Course detail

Discrete Mathematics

FIT-IDAAcad. year: 2018/2019

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

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.

Recommended optional programme components

Not applicable.

Recommended or required 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.

  • 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.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Pass out the practices in the prescribed range.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. The formal language of mathematics. A set intuitively. Basic set operations. The power set. Cardinality. The sets 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 binary relations and mappings. Reflective, symmetric and transitive closure. Equivalences and partitions. The partially ordered sets and lattices. The Hasse diagrams.
3. General algebras and algebras with one and two operations. Lattices as algebras with two operations. Boolean algebras.
4. Propositional logic, its syntaxis and semantics. The formal system of the propositional calculus. The completeness of propositional logic.
5. Predicate logic, its syntaxis and semantics. The formal system of the first-order predicate logic. The problem of completeness in predicate logic.
6. Demonstration of usage and utility of proposional and predicate logic in proofs.
7. Matrices and matrix operations. Systems of linear equations. Gaussian elimination. Determinant. Inverse and adjoint matrices. The Cramer's Rule.
8. The vector space and its subspaces. The basis and the dimension. The coordinates of a vector relative to a given basis. The transformation of the coordinates and the change of the basis. The sum and intersection of vector spaces. Linear mappings of vector spaces.
9. The inner product. Orthonormal systems of vectors. Orthogonal projection and approximation. The eigenvalues and eigenvectors. The orthogonal projections onto eigenspaces.
10. Quadratic forms and conic sections.
11. The elementary notions of the graph theory. Various representations of a graph.The Dijkstra 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.

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.

Classification of course in study plans

  • Programme IT-BC-3 Bachelor's

    branch BIT , 1. year of study, winter semester, 7 credits, compulsory

Type of course unit

 

Lecture

52 hours, optionally

Teacher / Lecturer

Syllabus


  1. The formal language of mathematics. A set intuitively. Basic set operations. The power set. Cardinality. The sets 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 binary relations and mappings. Reflective, symmetric and transitive closure. Equivalences and partitions. The partially ordered sets and lattices. The Hasse diagrams.
  3. General algebras and algebras with one and two operations. Lattices as algebras with two operations. Boolean algebras.
  4. Propositional logic, its syntaxis and semantics. The formal system of the proposional calculus. The completeness of propositional logic.
  5. Predicate logic, its syntaxis and semantics. The formal system of the first-order predicate logic. The problem of completeness in predicate logic.
  6. Demonstration of usage and utility of propositional and predicate logic in proofs.
  7. Matrices and matrix operations. Systems of linear equations. Gaussian elimination. Determinant. Inverse and adjoint matrices. The Cramer's Rule.
  8. The vector space and its subspaces. The basis and the dimension. The coordinates of a vector relative to a given basis. The transformation of the coordinates and the change of the basis. The sum and intersection of vector spaces. Linear mappings of vector spaces.
  9. The inner product. Orthonormal systems of vectors. Orthogonal projection and approximation. The eigenvalues and eigenvectors. The orthogonal projections onto eigenspaces.
  10. Quadratic forms and conic sections.
  11. The elementary notions of the graph theory. Various representations of a graph.The Dijkstra 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.

seminars in computer labs

20 hours, compulsory

Teacher / Lecturer

Syllabus


  • Practising and modelling of selected items of lectures.

Projects

6 hours, compulsory

Teacher / Lecturer

Syllabus

Three individual, structured home-tasks (works) - an instructor will inform.

eLearning