Mathematical Structures in Computer Science

Completion:examination (written)
Type of
Hour/semLecturesSem. ExercisesLab. exercisesComp. exercisesOther
Guarantee:Šlapal Josef, prof. RNDr., CSc., DADM
Lecturer:Šlapal Josef, prof. RNDr., CSc., DADM
Instructor:Hrdina Jaroslav, doc. Mgr., Ph.D., DADM
Šlapal Josef, prof. RNDr., CSc., DADM
Faculty:Faculty of Mechanical Engineering BUT
Department:Department of Algebra and Discrete Mathematics FME BUT
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Learning objectives:
  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.
  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.
Learning outcomes and competences:
  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.
Syllabus of lectures:
  • 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.
Fundamental literature:
  • 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
Study 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
Progress assessment:
  Middle-semester written test.