Title:  Mathematical Structures in Computer Science 

Code:  MAT 

Ac.Year:  2017/2018 

Term:  Winter 

Curriculums:  

Language of Instruction:  Czech 

Credits:  5 

Completion:  examination (written) 

Type of instruction:  Hour/sem  Lectures  Sem. Exercises  Lab. exercises  Comp. exercises  Other 

Hours:  39  13  0  0  0 

 Examination  Tests  Exercises  Laboratories  Other 

Points:  80  20  0  0  0 



Guarantor:  Šlapal Josef, prof. RNDr., CSc., DADM 

Lecturer:  Šlapal Josef, prof. RNDr., CSc., DADM 
Instructor:  Duránik Lukáš, DEAN 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 

 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.  Description: 

  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 FloydWarshall'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, SpringerVerlag, 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, SpringerVerlag, 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, SpringerVerlag, Berlin  Heidelberg  New York, 1971, Algebra I. ISBN 0387406247, Algebra II. ISBN 0387406255
 Nerode, A., Shore, R.A.: Logic for Applications, SpringerVerlag, 1993, ISBN 0387941290
 Progress assessment: 

  Middlesemester written test.  
