Course detail

Discrete Mathematics

FIT-IDAAcad. year: 2014/2015

The sets, relations and mappings. Equivalences and partitions. Posets. The structures with one and two operations. Lattices and Boolean algebras.The propositional calculus. The normal forms of formulas. Matrices and determinants. Vector spaces. Systems of linear equations.The elementary notions of the graph theory. Connectedness. Subgraphs and morphisms of graphs. Planarity. Trees and their properties. Simple graph algorithms.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

doc. RNDr. Martin Kovár, Ph.D.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

The students will obtain the basic orientation in discrete mathematics and linear algebra, and an ability of orientation in related mathematical structures.

Prerequisites

Secondary school mathematics.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

Course curriculum

    Syllabus of lectures:
    1. The formal language of mathematics. A set intuitively. Basic set operations. The power set. Cardinality. The set of numbers. Combinatoric properties of sets. The principle of inclusion and exclusion. Proof techniques and their illustrations.
    2. Binary relations and mappings. The composition of a binary relation and mapping. Abstract spaces and their mappings. Real functions and their basic properties. Continuity and discontinuity. The functions defined by recursion.
    3. More advanced properties of binary relations. Reflective, symmetric and transitive closure. Equivalences and partitions. The partially ordered sets and lattices. The Hasse diagrams.
    4. Algebras with one and two operations. Morphisms. Groups and fields. The lattice as a set with two binary operations. Boolean algebras.
    5. The basic properties of Boolean algebras. The duality and the set representation of a finite Boolean algebra.
    6. Predicates, formulas and the semantics of the propositional calculus. Interpretation  and  classification of formulas. The structure of the algebra of non-equivalent formulas. The syntaxis of the propositional calculus. Prenex normal forms of formulas. 
    7. Matrices and matrix operations. Determinant. Inverse and adjoint matrices. Determinant calculation methods.
    8. The vector space. Subspaces. The basis and the dimension. The coordinates of a vector. The transformation of the coordinates and the change of the basis. Linear mappings of vector spaces.
    9. Systems of linear equations. The Gauss and Gauss-Jordan elimination. The Frobenius theorem. The Cramer's Rule.
    10. The inner product. Orthonormal systems of vectors.  The orthogonal projection on a vector subspace. The cross product and the triple product.
    11. The elementary notions of the graph theory. Various representations of a graph.The Shortest path algorithm. The connectivity of graphs.
    12. The subgraphs. The isomorphism and the homeomorphism of graphs. Eulerian and Hamiltonian graphs. Planar and non-planar graphs.
    13. The trees and the spanning trees and their properties. The searching of the binary tree. Selected searching algorithms. Flow in an oriented graph.

    Syllabus of numerical exercises:
  1. Practising and modelling of selected items of lectures.

    2. Syllabus of computer exercises:
    Practising and modelling of selected items of lectures 8, 9 and 10.
    Syllabus - others, projects and individual work of students:
    Five individual home-tasks/works - an instructor will inform.

Work placements

Not applicable.

Aims

The modern conception of the subject yields a fundamental mathematical knowledge which is necessary for a number of related courses. The student will be acquainted with basic facts and knowledge from the set theory, topology and especially the discrete mathematics with focus on the mathematical structures applicable in computer science.

Specification of controlled education, way of implementation and compensation for absences

