## 2 Introduction to Multidimensional Type-1 Fuzzy Arithmetic (MT1 EFA)  
Currently existing fuzzy arithmetic is 2-dimensional in the sense that it assumes that the result of arithmetic operations on 2D-T1 FNs is also 2D-T1 FN. At first glance, it seems obvious, especially as it is in line with the mathematical closure principle [41]. We are used to the fact that the arithmetic operation on crisp numbers gives the same mathematical object, i.e., a crisp number (\(2+3=5\)). However, is it the same in the case of uncertain numbers (interval, fuzzy, probabilistic numbers, etc.)? Let us consider an example that illustrates the problem. In the tank \(A\), we have \(a=5\) m\({}^{3}\) of water and the measurement was made perfectly precise (\(a=5.000000\ldots[\text{m}^{3}]\)). There is an unknown amount of water \(x[\text{m}^{3}]\) in the tank \(X\). Water from the \(A\) and \(X\) tanks are poured into the \(B\) tank and then we measure the perfectly precise total amount of water, obtaining the information \(b=8.000000\ldots[\text{m}^{3}]\). If we want to know the unknown amount of water in the tank \(X\), we have to solve Equation (1) with one unknown \(x\). \[5.000000\ldots+x=8.000000\ldots \tag{1}\]  
After solving it, we will get the precise result \(x=3.000000\ldots[\text{m}^{3}]\). Let us now consider a more realistic example. We have a cylindrical \(A\) tank with a slightly deformed side wall. However, we can determine the outer diameter \(D_{1}\) and inner diameter \(D_{2}\) of this tank and the height \(h\) of the water level. Based on these data, we have established that there is \(a\in[4,6][\text{m}^{3}]\) of water in the tank. There is also an unknown amount of water in the tank \(X\). We poured the water from the tank \(A\) and \(X\) into the cylindrical tank \(B\), which also has slightly distorted side walls. Based on the measurement of the external and internal diameter and the water level in this tank, we found that it contains the amount of water \(b\in[7.5,8.5][\text{m}^{3}]\). The problem is described by Equation (2). \[[4,6]+X=[7.5,8.5] \tag{2}\]  
At this point, however, one should ask: is Equation (2) really an equation with only one unknown? The answer is this: it is an equation with three unknowns, because we do not know not only the exact amount of water \(x\) in tank \(X\), but also the exact amount \(a\) in \(A\) and \(b\) in \(B\)! So, in Equation (2), we have 3 unknowns \(a\), \(x\), \(b\). Can you solve one equation with 3 unknowns? It would seem that you cannot. It turns out, however, that it is possible. However, we must adopt a broader sense of the concept of \"solution\" than in the arithmetic of crisp numbers. In this case, the solution will not be one number, but a set of triples \((a,b,x)\), each of which is one possible state of the addition system in which the relationship \(a+x=b\) or \(x=b-a\) applies, of which we have an additional knowledge \(a\in[4,6]\), \(b\in[7.5,8.5]\). Below are some examples of possible states of the considered system. All possible states constitute a set of possible solutions, which we will denote with \(X\), Equation (3). \[X=\{(a,b,x)|x=b-a,a\in[4,6],b\in[7.5,8.5]\}=\{(4,7.5,3.5),(4.1,7.6,3.5)\ldots(6,8.5,2.5)\} \tag{3}\]  
The above example shows that the set of solutions to the interval Equation (2) is 3-dimensional. Additionally, it is easy to check that there is no 1-dimensional interval set of solutions that would satisfy all 4 possible forms \(([4,6]+X=[7.5,8.5],[4,6]=[7.5,8.5]-X,\)\(X=[7.5,8.5]-[4,6]\), \([4,6]+X-[7.5,8.5]=0)\), of Equation (2). There is, however, a 3-dimensional set \(X\) defined by the Formula (3) which satisfies these equations. As in the case of interval numbers, also in the case of fuzzy numbers T1 FNs and T2 FNs, sets of solutions of the results of arithmetic operations \((+,-,\cdot,/)\), solutions of equations with unknowns, etc., are multidimensional sets. Each uncertain number is a mathematical record (model) of our approximate knowledge of the true value of the variable.