Course detail

# Mathematical Structures in Computer Science

FIT-MATAcad. year: 2018/2019

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 and networks.

Supervisor

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.

Recommended or required reading

- 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, E.: 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.

Course curriculum

- 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, minimal spanning trees (the Kruskal's and Prim's algorithms), vertex and edge colouring.
- Directed graphs, directed Eulerian graphs, networks, the critical path problem (Dijkstra's and Floyd-Warshall's algorithms).
- Networks, flows and cuts in networks, maximal flow and minimal cut problems, circulation in networks.

**Syllabus of lectures:**

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 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

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, minimal spanning trees (the Kruskal's and Prim's algorithms), vertex and edge colouring.

- Directed graphs, directed Eulerian graphs, networks, the critical path problem (Dijkstra's and Floyd-Warshall's algorithms).

- Networks, flows and cuts in networks, maximal flow and minimal cut problems, circulation in networks.