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1. |
Name of the discipline |
INTRODUCTION TO MATHEMATICS |
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2. |
Course of Study |
3, mathematics (economic activity) |
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3. |
Semester of training |
1 |
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4. |
Amount of credits |
2 |
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5. |
Full name of lecturer |
Zabreiko Petr Petrovich |
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6. |
Objectives of studying the discipline |
1) acquaintance of students starting their mathematical education with the language of higher mathematics, terminology and general mathematical constructs underlying various mathematical disciplines; 2) a presentation of the elements of mathematical logic and the basic methods used in the proofs of mathematical statements; 3) demonstration of the axiomatic method of constructing mathematical theories by the example of the axiomatics of natural numbers and Euclidean planimetry; 4) an exposition of the elements of combinatorics used in various disciplines, in particular, in the courses “Mathematical Analysis” and “Algebra and Number Theory” As a result of studying the discipline “Introduction to mathematics” the student must: be able to: – to use the basic objects of the theory of sets and mappings and their properties; – carry out rigorous proofs of mathematical statements using logical operations and laws; – solve the simplest combinatorial problems |
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7. |
Prerequisites |
elementary school mathematics |
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8. |
Contents of the discipline |
Introduction. Features of mathematics as a science. Its content and research methods. Elements of mathematical logic. Mathematical statements. Logical operations: negation, con-junction, disjunction, implication, equivalence. Logical laws. Subjects, predicates and expressive forms. The antinomy of the “naive” set theory. An empty set, a universal set. The beginning of Zermelo-Frenkel axiomatics of set theory. Operations on sets: union, intersection, subtraction. Addition of the set. Cartesian product of sets. Binary relations. Properties of reflexivity, symmetry, anti-symmetry, transitivity of binary relations. Equivalence relation, classes of equivalent elements, quotient set. Mappings. The concept of a function (mapping). Terminology and examples. The notions of family, sequences, equations. Images and preimages of elements and subsets. Composition of mappings (complex function), associativity property of a composition of mappings. Injective, surjective, bijective mappings. Inverse mapping, one-sided inverse mappings. Cartesian product of a family of sets. Binary algebraic operations as mappings. Elements of combinatorics Rules of the sum and product in combinatorics. Combinations and permutations. Binomial theorem. The rule of inclusions and exceptions in combinatorics and its applications. Natural, integer and rational numbers Axiomatics Peano of natural numbers. Definition of addition, multiplication of natural numbers and natural order in the set of natural numbers. Advanced natural series. Number systems. The axiomatics of Euclidean planimetry Primary concepts and primary relations in the Hilbert axiomatics of the Euclidean plane. Axioms of connection and parallelism. Axioms of order and consequence from them. Axioms of congruence of segments and angles. Measurement of segments and angles. Construction of a bijection between the set of points of a line and the set of real numbers. Powers and orders Comparison of sets by their cardinality, the Cantor-Bernstein theorem. Countable sets: examples and basic properties. The sets of cardinality of the continuum. Continuum problem. Ordered sets: partial, linear and complete order |
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9. |
Recommended literature |
1. Kononov SG, Tyshkevich RI, Yanchevsky VI Introduction to mathematics: a tutorial. – Minsk: BSU, 2003 – Parts 1-3. Additional literature 1. R.Volvachev. Elements of mathematical logic and set theory: a teaching aid. – Minsk: Universitetskaya, 1986. – 112 p. 2. Vilenkin N. Ya. Stories about sets. M .: Nauka, 1969. (3rd edition, MCSMNO, 2005) .- 150 p. 3. Kuratowski K., Mostowski A. The theory of sets. M. Mir, 1970.- 416 p. 4. Cohen P. Theory of sets and the continuum hypothesis. M. Mir, 1969. – 347 p. |
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10. |
Teaching Methods |
Lectures, practical exercises, UIRS |
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11. |
Language of education |
Russian |
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12. |
Conditions (requirements), current control |
test papers |
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13. |
Form of certification |
credit |