# Fuzzy Relations Matrixes of Damages and Technical Wear Related to Apartment Houses
## 2 Method of Research
### Setting the Problem--Fuzzy Relations
For all the fuzzy relations \(R\subseteq Z\times U\), which are determined later in the paper for the fuzzy sets that represent satisfactory, average, and poor maintenance conditions of residential buildings, the following were determined [14]:  
the domain of the fuzzy relation \(R\subseteq Z\times U\), which is called the first projection of the fuzzy relation, and denoted by domR:  
\[\mu_{domR(Z)}=\bigvee_{z\in Z}\mu_{R(z,u)}\Leftrightarrow\bigvee_{i=1}^{n}z_{i}= max\{z_{1},z_{2},\dots,z_{n}\} \tag{4}\]  
the scope of the fuzzy relation \(R\subseteq Z\times U\), which is called the second projection of the fuzzy relation, and denoted by ranR:  
\[\mu_{ranR(u)}=\bigvee_{u\in U}\mu_{R(z,u)}\Leftrightarrow\bigvee_{j=1}^{m}u_{j }=max\{u_{1},u_{2},\dots,u_{m}\} \tag{5}\]  
the height of the fuzzy relation \(R\subseteq Z\times U\), which is called the global projection of the fuzzy relation, and marked as h(R)  
\[h(R)=\bigvee_{z\in Z}\mu_{domR(z)}=\bigvee_{u\in U}\mu_{ranR(u)}=\bigvee_{z\in Z \cup U}\mu_{R(z,u)} \tag{6}\]  
where if h(R) = 1, the fuzzy relation is normal, and if h(R) < 1, it is subnormal. The domain, scope, and height of the maximal and minimal fuzzy relations \(R\subseteq Z\times U\) for 10 selected elements in the II, III, and IV maintenance class of a residential building are given in Table 4. The table presents quantitative and measurable values that can be easily quantified. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Group Number** & **Building Element** & **Parameters of Fuzzy** & **Characteristics of Maximal Fuzzy** & **Characteristics of Minimal** \\  & & **Relations** & **Relations** & **Fuzzy Relations** \\ \hline \multirow{3}{*}{Z2} & \multirow{3}{*}{Foundations} & dom R(Z) & 1.00/ZII + 1.00/ZII + 1.00/ZIV & 0.50/ZIII + 0.61/ZIV \\  & & ran R(U) & 1.00/UII + 0.61/UIII + 0.61/UIV & 0.61/UIII + 0.61/UIV \\  & & h[R(Z,U)] & 1.00 & 0.61 \\ \hline \multirow{3}{*}{Z3} & \multirow{3}{*}{Basement walls} & dom R(Z) & 1.00/ZII + 1.00/ZII + 1.00/ZIV & 0.48/ZIII + 0.61/ZIV \\  & & ran R(U) & 1.00/UII + 0.61/UIII + 1.00/UIV & 0.61/UIII \\  & & h[R(Z,U)] & 1.00 & 0.61 \\ \hline \multirow{3}{*}{Z4} & \multirow{3}{*}{Solid floors above} & dom R(Z) & 0.28/ZII + 0.46/ZIII + 1.00/ZIV & 0.28/ZII + 0.46/ZIII \\  & & ran R(U) & 1.00/UII + 1.00/UII + 1.00/UIV & 0.41/UII + 0.46/UIII \\  & & h[R(Z,U)] & 1.00 & 0.46 \\ \hline \multirow{3}{*}{Z7} & \multirow{3}{*}{Structural walls} & dom R(Z) & 1.00/ZII + 1.00/ZII + 1.00/ZIV & 0.48/ZIII + 0.62/ZIV \\  & & ran R(U) & 1.00/UII + 0.62/UIII + 1.00/UIV & 0.62/UIII + 0.43/UIV \\  & & h[R(Z,U)] & 1.00 & 0.62 \\ \hline \multirow{3}{*}{Z8} & \multirow{3}{*}{Inter-story wooden floors} & dom R(Z) & 1.00/ZII + 1.00/ZII + 1.00/ZIV & 0.36/ZIII + 0.52/ZIV \\  & & ran R(U) & 1.00/UII + 1.00/UIII + 1.00/UIV & 0.45/UIII + 0.48/UIV \\  & & h[R(Z,U)] & 1.00 & 0.54 \\ \hline \multirow{3}{*}{Z9} & \multirow{3}{*}{Stairs} & dom R(Z) & 0.28/ZII + 1.00/ZIII + 1.00/ZIV & 0.28/ZII + 0.45/ZIII \\  & & ran R(U) & 1.00/UII + 1.00/UIII + 1.00/UIV & 0.43/UII + 0.45/UIII \\  & & h[R(Z,U)] & 0.45 \\ \hline \multirow{3}{*}{Z10} & \multirow{3}{*}{Rood construction} & dom R(Z) & 1.00/ZII + 1.00/ZII + 1.00/ZIV & 0.30/ZII + 0.44/ZIII \\  & & ran R(U) & 1.00/UII + 1.00/UIII + 1.00/UIV & 0.41/UIII + 0.44/UIV \\  & & h[R(Z,U)] & 1.00 & 0.44 \\ \hline \multirow{3}{*}{Z13} & \multirow{3}{*}{Window joinery} & dom R(Z) & 0.29/ZII + 0.50/ZIII + 1.00/ZIV & 0.29/ZII + 0.50/ZIII + 0.65/ZIV \\  & & ran R(U) & 1.00/UII + 0.65/UIII + 1.00/UIV & 0.40/UII + 0.65/UIII + 0.50/UIV \\  & & h[R(Z,U)] & 1.00 & 0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The domain, scope, and height of fuzzy relations for 10 selected elements in the II, III, and IV maintenance class of a residential building.