FIT-IDMAcad. year: 2019/2020
Sets, relations and mappings. Equivalences and partitions. Posets.
Structures with one and two operations. Lattices and Boolean algebras.
Propositional and predicate calculus. Elementary notions of graph
theory. Connectedness. Subgraphs and morphisms of graphs. Planarity.
Trees and their properties. Basic graph algorithms. Network flows.
Learning outcomes of the course unit
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 claims and their proofs.
Secondary school mathematics.
Recommended optional programme components
Recommended or required reading
Hliněný, P., Úvod do informatiky. Elportál, Brno, 2010. (CS)
Kovár, M., Diskrétní matematika, FEKT VUT, Brno, 2013 (CS)
Anderson I., A First Course in Discrete Mathematics, Springer-Verlag, London 2001.
Grimaldi R. P., Discrete and Combinatorial Mathematics, Pearson Addison Valley, Boston 2004.
Grossman P., Discrete mathematics for computing, Palgrave Macmillan, New York 2002.
Kolibiar, M. a kol., Algebra a príbuzné disciplíny, Alfa, Bratislava, 1992. (CS)
Kolman B., Busby R. C., Ross S. C., Discrete Mathematical Structures, Pearson Education, Hong-Kong 2001.
Klazar M., Kratochvíl J, Loebl M., Matoušek J. Thomas R., Valtr P., Topics in Discrete Mathematics, Springer-Verlag, Berlin 2006.
Matoušek J., Nešetřil J., Kapitoly z diskrétní matematiky, Karolinum, Praha 2007.
Matoušek J., Nešetřil J., Invitation to Discrete Mathematics, Oxford University Press, Oxford 2008.
O'Donnell, J., Hall C., Page R., Discrete Mathematics Using a Computer, Springer-Verlag, London 2006.
Sochor, A., Klasická matematická logika, Karolinum, Praha 2001. (CS)
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
- Evaluation of the two home assignments solved in the groups (max 10 points).
- Evaluation of the two mid-term exams (max 30 points).
The minimal total score of 10 points gained out of the mid-term exams. Plagiarism and not allowed cooperation will cause that involved students are not classified and disciplinary action may be initiated.
Language of instruction
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.
Specification of controlled education, way of implementation and compensation for absences
- The knowledge of students is tested at exercises; including two homework assignments worth for 5 points each, at two midterm exams for 15 points each, and at the final exam for 60 points.
- If a student can substantiate serious reasons for an absence from an exercise, (s)he can either attend the exercise with a different group (please inform the teacher about that).
- Passing boundary for ECTS assessment: 50 points.
Type of course unit
26 hours, optionally
Teacher / Lecturer
- The formal language of mathematics. A set intuitively. Basic set operations. Power set. Cardinality. Sets of numbers. The principle of inclusion and exclusion.
- Binary relations and mappings. The composition of binary relations and mappings.
- Reflective, symmetric, and transitive closure. Equivalences and partitions.
- Partially ordered sets and lattices. Hasse diagrams. Mappings.
- Binary operations and their properties.
- General algebras and algebras with one operation. Groups as algebras with one operation. Congruences and morphisms.
- General algebras and algebras with two operations. Lattices as algebras with two operations. Boolean algebras.
- Propositional logic. Syntax and Semantics. Satisfiability and
validity. Logical equivalence and logical consequence. Ekvivalent
formulae. Normal forms.
- Predicate logic. The language of first-order predicate logic. Syntax,
terms, and formulae, free and bound variables. Interpretation.
- Predicate logic. Semantics, truth definition. Logical validity,
logical consequence. Theories. Equivalent formulae. Normal forms.
- A formal system of logic. Hilbert-style axiomatic system for
propositional and predicate logic. Provability, decidability,
- Basic concepts of graph theory. Graph Isomorphism. Trees and their properties. Trails, tours, and Eulerian graphs.
- Finding the shortest path. Dijkstra's algorithm. Minimum spanning tree problem. Kruskal's and Jarnik's algorithms. Planar graphs.
26 hours, compulsory
Teacher / Lecturer
Examples at tutorials are chosen to suitably complement the lectures.