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TECHNICAL SCIENCE
UDC 519.6
Bagirova Leyla Faiq Quliyeva Nazrin Ceyhun Ibrahimova Ilaha Qadir Ganja State University DOI: 10.24412/2520-6990-2023-28187-21-23 ELEMENTS OF SET THEORY AND NEW METHODS OF THEIR STUDY
Abstract:
Set-theoretic operations according to the definition of G. Kantor, the founder of the theory of sets, a set is any collection of certain and distinguishable objects of our intuition or intellect, which we think of as a single whole. There is an affiliation relation between individual objects and sets. If the object x belongs to the set A, then it is written in theform xeA, if it does not belong to the set A, then write x0Á. A pair ofcurly brackets {.... serves to denote the set.}, inside which the elements of the set are listed.
In the curriculum of schoolchildren, many begin to study already in elementary school. This usually happens in the 2nd or 3rd grade. At this stage, children learn to identify the set, distinguish the elements of the set and establish a connection between the sets. It is important that students understand b asic concepts such as empty set and equality of sets.
Keywords: empty sets, subsets, interactive lesson, Problem-oriented learning
Introduction: According to the theory of G. Kantor, the founder of set theory, operations with sets include union, intersection, difference and complement.
Union: Union of two sets A and B (denoted as A U B) consists of all elements that belong to at least one of the sets A and B.
Intersection: The intersection of two sets A and B (denoted as A 0 B) consists of all elements that belong to both set A and set B.
Difference: The difference of two sets A and B (denoted as A \ B) consists of all elements that belong to the set A, but are not the set B.
Complement: The complement of the stream A (denoted as A') consists of all elements that are not part of the set A, but are an exceptional universal set U.
These operations can be combined to create complex multiple operations. For example, you can use intersection and difference to determine the symmetric difference of two sets (A A B), consisting of all elements that belong to only one of the sets A and B.
Set-theoretic operations allow reasoning and proving conclusions in set theory. They are also essential tools in other areas of mathematics such as logic, algebra and analysis.
There are three ways to define a set:
Enumeration of elements: A trace can be specified by explicitly enumerating its elements. For example, the set of natural numbers from 1 to 5 can be set as follows: {1, 2, 3, 4, 5}.
Description path of the characteristic elements: Next, you can specify a description of the characteristic elements. For example, the set of four of all numbers can be described as: {x | x is an even number}.
In the formula where the conditions for the elements are specified: The following can be set using a formula that specifies the conditions that the elements must meet. For example, the set of all numbers that represent simple equations x^2 - 4 = 0 can be set as follows: {xjx^2 - 4 = 0}.
Also, when solving problems on sets, the concept of Inductive magnification, Empty set, subsets is often encountered
Inductive magnification in mathematical science using inductive constructions or inductive rules. This process creates more complex objects or sets from simpler elements or sets. For example, the set of natural numbers can be determined inductively, starting from zero and adding consecutive physical numbers.
Both connections, recursive and inductive, are necessary in mathematics and are widely used in various fields, including many experiments, graphs, experiments and others.
An empty set is special because all its properties and operations develop its emptiness.
Some properties of empty usage:
The power (number of elements) of an empty value is 0.
An empty set is a subset of any number, including itself.
Intersection of an empty volume with all volumes equal to an empty set.
Combining an empty distribution with any number equal to this set.
The presence of an empty distribution in mathematics allows us to formalize some concepts, for example, an empty set is an initial number for constructing other structures based on it (for example, natural numbers or functions). In addition, an empty set is used to define empty sets in other mathematical objects, such as an empty usage or an empty graphical representation.
Thus, let's summarize the study:
• Empty set. An empty set is denoted by {} or 0. This is a subset of any number.
• The set itself. Each set is a subset of itself.
• Proper subset. A subset is called its own if it contains at least one element that is missing from this set.
• Distribution power. Power is the number of elements in a given set. The power of an empty value is 0.
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• Empty s et and power. An empty set is a subset of any value, and its cardinality is 0.
• Inclusion of a subset. If each element of the extension A belongs to the set B, then the set A is a subset of the extension B and means A £ B.
• Direct inclusion of a subset. If the set A is a subs et of the quantity B, and there is an element that does not belong to the set A, then the set A is called a subset of the quantity B and means A c B.
These are just some basic concepts and properties in mathematics.
The main part. The school curriculum usually includes the following types of tasks on set-theoretic operations:
1. Propagation operations: union, intersection, difference, symmetric difference. Examples of tasks: find the union and intersection of two sets, find the difference of two sets, determine the symmetric difference.
2. Finding the number of elements in the quantity. Examples of tasks: how many elements are in a given set, how many elements are in the union of two sets, how many elements are in the intersection of two sets.
3. Venn diagrams. Examples of tasks: build a Venn diagram for two sets, find the number of elements represented in the inclined parts of the diagram
4. Equality and the content of the set. Examples of tasks: checking whether there are two levels of conditions, checking whether one set contains the other set.
5. Definition of definition and images with prevalence. Examples of problems: solve an equation of the form A U B = C, solve an equation of the form A 0 B = C, solve an equation of the form A £ B.
6. Application of set-theoretic operations in other mathematical fields. Examples of problems: use set-theoretic operations to solve problems in combinatorics, probability theory, algebra, etc.
These are just examples of some typical tasks that can be included in the school curriculum on set-theoretic operations. Depending on the level of education and the training plan, the list of tasks may vary.
To date, the following difficulties may occur in solving typical problems in set theory:
1. Understanding concepts and definitions: set, increase of elements, subset, intersection, union, difference, etc. These concepts may be new to the rack and it may take them some time to fully understand their meaning.
