# Fuzzy Relations Matrixes of Damages and Technical Wear Related to Apartment Houses
## 2 Method of Research
### Fuzzy Relational Equations
#### 2.2.2 The Minimal Relational Equation  
The last empirical stage of the model for determining the minimal cause-effect relationship was based on the same adjustments as in the case of determining the maximal solution (see Section 2.2.1). However, the assumptions and the defined composition of the membership function are different. The minimal cause-effect solution, which satisfies Equation (7), is determined by relationship:  
\[\overset{\Delta}{\mathrm{R}}\ =\ \mathrm{A}\ -\ 1\ \sigma\ \mathrm{B}\ =\ \mathrm{A}\ \sigma\ \mathrm{B} \tag{16}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Fuzzy Relational Equations of Damage (U), Which Correspond to the II, III, and IV Conditions of the Maintenance of the**} & \multicolumn{3}{c}{**Element and Its Technical Wear (Z)**} & \multicolumn{3}{c}{**Minimal Fuzzy Relational Equation**} \\ \hline \multirow{3}{*}{Z2} & \multirow{3}{*}{Foundations} & & UII & UIII & UIV & UII & UIII & UIV \\  & & ZII & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & & ZIII & 1.00 & 0.50 & 0.00 & 0.00 & 0.50 & 0.00 \\  & & ZIV & 1.00 & 0.61 & 0.61 & 0.00 & 0.61 & 0.61 \\ \hline \multirow{3}{*}{Z3} & \multirow{3}{*}{Basement walls} & & UII & UIII & UIV & UII & UIII & UIV \\  & & ZII & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & & ZIII & 1.00 & 0.48 & 0.00 & 0.00 & 0.48 & 0.00 \\  & & ZIV & 1.00 & 0.61 & 1.00 & 0.00 & 0.61 & 0.00 \\ \hline \multirow{3}{*}{Z4} & \multirow{3}{*}{Solid floors above} & & UII & UIII & UIV & UIII & UIII & UIV \\  & & ZII & 0.28 & 0.00 & 0.00 & 0.28 & 0.00 & 0.00 \\  & & basements & ZIII & 0.41 & 0.46 & 0.00 & 0.41 & 0.46 & 0.00 \\  & & ZIV & 1.00 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 \\ \hline \multirow{3}{*}{Z7} & \multirow{3}{*}{Structural walls} & & UII & UIII & UIV & UII & UIII & UIV \\  & & ZII & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & & ZIII & 1.00 & 0.48 & 0.43 & 0.00 & 0.48 & 0.43 \\  & & ZIV & 1.00 & 0.62 & 1000 & 0.00 & 0.62 & 0.00 \\ \hline \multirow{3}{*}{Z8} & \multirow{3}{*}{Inter-story wooden} & & UII & UIII & UIV & UII & UIII & UIV \\  & & ZII & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & & ZIII & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & & ZIV & 1.00 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 \\ \hline \multirow{3}{*}{Z9} & \multirow{3}{*}{Stairs} & & UII & UIII & UIV & UII & UIII & UIV \\  & & ZII & 0.28 & 0.00 & 0.00 & 0.28 & 0.00 & 0.00 \\  & & ZIII & 0.43 & 0.45 & 1.00 & 0.43 & 0.45 & 0.00 \\  & & ZIV & 1.00 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 \\ \hline \multirow{3}{*}{Z10} & \multirow{3}{*}{Roof construction} & & UII & UIII & UIV & UIII & UIII & UIV \\  & & ZII & 1.00 & 0.00 & 0.30 & 0.00 & 0.00 & 0.30 \\  & & ZIII & 1.00 & 0.41 & 0.44 & 0.00 & 0.41 & 0.44 \\  & & ZIV & 1.00 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 \\ \hline \multirow{3}{*}{Z13} & \multirow{3}{*}{Window joinery} & & UII & UIII & UIV & UIII & UIII & UIV \\  & & ZII & 0.29 & 0.00 & 0.00 & 0.29 & 0.00 & 0.00 \\  & & ZIII & 0.4 & 0.50 & 0.50 & 0.40 & 0.50 & 0.50 \\  & & ZIV & 1.00 & 0.65 & 1.00 & 0.00 & 0.65 & 0.00 \\ \hline \multirow{3}{*}{Z15} & \multirow{3}{*}{Inner plasters} & & UII & UIII & UIV & UII & UIII & UIV \\  & & ZII & 0.30 & 0.00 & 0.00 & 0.30 & 0.00 & 0.00 \\ \cline{1-1}  & & ZIII & 1.00 & 0.44 & 0.00 & 0.00 & 0.44 & 0.00 \\ \cline{1-1}  & & ZIV & 1.00 & 0.65 & 1.00 & 0.00 & 0.65 & 0.00 \\ \hline \multirow{3}{*}{Z20} & \multirow{3}{*}{Facades} & & UII & UIII & UIV & UII & UIII & UIV \\ \cline{1-1}  & & ZII & 0.28 & 0.00 & 0.00 & 0.28 & 0.00 & 0.00 \\ \cline{1-1}  & & ZIII & 0.45 & 0.44 & 0.00 & 0.45 & 0.44 & 0.00 \\ \cline{1-1}  & & ZIV & 1.00 & 1.00 & 0.68 & 0.00 & 0.00 & 0.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Maximal and minimal equations of relational damage to elements in the II, III, and IV maintenance class of a residential building.