## Scientific reports of Bukhara State University

#### Article Title

#### Abstract

Introduction. As you know, the concept of a set is a basic concept in mathematics, and many mathematical problems have been solved using the theory developed around it. Building a set theory apparatus begins with defining the operations that can be performed on sets. Most of us know about operations on sets, such as joining, intersecting, subtracting, symmetric subtraction, and we also have an understanding of the practical problems they can solve. With the development of mathematics, including the science of geometry, the idea of adding other operations to the sets in addition to the above operations arose, and there was a need to enrich the content of set theory and apply them to new practical problems.

Research methods. The Minkowski sum and difference of sets is one such operation, which is used to solve problems in various fields of mathematics, from elementary mathematics, and to enrich the content of set theory. This paper uses set theory and methods of orthogonal projection of vectors.

Results and discussions. This work describes the Minkowski sum and difference of sets and some of their important geometric properties. At the beginning of the article, several methods for calculating the Minkowski sum of polygons in a plane are given. In particular, methods for finding the sum using geometric inequalities, using polygon ends, and vectors corresponding to the sides of a polygon are given. As a basic result, necessary and sufficient condition have created for the existence of the Minkowski difference of the squares given on the plane . Also, the calculation formula and the exact method of finding the Minkowski difference of the squares given by the vectors corresponding to the side on the plane are introduced. At the end of the article, Minkowski difference on sets is applied to linear differential games.

Conclusion. The exact way and formula for finding the Minkowski difference of squares given by the corresponding vectors were created. The basis for the problem of finding the Minkowski difference of cubes in three-dimensional space was laid.

#### First Page

13

#### Last Page

29

#### DOI

10.52297/2181-1466/2021/5/3/2

#### References

1. Minkowski H, Verhandlungen des III internationalen Mathematiker-Kongresses in Heidelberg (Berlin,1904).

2. Pontryagin L.S. 1981, Linear differential games of pursuit, Math. USSR-Sb. 40.3, 285-303.
3. Mamatov M.Sh., On the theory of differential pursuit games in distributed parameter systems, Automatic Control and Computer Sciences. 43.1, (2009), 1-8.

4. Mamatov M.Sh, Tukhtasinov M, Pursuit problem in distributed control systems, Cybernetics and Systems Analysis. 45.2, (2009), 229-239.

5. Mamatov M.Sh, Nuritdinov J.T, Minkovskiy yig‘indisini va ayirmasini hisoblashga doir ba’zi qonuniyatlar haqida, Mat. Inst. Byul. 3, (2020), 49-59.

6. Mamatov M, Nuritdinov J, Some Properties of the Sum and Geometric Differences of Minkowski, Journal of Applied Mathematics and Physics, 8, (2020), 2241-2255.

7. Panina G.Yu., Arifmetika mnogogrannikov, Jurnal “Kvant”, 4, (2009), 8-14.

8. Uxanov M.V., Algoritm postroeniya summi mnogogrannikov, Vestnik YuUrGU, 7, (2001), 39-44.

9. Panyukov A.V. Predstavlenie summi Minkovskogo dlya dvux poliedrov sistemoy lineynix neravenstv, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie., 14, (2012),108-119

#### Recommended Citation

Nuritdinov, Jalolkhon Tursunboy ugli
(2021)
"ABOUT THE MINKOWSKI DIFFERENCE OF SQUARES ON A PLANE,"
*Scientific reports of Bukhara State University*: Vol. 5
:
Iss.
3
, Article 2.

DOI: 10.52297/2181-1466/2021/5/3/2

Available at:
https://uzjournals.edu.uz/buxdu/vol5/iss3/2