Pass out the practices.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Anderson I., A First Course in Discrete Mathematics, Springer-Verlag, London 2001. Acharjya D. P., Sreekumar, Fundamental Approach to Discrete Mathematics, New Age International Publishers, New Delhi, 2005. Faure R., Heurgon E., Uspořádání a Booloeovy algebry, Academia, Praha 1984. Gantmacher, F. R., The Theory of Matrices, Chelsea Publ. Comp., New York, 1960. Garnier R.,  Taylor J., Discrete Mathematics for New Technology, Institute of Physics Publishing, Bristol and Philadelphia 2002. Gratzer G., General Lattice Theory, Birkhauser Verlag, Berlin 2003. Grimaldi R. P., Discrete and Combinatorial Mathematics, Pearson Addison Valley, Boston 2004. Grossman P., Discrete mathematics for computing, Palgrave Macmillan, New York 2002. Johnsonbaugh, R., Discrete mathematics, Macmillan Publ. Comp., New York, 1984. Kolář, J., Štěpánková, O., Chytil, M., Logika, algebry a grafy, STNL, Praha 1989. Kolibiar, M. a kol., Algebra a príbuzné disciplíny, Alfa, Bratislava, 1992. Kolman B., Elementary Linear Algebra, Macmillan Publ. Comp., New York 1986. Kolman B., Introductory Linear Algebra, Macmillan Publ. Comp., New York 1993. Kolman B., Busby R. C., Ross S. C., Discrete Mathematical Structures, Pearson Education, Hong-Kong 2001. Klazar M., Kratochvíl J, Loebl M., Matoušek J. Thomas R., Valtr P., Topics in Discrete Mathematics, Springer-Verlag, Berlin 2006. Kučera, L., Kombinatorické algoritmy, SNTL, Praha 1983. Lipschutz, S., Lipson, M.L., Theory and Problems of Discrete Mathematics, McGraw-Hill, New York, 1997. Lovász L., Pelikán J., Vesztergombi, Discrete Mathematics, Springer-Verlag, New York 2003. Mannucci M. A., Yanofsky N. S., Quantum Computing For Computer Scientists, Cambridge University Press, Cambridge 2008. Mathews, K., Elementary Linear Algebra, University of Queensland, AU, 1991. Matoušek J., Nešetřil J., Kapitoly z diskrétní matematiky, Karolinum, Praha 2000. Matoušek J., Nešetřil J., Invitation to Discrete Mathematics, Oxford University Press, Oxford 2008. Nahara M., Ohmi T., Qauntum Computing: From Linear Algebra to Physical Realizations, CRC Press, Boca Raton 2008. O'Donnell, J., Hall C., Page R., Discrete Mathematics Using a Computer, Springer-Verlag, London 2006. Preparata, F.P., Yeh, R.T., Úvod do teórie diskrétnych štruktúr, Alfa, Bratislava, 1982. Rosen, K.H., Discrete Mathematics and its Applications, AT & T Information systems, New York 1988. Rosen, K. H. et al., Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton 2000. Ross, S. M. Topics in Finite and Discrete Mathematics, Cambridge University Press, Cambridge 2000. Sochor, A., Klasická matematická logika, Karolinum, Praha 2001. Švejdar, V., Logika, neúplnost, složitost a nutnost, Academia, Praha 2002. Vickers S., Topology via Logic, Cambridge University Press, Cambridge 1990.

Recommended reading

Demlová M., Nagy J., Algebra, SNTL, Praha 1982. Havel, V., Holenda, J., Lineární algebra, STNL, Praha 1984. Jablonskij, S.V., Úvod do diskrétnej matematiky, Alfa, Bratislava, 1984. Kolář, J., Štěpánková, O., Chytil, M., Logika, algebry a grafy, STNL, Praha 1989. Matoušek J., Nešetřil J., Kapitoly z diskrétní matematiky, Karolinum, Praha 2000. Peregrin J., Logika a logiky, Academia, Praha 2004. Preparata, F.P., Yeh, R.T., Úvod do teórie diskrétnych štruktúr, Alfa, Bratislava, 1982.

Classification of course in study plans

  • Programme IT-BC-3 Bachelor's

    branch BIT , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

  1. The formal language of mathematics. A set intuitively. Basic set operations. The power set. Cardinality. The set of numbers. Combinatoric properties of sets. The principle of inclusion and exclusion. Proof techniques and their illustrations.
  2. Binary relations and mappings. The composition of a binary relation and mapping. Abstract spaces and their mappings. Real functions and their basic properties. Continuity and discontinuity. The functions defined by recursion.
  3. More advanced properties of binary relations. Reflective, symmetric and transitive closure. Equivalences and partitions. The partially ordered sets and lattices. The Hasse diagrams.
  4. Algebras with one and two operations. Morphisms. Groups and fields. The lattice as a set with two binary operations. Boolean algebras.
  5. The basic properties of Boolean algebras. The duality and the set representation of a finite Boolean algebra.
  6. Predicates, formulas and the semantics of the propositional calculus. Interpretation  and  classification of formulas. The structure of the algebra of non-equivalent formulas. The syntaxis of the propositional calculus. Prenex normal forms of formulas. 
  7. Matrices and matrix operations. Determinant. Inverse and adjoint matrices. Determinant calculation methods.
  8. The vector space. Subspaces. The basis and the dimension. The coordinates of a vector. The transformation of the coordinates and the change of the basis. Linear mappings of vector spaces.
  9. Systems of linear equations. The Gauss and Gauss-Jordan elimination. The Frobenius theorem. The Cramer's Rule.
  10. The inner product. Orthonormal systems of vectors.  The orthogonal projection on a vector subspace. The cross product and the triple product.
  11. The elementary notions of the graph theory. Various representations of a graph.The Shortest path algorithm. The connectivity of graphs.
  12. The subgraphs. The isomorphism and the homeomorphism of graphs. Eulerian and Hamiltonian graphs. Planar and non-planar graphs.
  13. The trees and the spanning trees and their properties. The searching of the binary tree. Selected searching algorithms. Flow in an oriented graph.

Fundamentals seminar

12 hod., compulsory

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

  • Practising and modelling of selected items of lectures.

Exercise in computer lab

4 hod., optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.

Syllabus

Practising and modelling of selected items of lectures 8, 9 and 10.

Project

10 hod., optionally

Teacher / Lecturer

doc. RNDr. Martin Kovár, Ph.D.