1. komponenta

Lecture type	Total
Lectures	60
Exercises	45

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The main aim of this course is to make "connection" between secondary school and university mathematics. The students are introduced to basic of mathematical language and ideas. The sets of numbers, functions and relations are considered at the beginning. The second aim is to repeat polynomials (in one and several variables) and fractional rational functions very carefully. Also, other elementary functions such as exponential, logarithmical, hyperbolic and area hyperbolic functions will be considered.

COURSE DESCRIPTION AND SYLLABUS:
1. Introduction. A short overview of history and parts of mathematics. Greek alphabet.
2. Introduction to propositional logic. Propositions. Logical operations. Tautologies. Necessary and sufficient condition. The contrapositive. Opposite of proposition. Negation of implication.
3. Predicates and quantificators. Predicates. Universal and existential quantificator. Negation of quantificators.
4. Forms of mathematical opinions. Axiomatic construction of mathematical theories. Mathematical concepts. Definition of a concept. Theorem and its converse. Fundamental rules of deduction. Basic types of proofs.
5. Sets. Set-theoretical terminology and symbols. Operations on sets (union, intersection, complement, difference, symmetric difference). Boolean algebra. Partition of a set. Cartesian product.
6. Relations. Notion of a relation. Partial order. Equivalence relation. Equivalence classes and quotient set. Examples of relations (divisibility, congruence relations, some relations in geometry) and their properties.
7. Functions. The concept of a function. Domain and range of a function. Inverse of a set. Graph. Equality of functions. Restriction and extension of a function. Injective, surjective and bijective function. Permutation. Composition of functions. Inverse of a function.
8. Sets of numbers. Natural numbers, integers, rational, real and complex numbers. Principle of mathematical induction. Binomial formula. Trigonometric form of a complex number. Moivre's formula.
9. Equivalence of sets. Concepts of equivalence of sets. Cardinal numbers. Finite and infinite sets. Countable and uncountable sets. Connections between cardinal numbers and set operations.
10. Ring of polynomials in one variable. Quadratic function. The ring of polynomials. Theorem on null-polynomial. Divisibility of polynomials. The Horner method. The greatest common divisor of polynomials. Roots of polynomials and algebraic equations. Fundamental theorem of algebra. Interpolation polynomial. Integer and rational roots of algebraic equations. Complex roots of algebraic equations. Reduciblity and irreducibility of polynomials over the fields C and R. Viete formulae.
11. Polynomials in two or several variables. Ring of polynomials in two variables. Symmetric polynomials. Fundamental theorem on symmetric polynomials. Symmetric equations. Polynomials in several variables.
12. Fractional rational functions and roots. Definition of a fractional rational function and rooting. Decomposition of fractional rational function into a sum of partial fractions. Equations and inequations involving roots.
13. Exponential and logarithmical functions. Powers. Definition, properties and graph of exponential function. Logarithmic function as an inverse function of an exponential function. Properties and graphs of logarithmic functions. Exponential and logarithmic equations and inequations.
14. Hyperbolic and area hyperbolic functions. Definitions, properties and graphs of hyperbolic functions and their inverse functions.

Mandatory course - Regular study - Mathematics and Physics Education