Course detail

Discrete Mathematics

FIT-IDMAcad. year: 2022/2023

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. Directed graphs.

Language of instruction

Czech, English

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Dana Hliněná, Ph.D.

Department

Department of Mathematics (UMAT)

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 propositions and their proofs.

Prerequisites

Secondary school mathematics.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Five written tests (max 20 points).

Course curriculum

 

Work placements

Not applicable.

Aims

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 at five written tests for 4 points each and at the final exam for 80 points.
  • If a student can substantiate serious reasons for an absence from an exercise, (s)he can attend the exercise with a different group (please inform the teacher about that). 
  • Passing boundary for ECTS assessment: 50 points.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme BIT Bachelor's 1 year of study, winter semester, compulsory
  • Programme BIT Bachelor's 1 year of study, winter semester, compulsory

  • Programme IT-BC-3 Bachelor's

    branch BIT , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Dana Hliněná, Ph.D.

Syllabus

  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.

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Petr Fuchs, Ph.D.
doc. RNDr. Dana Hliněná, Ph.D.
Mgr. Martin Tůma, Ph.D.

Syllabus

Examples at tutorials are chosen to complement suitably the lectures.