# Damage and Technical Wear of Tenement Houses in Fuzzy Set Categories
## 3 Results  
The analysis of the results of the research concerning the impact of damage to building elements on their technical wear with regard to fuzzy sets leads to the following conclusions (Table 1):  
* in the field of assessing the degree of fuzzy damage to elements of downtown tenement houses--S(U) II, III, IV:
* the development of the model presented in the article allowed the fundamental question of to what extent a building element is worn (damaged), when knowing that it is (more or less) satisfactorily, moderately or poorly maintained, to be answered;
* the use of simple operations in the fuzzy set calculus enabled the influence of both elementary damage that occurs with a specific frequency (probability) and the measure of its interdependence (correlation) on the observed technical wear of building elements to be considered;
* as a result of the proposed model, which is based on fuzzy set theory, it was possible to identify the elementary damage that determines the degree of destruction of the building's elements;
* when determining the degree of damage of 10 selected building elements according to fuzzy criteria, it was indicated that there is a need for an individual approach to each of the elements (especially structural) during the process of their technical assessment. However, several regularities can be identified:
* the degree of damage to the element increases with the deterioration of its maintenance conditions (although not proportionally to the maintenance conditions and not equally for different types of elements). For instance, degrees of fuzzy damage set S(U) corresponding to the maintenance states II, III and IV grow in the following way: Z3--basement walls--u\({}_{3}\)--brick losses: 0.05; 0.25; 0.67. It most often differs from the observed values of the degree of the technical wear that was determined using the probabilistic approach [1]--in particular, in poor conditions of building maintenance, the degree of damage exceeds 70% of its technical wear threshold;
* elementary damage that determines the degree of destruction of an element comes much more often from group I (mechanical damage to the structure and texture of elements) than was the case in the analysis of the observed states. Only under poor conditions of building maintenance does the analysis of the observed random [1] and fuzzy [15] phenomena show a great similarity--the decisive damage is the destruction of the element caused by water penetration and moisture penetration (group II);
* at the level of the greatest detail, the type of damage and the degrees of fuzzy damage to the elements of the downtown tenement houses were determined. In the most representative, i.e., average/satisfactory condition of maintenance--S (U) III--the degrees were as follows:
* for foundations: brick decay 0.59 for basement walls: brick decrements 0.25 for solid floors above basements: brick decrements 0.22 for structural walls: mortar decrements 0.93 for wooden inter-storey floors: weeping 0.64 for internal stairs: mechanical damage 0.56 for roof constructions: weeping on wooden elements 0.43 for window joinery: mechanical damage 0.85 for inner plasters: plaster decay 0.85 for facades: cracks on plaster 0.94