Course detail

Mathematical Structures in Computer Science

FIT-MATAcad. year: 2020/2021

Formal theories, propositional logic, predicate logic, universal algebra, algebraic structures with one and with two binary operations, topological and metric spaces, Banach and Hilbert spaces, undirected graphs, directed graphs.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Dept. of Algebra and Discrete Mathematics (ÚM OAAG)

Learning outcomes of the course unit

The students will improve their knowledge of the algebraic structures that are most often employed in informatics. These will be mathematical logic, algebra, functional alalysis and graph theory. This will enable them to better understand the theoretical foundations of informatics and also conduct research work in the field.

Prerequisites

Not applicable.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Literature

Birkhoff, G., MacLane, S.: Aplikovaná algebra, Alfa, Bratislava, 1981
Procházka, L.: Algebra, Academia, Praha, 1990
Lang, S.: Undergraduate Algebra, Springer-Verlag, New York - Berlin - Heidelberg, 1990, ISBN 038797279
Polimeni, A.D., Straight, H.J.: Foundations of Discrete Mathematics, Brooks/Cole Publ. Comp., Pacific Grove, 1990, ISBN 053412402X
Shoham, Y.: Reasoning about Change, MIT Press, Cambridge, 1988, ISBN 0262192691
Van der Waerden, B.L.: Algebra I, II, Springer-Verlag, Berlin - Heidelberg - New York, 1971, Algebra I. ISBN 0387406247, Algebra II. ISBN 0387406255
Nerode, A., Shore, R.A.: Logic for Applications, Springer-Verlag, 1993, ISBN 0387941290
Mendelson, M.: Introduction to Mathematical Logic, Chapman Hall, 1997, ISBN 0412808307
Cameron, P.J.: Sets, Logic and Categories, Springer-Verlag, 2000, ISBN 1852330562
Biggs, N.L.: Discrete Mathematics, Oxford Science Publications, 1999, ISBN 0198534272

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Middle-semester written test.

Language of instruction

Czech, English

Work placements

Not applicable.

Aims

The aim of the subject is to improve the students' knowlende of the basic mathematical structures that are often utilized in different branches of informatics. In addition to universal algebra and the classical algebraic structures, foundations will be discussed of the mathematical logic, the theory of Banach and Hilbert spaces, and the theory of both udirected and directed graphs.

Classification of course in study plans

  • Programme MITAI Master's

    specialization NADE , any year of study, winter semester, 5 credits, elective
    specialization NBIO , any year of study, winter semester, 5 credits, elective
    specialization NGRI , any year of study, winter semester, 5 credits, elective
    specialization NNET , any year of study, winter semester, 5 credits, elective
    specialization NVIZ , any year of study, winter semester, 5 credits, elective
    specialization NCPS , any year of study, winter semester, 5 credits, elective
    specialization NSEC , any year of study, winter semester, 5 credits, elective
    specialization NEMB , any year of study, winter semester, 5 credits, elective
    specialization NHPC , any year of study, winter semester, 5 credits, elective
    specialization NISD , any year of study, winter semester, 5 credits, elective
    specialization NIDE , any year of study, winter semester, 5 credits, elective
    specialization NISY , any year of study, winter semester, 5 credits, elective
    specialization NMAL , any year of study, winter semester, 5 credits, elective
    specialization NMAT , any year of study, winter semester, 5 credits, elective
    specialization NSEN , any year of study, winter semester, 5 credits, elective
    specialization NVER , any year of study, winter semester, 5 credits, elective
    specialization NSPE , any year of study, winter semester, 5 credits, elective

  • Programme IT-MGR-2 Master's

    branch MBI , 1. year of study, winter semester, 5 credits, compulsory
    branch MPV , 1. year of study, winter semester, 5 credits, compulsory
    branch MGM , 1. year of study, winter semester, 5 credits, compulsory
    branch MSK , 1. year of study, winter semester, 5 credits, compulsory
    branch MIS , 1. year of study, winter semester, 5 credits, compulsory
    branch MBS , 1. year of study, winter semester, 5 credits, compulsory
    branch MIN , 1. year of study, winter semester, 5 credits, compulsory
    branch MMI , 1. year of study, winter semester, 5 credits, compulsory
    branch MMM , 1. year of study, winter semester, 5 credits, compulsory

Type of course unit

 

Lecture

39 hours, optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

  • Propositional logic, formulas and their truth, formal system of propositional logic, provability, completeness theorem. 
  • Language of predicate logic (predicates, kvantifiers, terms, formulas) and its realization, truth and validity of formulas.
  • Formal system of 1st order predicate logic, correctness, completeness and compactness theorems, prenex  form of formulas.
  • Universal algebras and their basic types: groupoids, semigroups, monoids, groups, rings, integral domains, fields, lattices and Boolean lattices.
  • Basic algebraic methods: subalgebras, homomorphisms and isomorphisms, congruences and direct products of algebras.
  • Congruences on groups and rings, normal subgroups and ideals.
  • Polynomial rings, divisibility in integral domains, Gauss and Eucledian rings.
  • Field theory: minimal fields, extension of fields, finite fields. 
  • Metric spaces, completeness, normed and Banach spaces.
  • Unitar and Hilbert spaces, orthogonality, closed orthonormal systems and Fourier series.
  • Trees and spanning trees, a minimum spanning tree (the Kruskal's and Prim's algorithms), vertex and edge colouring.
  • Eulerian and Hamiltonian graphs, vertex coloring, planarity. 
  • Directed graphs, directed Eulerian graphs, a minimum length path (Dijkstra's and Floyd-Warshall's algorithms).  

Fundamentals seminar

13 hours, compulsory

Teacher / Lecturer

Lukáš Duránik
doc. Mgr. Jaroslav Hrdina, Ph.D.
Mgr. Jan Pavlík, Ph.D.
prof. RNDr. Josef Šlapal, CSc.