# Damage and Technical Wear of Tenement Houses in Fuzzy Set Categories
## 2 Research Methodology
### Research Model
The result of the proposed Formula (13) is the following change in the membership function:  
* r(Z) 0 \(\Rightarrow\)\(\mu_{\text{uj}}\) \(\to\) 0;
* r(Z) \(\to\) 1 \(\Rightarrow\)\(\mu_{\text{uj}}\) \(\rightarrow\)\(\mu_{\text{uj}}\):  
The application of the original procedures of the intensification and dilution of membership functions, according to Formulas (12) and (13), to the general Formula (11) of the model for assessing the degree of damage to the elements of the analyzed tenement houses in terms of fuzzy sets allowed for the transition from the data recorded using non-measurable variables to results defined by measurable values. The proposed model gives a numerical answer to the question of to what extent is a building element damaged. The total degrees of damage to the ten selected elements of the analyzed buildings S(U) in the maintenance states II, III and IV are presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Group Number**} & \multirow{2}{*}{
\begin{tabular}{c} **Building** \\ **Element** \\ \end{tabular} } & \multirow{2}{*}{**Damage Number**} & \multirow{2}{*}{**Damage Description**} & \multicolumn{3}{c}{**Degree of Fuzzy Damage Set S(U) Corresponding to the Maintenance States II, III and IV**} \\ \cline{5-8}  & & & & & S(U)II & S(U)III & S(U)IV \\ \hline \multirow{8}{*}{Z2} & \multirow{8}{*}{Foundations} & \(u_{3}\) & brick losses & 024 & 0 & 0 \\  & & \(u_{5}\) & brick decay & 0 & 0.59 & 0 \\  & & \(u_{9}\) & brick cracks & 0 & 0 & 0 \\  & & \(u_{15}\) & dampness of & 0 & 0 & 0 \\  & & \(u_{16}\) & foundations & 0 & 0 & 0.97 \\  & & \(u_{17}\) & biological corrosion of & 0 & 0 & 0 \\  & & \(u_{19}\) & bricks & 0 & 0 & 0 \\  & & \(u_{19}\) & mold and rot on & 0 & 0 & 0 \\  & & & \(u_{19}\) & foundations & 0 & 0 \\ \hline \multirow{8}{*}{Z3} & \multirow{8}{*}{Basement walls} & \(u_{3}\) & brick losses & 0.05 & 0.25 & 0.67 \\  & & \(u_{4}\) & mortar losses & 0 & 0 & 0 \\  & & \(u_{5}\) & brick decay & 0 & 0 & 0 \\  & & \(u_{6}\) & mortar decay & 0 & 0 & 0 \\  & & \(u_{9}\) & cracks in bricks & 0 & 0 & 0 \\  & & \(u_{10}\) & cracks in mortar & 0 & 0 & 0 \\  & & \(u_{15}\) & dampness of walls & 0 & 0 & 0 \\  & & \(u_{16}\) & weeping on walls & 0 & 0 & 0 \\  & & \(u_{17}\) & biological corrosion of & 0 & 0 & 0 \\  & & \(u_{19}\) & mold and rot on walls & 0 & 0 & 0 \\ \hline \multirow{8}{*}{Z4} & \multirow{8}{*}{Solid floors above} & \(u_{3}\) & brick losses & 0.01 & 0.22 & 0 \\  & & \(u_{5}\) & brick decay & 0 & 0 & 0 \\ \cline{1-1}  & & \(u_{9}\) & cracks in bricks & 0 & 0 & 0 \\ \cline{1-1}  & & \(u_{15}\) & dampness of floors & 0 & 0 & 0 \\ \cline{1-1}  & & \(u_{16}\) & weeping on floors & 0 & 0 & 0.50 \\ \cline{1-1}  & & \(u_{20}\) & corrosion raid on steel beams & 0 & 0 & 0 \\ \cline{1-1}  & & \(u_{21}\) & surface corrosion of & 0 & 0 & 0 \\ \cline{1-1}  & & \(u_{22}\) & deep corrosion of steel beams & 0 & 0 & 0 \\ \cline{1-1}  & & \(u_{23}\) & flooding of floors with water & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: The degree of fuzzy damage to building elements in their middle maintenance states. (grey backgroud is necessary to distinguish extreme values)