Title:  Mathematical Structures in Computer Science 

Code:  MAT 

Ac.Year:  2007/2008 

Sem:  Winter 

Curriculums:  

Language of Instruction:  Czech 

Credits:  5 

Completion:  examination (written) 

Type of instruction:  Hour/sem  Lectures  Seminar Exercises  Laboratory Exercises  Computer Exercises  Other 

Hours:  39  13  0  0  0 

 Exams  Tests  Exercises  Laboratories  Other 

Points:  80  20  0  0  0 



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

Faculty:  Faculty of Mechanical Engineering BUT 

 Learning objectives: 

  The aim of the subject is to improve the student's knowlende of the basic mathematical structures that are often utilized in different branches of informatics. In addition to the classical algebraic structures there will be discussed also foundations of the mathematical logic, the theory of Banach and Hilbert spaces, and the graph theory. An introduction to the category theory will be dealt with too.  Description: 

  Formal theories, predicate logic, intuitionistic, modal and temporal logics, algebraic structures with one and with two binary operations, universal algebras, topological and metric spaces, Banach and Hilbert spaces, undirected graphs, directed graphs and networks, basic concepts of the category theory.  Learning outcomes and competencies: 

  The students will improve their knowledge of the basic algebraic structures employed in informatics. It will enable them to understand better the theoretical foundations of informatics and to take an active part in the research work in the field.  Syllabus of lectures: 


 Predicates, kvantifiers, terms, formulas, firstorder language anf its interpretation, models of formulas.
 Firstorder predicate calculus and and its properties, completeness and compactness theorems, extensions of the theory.
 Prenex normal forms, skolemization and the Herbrand theorem, foundations of the intuitionistic, modal and temporal logics.
 Universal algebras and their types, subalgebras and homomorphisms, congruences and factoralgebras, products, terms and free algebras.
 Partial and manysorted algebras, hyperalgebras, grupoids, subgroupoids and homomorphisms, cartesian products, factorgroupoids and free groupoids.
 Semigroups and free semigroups, groups, subgroups and homomorphisms, factorgroups and cyclic groups, free and permutation groups.
 Rings, ideals, homomorphisms, integral rings and fields.
 Finite fields, polynomials and divisibility.
 Topological and 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), transportation networks, flows and cuts.
 Categories and concrete categories, subobjects, factorobjects and free objects, products and sums, functors and natural transformations.
 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: 

  Test during the semester and submission of homework.  
