The fuzzy set concept is often used in solution of problems in which the initial data is difficult or impossible to represent in the form of specific numbers or sets. Geo-information objects are distinguished by their uncertainty, their characteristics are often vague and have some error. Therefore, in the study of such objects is introduced the concept of "fuzziness" — fuzzy sets, fuzzy logic, linguistic variables, etc. The fuzzy set concept is given in the form of membership function. An ordinary set is a special case of a fuzzy one. If we consider a fuzzy object on the map, for example, a lake that changes its shape depending on the time of year, we can build up for it a characteristic function from two variables (the object’s points coordinates) and put a certain number in accordance with each point of the object. That is, we can describe a fuzzy set using its two-dimensional graphical image. Thus, we obtain an approximate view of a surface z = μ(x, y) in three-dimensional space. Let us now draw planes parallel to the plane. We’ll obtain intersections of our surface with these planes at 0 ≤ z ≤ 1. Let's call them as isolines. By projecting these isolines on the OXY plane, we’ll obtain an image of our fuzzy set with an indication of intermediate values μ(x, y) linked to the set’s points coordinates. So we’ll construct generalized Euler — Venn diagrams which are a generalization of well-known Euler — Venn diagrams for ordinary sets. Let's consider representations of operations on fuzzy sets A a n d B. Th e y u s u a l l y t a k e : μA B = min (μA,μB ), μA B = max (μA,μB ), μA = 1 − μA. Algebraic operations on fuzzy sets are defined as follows: μ A B x μ A x μ B x ( ) = ( ) + ( ) − −μ A (x)μ B (x), μ A B x μ A x μ B x ( ) = ( ) ( ), μ A (x) = 1 − μ A (x). Let's construct for a particular problem a generalized Euler — Venn diagram corresponding to it, and solve subtasks graphically, using operations on fuzzy sets, operations of intersection and integrating of the diagram’s bars.
fuzzy set, generalized Euler — Venn diagram, membership function, isoline, kernel, plume
1. Bozhenjuk A.V., Rozenberg I.A., Jastrebinskaja D.N. Nahozhdenie zhivuchesti nechetkih transportnyh setej s primeneniem geoinformacionnyh sistem [Finding the survivability of fuzzy networks with the use of geographic information systems]. Moscow, Nauchnyj mir Publ., 2012. 176 p. (in Russian)
2. Brylkin Ju.V. Racionalizacija algoritma modelirovanija poverhnosti metodom brounovskogo dvizhenija po kriteriju minimizacii kolichestva iteracij [Rationalization of the surface modeling algorithm by the Brownian motion method based on the criterion of minimizing the number of iterations]. Geometrija i grafika [Geometry and graphics]. 2017, V. 5, I. 1, pp. 43-50. (in Russian)
3. Bulgakov S.V. Osnovy geoinformacionnogo modelirovanija [Basics of geoinformation modeling]. Izvestija vysshih uchebnyh zavedenij. Geodezija i ajerofotos’emka [News of higher educational institutions. Surveying and aerial photography]. 2013, I. 3, pp. 77-80. (in Russian)
4. Ivanov V.N. Osnovy razrabotki i vizualizacii ob#ektov analiticheskih poverhnostej i perspektivy ih ispol'zovanija v arhitekture i stroitel'stve [Fundamentals of development and visualization of analytical surface objects and prospects for their use in architecture and construction]. Geometrija i grafika [Geometry and graphics]. 2017, V. 5, I. 4, pp. 3-14. DOI:https://doi.org/10.12737/article_5a17f590be3f51.37534061. (in Russian)
