Discrete Mathematics

• 26 hrs lectures
• 26 hrs exercises

• 80 pts final exam
• 20 pts numeric exercises

This course provides basic knowledge of mathematics necessary for a number of following courses. The students will learn elementary knowledge of algebra and discrete mathematics with an emphasis on mathematical structures that are needed for later applications in computer science. The students will acquire basic knowledge of discrete mathematics  and the ability to understand the logical structure of a mathematical text. They will be able to explain mathematical structures and to formulate their own mathematical propositions and their proofs.

Secondary school mathematics.

1. The formal language of mathematics. Basic formalisms - statements, proofs, propositional and predicate logic.
2. Intuitive set concepts. Basic set operations. Cardinality. Sets of numbers. The principle of inclusion and exclusion.
3. Proof techniques.
4. Binary relations, their properties and composition.
5. Reflective, symmetric, and transitive closure. Equivalences and partitions.
6. Partially ordered sets, lattices. Hasse diagrams. Mappings.
7. Basic concepts of graph theory. Graph Isomorphism, trees, trails, tours, and Eulerian graphs.
8. Finding the shortest path, Dijkstra's algorithm. Minimum spanning tree problem. Kruskal's and Jarnik's algorithms. Planar graphs.
9. Directed graphs.
10. Binary operations and their properties.
11. Algebras with one operation, groups.
12. Congruences and morphisms.
13. Algebras with two operations, lattices as algebras. Boolean algebras.

Problems discussed at numerical classes are chosen so as to complement suitably the lectures.

Written tests during the semester (maximum 20 points). Classes are compulsory. Presence at lectures will not be controlled, absence at numerical classes has to be excused.