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# Mathematical Structures in Computer Science

MAT Acad. year 2016/2017 Winter semester 5 credits

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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

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

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.

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