# Fuzzy Relations Matrixes of Damages and Technical Wear Related to Apartment Houses
## 2 Method of Research
### Setting the Problem--Fuzzy Relations
The problem arose when there was a need to express the interdependencies that are not very strictly defined, e.g., what is the impact (cause) of the ways of maintaining residential buildings on their technical wear (effect) or damage (to the elements), which is its symptom. The concept of a fuzzy relation was used to define the problem and to record the parameters of this phenomenon. Each pair of arguments (x,y)\(\Leftrightarrow\)(z,u) had a membership degree (measure) assigned to it, which expressed the intensity of the relationship between Z and U; that is, how Z depends on U, i.e., what the interrelationships (correlations) between them are. It was assumed that Z and U, as non-crisp sets, which are defined in fuzzy conditions, may have a certain relationship with each other. This led to the determination of the concept of a fuzzy relation in the following form [8; 9; 10; 11; 14]:  
A fuzzy binary relation R between two sets Z = {z} and U = {u} is a relation defined as a fuzzy set that is determined by the Cartesian product Z \(\times\) U:  
\[\text{R}\subseteq\text{Z}\times\text{U}=\{(\text{z, u})\text{: z}\in\text{Z},\text{u}\in\text{U}\} \tag{1}\]  
and, therefore, it is the set of pairs:  
\[\text{R}=\{(\text{uR(z, u),(z, u)})\},\,\forall\,\text{z}\in\text{Z},\,\forall\,\text{u}\in\text{U} \tag{2}\]  
where uR: Z \(\times\) U\(\rightarrow\)[0, 1] is the membership function of fuzzy relation R, which assigns to each pair (z,u): z\(\in\)U its degree of membership uR(z,u)\(\in\)[0, 1], which in turn is a measure of the intensity of the fuzzy relation R between Z and U. Thus, the fuzzy relation can be written as:  
\[\text{R}=\sum_{z,u}\text{uR(z,u)}/(z,u) \tag{3}\]  
The basic fuzzy relation, which is defined in the research at a general level, is the relation concerning the technical wear Z and damage U of the entire residential building that consists of the 23 elements subjected to technical tests. Therefore, Z = {Z1, Z2,..., Z23} and U = {u1, u2,..., u30}. The measure of the relationship between Z (for 10 selected building elements) and U are the values of the point two-series correlation coefficient r(Z), which are identical to the degree of the membership of uR\(\Leftrightarrow\)uZu [9]. Fuzzy relation R = Z \(\times\) U, which was defined in this way, is presented in the form of a fuzzy relation matrix (Table 3). The main parameters of the matrix of fuzzy relation R = Z \(\times\) U are as follows:  
* domain of fuzzy relation: dom R(Z) = 0.70/Z2 + 0.74/Z3 + 0.67/Z4 + 0.67/Z7 + 0.45/Z8 + 0.61/Z9 + 0.57/Z10 + 0.83/Z13 + 0.70/Z15 + 0.84/Z20
* scope of fuzzy relation: ran R(u) = 0.29/u1 + 0.26/u2 + 0.23/u3 + 0.30/u4 + 0.17/u5 + 0.48/u6 + 0.55/u7 + 0.57/u8 + 0.11/u9 + 0.63/u10 + 0.21/u11 + 0.63/u12 + 0.81/u13 + 0.57/u14 + 0.84/u15 + 0.79/u16 + 0.67/u17 + 0.60/u18 + 0.56/u19 + 0.54/u20 + 0.61/u21 + 0.55/u22 + 0.45/u23 + 0.00/u24 + 0.12/u25 + 0.42/u26 + 0.04/u27 + 0.07/u28 + 0.45/u29 + 0.57/u30
* height of fuzzy relation: h[R(Z,u)] = 0.84  
The matrix's components, which represent the fuzzy relations between Z and U, provide the extent to which the maintenance of residential buildings (expressed as the intensity of the occurrence of damage to their elements) affects the amount (degree) of their technical wear.