2. Formulation of tasks: some tasks may be formulated ambiguously or indistinctly, which makes it difficult to determine the exact answer or method of solution.
3. Choosing the solution method: It may be difficult for students to determine which method or algorithm should be used to solve the problem For example, to solve problems on the intersection or union of sets, you can sometimes use Venn diagrams or an algorithm with highlighted steps.
4. The complexity of working with given data sets: some tasks may require working with an increase in the amount of data that a block can transmit, perform many operations, or use software to automate the process.
5. Errors in calculations and logical errors: when considering typical tasks on a wide range, errors in calculations or logical errors may occur, which may lead to an incorrect answer. This can be done due to attention to detail or incorrect application of the rules for working with volumes.
In order to successfully solve typical distribution tasks, students need to pay attention to the basic concepts, visualize data (for example, a diagram using a Venn diagram), study a selection of correct solutions and accurately perform calculations.
Conclusions: As we know, educational standards for the subject change every year and today a popular math lesson is an interactive and integration lesson. To improve the math lesson during the study, we can offer the following methods:
1. Interactive classes: The teacher uses various interactive methods, such as the use of excluded presentations, quizzes, discussions and group assignments, to engage students in the learning process and stimulate their understanding.
2. Problem-oriented learning: students solve practical problems or problems related to set theory in order to apply their knowledge in practice. This helps students see real-world applications, multiple changes, and critical thinking skills.
2. Computer programs and online courses: The use of computer programs and online courses allows students to study a variety of connections in an interactive form, independently completing tasks and receiving instant feedback. It also allows students to work at their own pace and create materials as needed.
3. Group Projects and Research: Students can conduct group projects or research related to multiple theory in order to study the topic more deeply and gain cooperation and communication skills.
4.Using game methods: Games and simulations can be used to facilitate the study of set theory. For example, students can play math games related to competitions to understand the basic principles and rules.
These new teaching methods allow students to actively participate in the educational process, realize real applied ideas and develop critical thinking and collaboration skills.
List of used literature
1. Б.М.Верников. Множества и отображения. Уральский федеральный университет,Институт математики и компьютерных наук,кафедра алгебры и дискретной математики.М.2017
2. Веретенников Б. М. Дискретная математика : учебное пособие : в 2-х частях : Часть 1 / Б. М. Веретенников, В. И. Белоусова ; [науч. ред. Н. В. Чуксина]. - Екатеринбург : Издательство Уральского университета, 2014. - Ч. 1.
3. В. В. Расин ЛЕКЦИИ ПО АЛГЕБРЕ. Элементы теории множеств. Натуральные и целые числа. Неравенства. Отображения множеств. Числовые функции Учебное пособие Екатеринбург 2011
4. Т.С. Онискевич. МАТЕМАТИКА В РАЗНОУРОВНЕВЫХ ЗАДАНИЯХ.Практикум для студентов-заочников специальности «Начальное образование» Часть 1. Брест 2006
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5. ОСНОВЫ1 НАЧАЛЬНОГО КУРСА МАТЕМАТИКИ.ЭЛЕМЕНТЫ ТЕОРИИ МНОЖЕСТВ.Учебно-методическое пособие для студентов, обучающихся по направлению подго-товки.Педагогическое образование профиля «Начальное образование». Тувинский государственный университет, 2018Кызыл
6. Элементы теории множеств и теории графов Сборник задач и упражнений с решениями по разделу математики "Дискретная математика. 2011
Reference
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UDC 635.21:631.563.9:633.002.6:664.64.016.3
Voitsekhivskyi Volodymyr, Ostrovska Mariia, Krysko Liliia, Nesterova Natalia,
National university of life and environmental sciences of Ukraine, Kiev,
Muliarchuk Oksana,
HEI «Рodillia State University», Kamianets-Podilskyi, Ukraine,
Chukhleb Lydia, Matus Valentina, Rudenko Oleksandr, Ukrainian institute for plant varieties examination, Kiev,
Khomiv Natalia,
SSS «Yevhen Khraplyvyy Zalishchyky Professional College of NUBiP of Ukraine»
FEATURES OF PHYSICAL PROCESSES IN PLANT RAW MATERIALS DURING TRANSPORTATION AND STORAGE
Abstract.
The article presents and systematizes an integrated approach to the influence of specific physical processes occurring in plant raw materials during transportation and storage. The main critical parameters affecting the safety of product marketability are identified.
Key words: plant material, physical characteristics, transport, storage, quality.
Introduction.
On the way from the field to the consumer, plant materials may undergo certain critical changes, in particular during harvesting, post-harvest processing, transportation and storage. These changes are caused by quantitative and qualitative changes in their physical, chemical properties, and sometimes various anatomical and morphological characteristics. Depending on the characteristics of the changes, such processes can be divided into: physical, chemical, biochemical, microbiological, biological and anatomical-morphological [1,2,3,4,5].
The purpose of research. The aim of our work was a generalization and systematization of information on the impact of physical processes to change the quality of plant raw materials for scientific advice on the transportation and storage.
Physical processes - these are processes that cause changes in the properties of plant products under the influence of external factors: climatic and mechanical. These include: sorption processes - sorption and desorption of water, oxygen, volatile substances; thermal processes - cooling, heating, freezing; deformation processes - mechanical damage (punctures, pressure, cuts, etc.), deformation, fight, crushing.
Deformation processes - processes caused by various loads and lead to changes in the internal structure and appearance. Varieties of such processes include crushing battle acquisition unusual shape, crumbs, chopping, pressing, pricking, etc.
Crushing product called destructive dispersed loads from exposure to products which completely lose their characteristic shape, integrity, and internal struc-
