1. komponenta

Lecture type	Total
Lectures	60
Exercises	30

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

COURSE AIMS AND OBJECTIVES:
- Prepare students to use mathematical language in other courses
- Master the basics of mathematical logic and the fundamental methods of reasoning in mathematics
- Reinforce and expand knowledge of sets, relations, and functions
- Cover the fundamentals of analytic geometry in three-dimensional space

COURSE DESCRIPTION AND SYLLABUS:
1. Introduction to Mathematical Logic. Basics of propositional logic. Propositions. Logical connectives and compound propositions. Tautologies. Necessary and sufficient conditions. Converse of a proposition. Contrapositive. Inverse proposition. Negation of implication. Predicates. Universal and existential quantifiers. Negation of quantifiers.
2. Forms of Mathematical Reasoning. Axiomatic construction of mathematical theory. Mathematical concepts. Definition of concepts. Axioms. Theorems and their converses. Basic rules of inference. Basic types of proofs.
3. Sets. The concept of a set. Subsets. Equality of sets. Universal set. Ways of defining sets. Power set. Boolean algebra. Partition of a set. Cartesian product of sets. Basics of number sets: natural, integer, rational, real, and complex numbers. Principle of mathematical induction. Binomial formula.
4. Relations. The concept of a relation. Properties of relations. Equivalence relations. Equivalence classes. Quotient sets. Partial order. Order relations. Examples of relations (divisibility, congruences, certain relations in geometry) and their properties. Divisibility in the set Z. Greatest common divisor.
5. Functions. The concept of a function. Domain, codomain, and image of a function. Preimage. Graph of a function. Equality of functions. Restriction and extension of a function. Injection. Surjection. Bijection. Permutations of a set. Composition of functions. Inverse function.
6. Equipotent Sets. The concept of equipotent sets. Cardinality of a set. Finite and infinite sets. Countable and uncountable sets.
7. Vectors. Vectors as equivalence classes of directed line segments on a line, in the plane, and in space. Modulus, direction, and orientation. Linear operations in the vector spaces V2 and V3. Vector spaces V2 and V3. Collinearity and coplanarity. Linear independence of vectors. Bases for V2 and V3. Scalar product of vectors. Orthogonal projection of a vector onto a line and a plane. Orthonormal basis. Cross product. Mixed product.
8. Analytic Geometry in Space. Cartesian coordinate system in space. Planes in space. Lines in space. Distance from a point to a plane and a line. Angles between two planes, between a line and a plane, and between two lines. Common normal and distance between lines.
9. Second-order Curves.

1. Elementarna matematika 1, B. Pavković, D. Veljan, Školska knjiga, Zagreb, 2004.
2. Elementarna matematika 2, B. Pavković, D. Veljan, Školska knjiga, Zagreb, 1995.
3. Polinomi, B. Pavković, B. Dakić, Školska knjiga, Zagreb, 1991.
4. Uvod u matematiku, S. Kurepa, Tehnička knjiga, Zagreb, 1984.

Mandatory course - Regular study - Mathematics