5. Kapralov E.G. Geoinformatika [Geoinformatics]. Moscow: «Akademija» Publ., 2010. - 400 p. (in Russian)
6. Kapralov E.G. Geoinformatika [Geoinformatics]. Moscow: «Akademija» Publ., 2010. - 432 p. (in Russian)
7. Markin L.V. Diskretnye geometricheskie modeli ocenki stepeni zatenennosti geliojenergetike [Discrete geometric models for estimating the degree of shading in solar energy]. Geometrija i grafika. [Geometry and graphics]. 2019, V. 7, I. 1, pp. 28-45. - DOI:https://doi.org/10.12737/article_5c9202d8d821b0.81468033. (in Russian)
8. Mironova Yu.N. Geoinformacionnye sistemy [Geoinformation systems]. Aktual'nye problemy gumanitarnyh i estestvennyh nauk [Actual problems of the humanities and natural sciences]. 2014, Moscow, pp. 63-65. (in Russian)
9. Mironova Yu.N. Matematicheskie aspekty geoinformatiki [Mathematical aspects of Geoinformatics]. Internet-zhurnal «NAUKOVEDENIE» [Internet journal "SCIENCE"]. V. 7, I. 5 (2015). Available at: http://naukovedenie.ru/PDF/93TVN515.pdf. . DOI:https://doi.org/10.15862/93TVN515. (in Russian)
10. Mironova Yu.N. Nekotorye aspekty geoinformatiki [Some aspects of Geoinformatics]. Moscow: «Pero» Publ., 2018. 98 p. (in Russian)
11. Sal'kov N.A. Geometricheskaja sostavljajushhaja tehnicheskih innovacij [Geometric component of technical innovations]. Geometrija i grafika [Geometry and graphics]. 2018, V. 18, I. 2, pp. 85-94. DOI: 10.12737/ article_5b55a5163fa053.0722109. (in Russian)
12. Sal'kov N.A. Formirovanie poverhnostej pri kineticheskom otobrazhenii [The formation of surfaces under kinetic display]. Geometrija i grafika [Geometry and graphics]. 2018, V. 6, I. 1, pp. 20-33. DOI:https://doi.org/10.12737/article_5ad094a0380725.32164760. (in Russian)
13. Sal'kov N.A. Nachertatel'naja geometrija - baza dlja komp'juternoj grafiki [Descriptive geometry-base for computer graphics]. Geometrija i grafika [Geometry and graphics]. 2016, V. 4, I. 2, pp. 37-47. DOI:https://doi.org/10.12737/19832. (in Russian)
14. Sal'kov N.A. Obshhie principy zadanija linejchatyh poverhnostej. Chast' 1 [General principles for setting linear surfaces. Part 1]. Geometrija i grafika [Geometry and graphics]. 2018, V. 6, I. 4, pp. 20-31. DOI:https://doi.org/10.12737/article_5c21f4a06dbb74.56415078. (in Russian)
15. Sal'kov N.A. Obshhie principy zadanija linejchatyh poverhnostej. Chast' 2 [General principles for setting linear surfaces. Part 2]. Geometrija i grafika [Geometry and graphics]. 2019, V. 7, I. 1, pp. 14-27. DOI:https://doi.org/10.12737/article_5c9201eb1c5f06.47425839. (in Russian)
16. Sbornik zadach i uprazhnenij po geoinformatike [Collection of problems and exercises in Geoinformatics]. Moscow: «Akademija» Publ., 2009. - 512 p. (in Russian)
17. Skvorcov A.V., Sarychev D.S. Tehnologija postroenija i analiza topologicheskih struktur dlja geoinformacionnyh sistem i sistem avtomatizirovannogo proektirovanija [Technology of construction and analysis of topological structures for geoinformation systems and computer-aided design systems]. Vestnik Tomskogo gosudarstvennogo universiteta [Bulletin of Tomsk State University]. 2002, I. 275, pp. 60-63. (in Russian)
18. Jahjaeva G.Je. Nechetkie mnozhestva i nejronnye sety [Fuzzy sets and neural networks]. Moscow: Internet-Universitet Informacionnyh tehnologij; BINOM Publ., 2006. 316 p. (in Russian)
19. Mironova Yu.N. Geographic information systems and their classification // International Journal Of Applied And Fundamental Research. - 2016. - I. 1. Available at: www.science-sd.com/463-24961 (accessed 26 April 2016).
