COURSE AIMS AND OBJECTIVES:
The main aim of this course is to make 'connection' between secondary school and faculty 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. A special care will be taken to equations and inequations. A systematic overview of algebraic equations (especially cubic and quartic algebraic equations), rational and irrational equations, equations with modulus, exponential and logarithmic equations, and trigonometric equations, will be done.
COURSE DESCRIPTION AND SYLLABUS:
1. Introduction. A short overview of history and parts of mathematics. Greek alphabet. Introduction to propositional logic. Propositions. Boolean connectives and complex propositions. Tautologies. Necessary and sufficient condition. The contrapositive. Opposite proposition. Negation of implication.
2. Predicates and quantifiers. Predicates. Universal and existential quantifiers. Negation of quantifiers.
3. Forms of mathematical opinions. Axiomatic construction of mathematical theories. Mathematical notions. Definition of a notion. Theorem and its converse. Fundamental rules of deduction. Basic types of proofs.
4. Sets. The notion of set. Subset. Equality of sets. Universal set. Setting of sets. Power set. Boolean algebras. Partition of set. Cartesian product.
5. Sets of numbers: Natural numbers, integers, rational, real, and complex numbers. Principle of mathematical induction. Binomial formula.
6. Relations. Notion of relation. Partial order. Equivalence relation. Equivalence classes.
Examples of relations (divisibility, congruence relations, some relations in geometry) and their properties.
7. Functions. The notion of function. Domain and range of function. Inverse of set. Graph. Equality of functions. Restriction and extension of function.
8. Injective and surjective function. Bijection. Permutation. Composition of functions. Inverse of function.
9. Equipotent sets. The notion of equipotent sets. Cardinal numbers. Finite and infinite sets.
Countable and uncountable sets. Connections between cardinal numbers and sets operations.
10. Polynomials in one variable. Quadratic function. Theorem on nullpolynomial. Divisibility of polynomials. Horner method. The greatest common divisor of polynomials.
11. Roots of polynomials and algebraic equations (specially, especially cubic and quartic algebraic equations). Trigonometric form of complex number. Moivre formula.
12. 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 formulas.
13. Fractional rational functions. The notion of fractional rational function. Decomposition of fractional rational function into a sum of partial fractions.
14. Polynomials in several variables. Polynomials in two and three variables. Ring of polynomials in two variables. Symmetric polynomials. Fundamental theorem on symmetric polynomials. Symmetric equations. Polynomials in several variables.
