# Fuzzy Relations Matrixes of Damages and Technical Wear Related to Apartment Houses
## 2 Method of Research  
### Setting the Problem--Fuzzy Relations  
The basic concept of the theory that was used in this chapter is the concept of a fuzzy set [8; 9; 10; 11; 12; 13; 14]. The definition of a fuzzy set can be formulated as follows: a fuzzy set is set A, the elements x of which are characterized by a lack of a clear boundary between the membership and non-membership of x to A. The degree of the membership of element x to fuzzy set A is described by function \(\mu_{\text{A}}(\text{x})\), which is called the membership function. The \(\mu_{\text{A}}(\text{x})\) function takes values from the interval of [0,1], where:  
\(\mu_{\text{A}}(\text{x})=0\) means that x is not a member of A;  
\(\mu_{\text{A}}(\text{x})=1\) means that x is a full member of A. Fuzzy set A in a certain space (in this paper, it is the area of considerations concerning the observed states) X = {x}, which is written as A \(\subseteq\) X, is called the set of pairs [8; 9; 10; 11; 14]: A = {(uA(x), x)}, \(\forall\) x\(\in\)X. Therefore, two basic fuzzy sets can be distinguished:  
* a fuzzy set of the technical wear of building elements A \(\subseteq\) Ze \(\Leftrightarrow\) Z (to simplify the designations): Z = {(uZ(Z(z), z)}, \(\forall\) z \(\in\) Z;
* a fuzzy set of damage to building elements B \(\subseteq\) U: U = {(uU(u), u)}, \(\forall\) u\(\in\) U. With the use of conventional (non-fuzzy) sets, certain strictly defined properties of theoretical and observed states were adequately expressed, and with the use of the (non \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**No.** & **Damage** & **p(u)2** & **p(u)3** & **p(u)4** & **p(u)7** & **p(u)8** & **p(u)9** & **p(u)10** & **p(u)13** & **p(u)15** & **p(u)20** \\ \hline u1 & Mechanical damage & & & & & & 0.86 & & 0.89 & 0.74 & 0.81 \\ u2 & Leaks & & & & & & 0.93 & & & \\ u3 & Brick losses & 0.78 & 0.66 & & 0.96 & & 0.78 & & & & \\ u4 & Mortar losses & & 0.68 & 0.71 & 0.91 & & & & & \\ u5 & Brick decay & 0.76 & 0.63 & 0.73 & 0.79 & & & & & & \\ u6 & Mortar decay & & & 0.49 & & 0.74 & & & & & 0.8 & 0.86 \\ u7 & Peeling off of paint coatings & & & & & & & & & 0.79 & 0.85 \\ u8 & Falling off of paint coatings & & & & & & & & 0.31 & 0.32 \\ u9 & Cracks in bricks & 0.78 & 0.52 & 0.56 & 0.25 & & & & & & 0.78 & 0.82 \\ u10 & Cracks on plaster & & 0.39 & & & 0.69 & & & & & 0.78 & 0.82 \\ u11 & Scraitching on walls & & & & 0.11 & & & & & & & 0.73 & 0.75 \\ u12 & Scraitching on plaster & & & & & 0.55 & 0.80 & & & & 0.44 & 0.56 \\ u13 & Looseing of plaster & & & & & & 0.49 & & & & 0.13 & 0.22 \\ u14 & Falling off of plaster sheets & & & & & & & & & 0.13 & 0.22 \\ u15 & Dampness & 0.35 & 0.48 & 0.22 & 0.66 & 0.81 & & & 0.62 & 0.47 & 0.14 & 0.54 \\ u16 & Weeping & 0.09 & 0.24 & 0.18 & 0.18 & 0.53 & 0.59 & 0.44 & 0.29 & 0.10 & 0.38 \\ u17 & Biological corrosion of bricks & 0.66 & 0.17 & & & 0.39 & & & & & \\ u18 & Fungus & & & & & & & & & 0.03 & 0.19 \\ u19 & Mold and rot & & 0.03 & 0.05 & & 0.05 & & & & 0.11 & 0.03 & 0.14 \\ u20 & Corrosion raid of steel beams & & & & 0.52 & & & & & 0.72 & & \\ u21 & Surface corrosion of steel beams & & & 0.68 & & & & 0.54 & & & \\ u22 & Deep corrosion of steel beams & & & 0.24 & & & & 0.16 & & & \\ u23 & Flooding with water & & & & 0.04 & & & & & & \\ u24 & Dynamic sensitivity of floor beams & & & & & & 0.64 & & & & \\ u25 & Deformations of wooden beams & & & & & 0.35 & & & & & \\ u26 & Skewing of window joinery & & & & & & & & & 0.75 & \\ u27 & Warping of window joinery & & & & & & & & & 0.59 & \\ u28 & Delamination of wooden elements & & & & & & & & 0.51 & & \\ u29 & Partial insect infestation of wooden elements & & & & & & & & & 0.07 & 0.15 & 0.10 & \\ u30 & Complete insect infestation of wooden elements & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Damage and the probability of its occurrence in 10 elements of the examined buildings. fuzzy) relation, interrelationships between the variables of these states, if they existed, were defined [39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54].