20. Mironova Yu.N. Virtual modeling in geoinformation technologies // Materials of conferences (Munich, Germany, 1-6 November 2016):«EDUCATION AND SCIENCE WITHOUT BORDERS», «FUNDAMENTAL AND APPLIED RESEARCH IN NANOTECHNOLOGY». International Journal Of Applied And Fundamental Research. - 2016. - I. 2. Available at: www.science-sd.com/464-24988 (accessed 09 August 2016).
21. Mironova Yu.N. The classification of geoinformation objects // Materials of conferences (Munich, Germany, 1-6 November 2016):«EDUCATION AND SCIENCE WITHOUT BORDERS», «FUNDAMENTAL AND APPLIED RESEARCH IN NANOTECHNOLOGY». International Journal Of Applied And Fundamental Research. - 2016. - I. 2. Available at: www.science-sd.com/464-25151 (accessed 29 September 2016).
22. Mironova Yu.N. The study of geoinformatics with the use of gaming moments // International Journal Of Applied And Fundamental Research. - 2016. - I. 3. Available at: www.science-sd.com/465-25000 (accessed 09 August 2016).
23. Mironova Yu.N. Geographic information systems and confidentiality of information // European Journal Of Natural History. - 2016, I. 5, pp. 48-49. Available at: www.world-science.ru/euro/519-33634 (accessed 01 September 2016).
24. Mironova Y.N. The use of consumers of Internet GIS // Materials of conferences (Munich, Germany, 31 October - 5 November 2017): «EDUCATION AND SCIENCE WITHOUT BORDERS» «FUNDAMENTAL AND APPLIED RESEARCH IN NANOTECHNOLOGY» / International Journal Of Applied And Fundamental Research. - 2017. - I. 3. Available at: www.science-sd.com/471-25218 (accessed 14 July 2017).
25. Mironova Yu.N. Decryption of space images by using GIS-technologies // Materials of conferences (Munich, Germany, 31 October - 5 November 2017): «EDUCATION AND SCIENCE WITHOUT BORDERS» «FUNDAMENTAL AND APPLIED RESEARCH IN NANOTECHNOLOGY» / International Journal Of Applied And Fundamental Research. - 2017. - I. 3. Available at: www.science-sd.com/471-25226 (accessed 31 July 2017).
26. Mironova Yu.N. Geo-information systems applied in competitive orienteering // Theory and Practice of Physical Culture № 3 2018. Available at: http://www.teoriya.ru/ru/node/8139
27. Mironova Y.N. The use of Internet GIS and the problems of information security // International Journal Of Applied And Fundamental Research. - 2018. - I. 1. Available at: www.science-sd.com/473-25383 (accessed 16 February 2018).
28. Mironova Yu.N. Use of fuzzy sets in modeling of GIS objects // International Conference Information Technologies in Business and Industry 2018 / Journal of Physics: Conference Series. 1015 (2018) 032094. Available at: http://iopscience.iop.org/article/10.1088/1742-6596/1015/3/032094. Doi https://doi.org/10.1088/1742-6596/1015/3/032094
29. Mironova Yu.N. Fuzzy information in Geoinformatics // Materials of conferences (Munich, Germany, 31 October - 5 November 2018): «EDUCATION AND SCIENCE WITHOUT BORDERS», «FUNDAMENTAL AND APPLIED RESEARCH IN NANOTECHNOLOGY» / International Journal Of Applied And Fundamental Research. - 2018. - I. 6. Available at: www.science-sd.com/478-25434 (accessed 30 October 2018).
30. Mironova Yu.N. Using fuzzy set theory in Geoinformatics // Materials of conferences (Munich, Germany, 31 October - 5 November 2018): «EDUCATION AND SCIENCE WITHOUT BORDERS», «FUNDAMENTAL AND APPLIED RESEARCH IN NANOTECHNOLOGY» / International Journal Of Applied And Fundamental Research. - 2018. - I. 6. Available at: www.science-sd.com/478-25462 (accessed 30 October 2018).
31. Mironova Yu. N. Notes on Geoinformatics // Materials of the International Conference «Scientific research of the SCO countries: synergy and integration» - Reports in English. Part 1. (May 31, 2019. Beijing, China) - Beijing, 2019 - pp. 92-95.
