B}\), its span can be determined from the Formula (37). \[SP_{\tilde{C}=\tilde{A}+\tilde{B}}=[\min_{\gamma_{\tilde{A}},\gamma_{\tilde{B} }}x_{\tilde{C}}(c,\gamma_{\tilde{A}},\gamma_{\tilde{B}},\mu),\max_{\gamma_{ \tilde{A}},\gamma_{\tilde{B}}}x_{\tilde{C}}(c,\gamma_{\tilde{A}},\gamma_{ \tilde{B}},\mu)] \tag{37}\]  
Figure 16: Type-2 fuzzy number \(\tilde{B}\) in 3D-space. Examination of the function \(x_{\mathcal{C}}(c,\gamma_{A},\gamma_{B},\mu)\), Formula (37) shows that \(\min x_{\mathcal{C}}\) is obtained for \(\gamma_{\bar{A}}=\gamma_{\bar{B}}=0\) and \(\max x_{\bar{C}}\) for \(\gamma_{\bar{A}}=\gamma_{\bar{B}}=1\). As a result, we obtain the span \(SP_{\bar{C}}\) defined by the Formula (38). \[SP_{\bar{C}=\bar{A}+\bar{B}}=\left[(11+4\mu+5c-c\mu),(30-6\mu-5c+2c\mu)\right] \tag{38}\]  
Figure 18 shows the visualization of the span function \(SP_{\bar{C}=\bar{A}+\bar{B}}\). Span \(SP_{\bar{C}=\bar{A}+\bar{B}}\) does not need to be visualized in 3D space. It can be more easily presented in 2D space in the form of cuts made for various \(c\)-values entered into the Formula (38), e.g., for values \(c=0,c=0.5,c=1\).