# 2 semester

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Name of the course

Mathematical Logic

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Year of study, speciality

1,

Mathematics (Science-Pedagogical Activity)

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Semester of study

2

4

Credits

2

5

Lecturer

Ivanov K.A.

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Course objectives

Increase the level of professional competence, the ability to navigate in the study of the foundations of mathematics, the structure of proofs, the logical foundations of programming, the logical design of microprocessor and computer technology.

Reporting to students knowledge about the subject and method of mathematical logic and its meaning for mathematics, mathematical cybernetics, programming and computing.

Improve students’ understanding of the structure of mathematical theories, the nature and structure of mathematical proofs and the concept of the algorithm, the logic of a computer.

As a result of studying the course, a student should be able to:

– to build normal forms of propositional logic;

– to build the prenex normal forms of predicate logic;

– to build conclusions of formulas in the propositional calculus;

– to build conclusions of formulas in the propositional calculus, obtained in the result of the application of deduction theorem;

– simplify contact schemes and build contact schemes corresponding to the functions of the logic statements.

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Prerequisites

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Contents

Propositional logic.

Elementary statements. Logical operations. Complicated statements. Equivilences. Perfect forms. Ligament dependencies.. Applications of the logic of statements.

Propositional calculus.

Propositional calculus: alphabet, language, correctly constructed formulas (c.c.f.), rules of inference, provability and proof. The concept of calculus. Propositional calculus of Church. Deduction theorem. Consequences of the deduction theorem. Derived rules of inference. The relationship between the propositional algebra and the propositional calculus. Completeness, consistency and solvability of the propositional calculus.

Logic and predicate calculus.

Multi-place functions and multi-place predicates. Algebraic systems, algebra, model. Language of calculus and logic of predicates of given signature, alphabet, correctly constructed formulas. Identical true and feasible formulas of predicate logic. Predicate calculus of a given signature, axioms, inference rules, provability in predicate calculus, proof from hypotheses, deduction theorem. Examples of provable formulas, prenex normal form.

The use of propositional logic and predicate logic to logical-mathematical practice.

Direct and inverse theorems, necessary and sufficient conditions, the law of contraposition, the rule of Gauber, a record in the language of logic predicates of various mathematical sentences, the principle of complete disjunction in the predicate form, the method of (full) mathematical induction.

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### RecommendedLiterature

1.  V. I. Igoshin. Mathematical logic and the theory of algorithms. Moscow. АСАDEMA, 2004.

2. . V. I. Igoshin. Tasks and exercises in mathematical logic and the theory of algorithms. . Moscow. АСАDEMA, 2007.

3. . E. Mendelssohn. Introduction to mathematical logic. M.: Nauka, 1971.

4.  Moshchensky V. A. Lectures on mathematical logic. Minsk, 1973.

5.  P. S. Novikov. Elements of mathematical logic. 1973.

6.  L. M. Likhtarnikov, T. G. Sukacheva. Mathematical logic. SPb:Lan’, 1999.

7.  V. A. Uspensky, N. K. Vereshchagin, V. E. Plisko. Introductory course of mathematical logic. Moskau: MCNMO, 2002.

8.  Lavrov I. A., Maksimova L. L. Problems on the set theory, mathematical logic and the theory of algorithms. M.: Fiz-math. lit., 1985.

9.  L. Ershov, E. A. Palyutin. Mathematical logic. M.: Fiz-math. lit., 2011.

10. A. N. Kolmogorov, A. G. Dragalin. Elements of mathematical logic.M.: Izd. Mosk. Univ., 1982.

11. I. A. Lavrov. Mathematical logic. M.: Izd. centr “Academiya”, 2006.

12. A. K. Guts. Mathematical logic and the theory of algorithms. Librokom, 2016

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Teaching methods

Verbal, visual, problem-based, practical, dialog-based and heuristic.

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Language of teaching

Russian

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Conditions (requirements), current control