Title:

# Discrete Mathematics

Code:IDM
Ac.Year:2002/2003
Sem:Winter
Curriculums:
ProgrammeFieldYearDuty
IT-BC-3BIT1stCompulsory
Language of Instruction:Czech
Credits:6
Completion:examination (written)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:39130130
ExamsTestsExercisesLaboratoriesOther
Points:8000200
Guarantor:Kovár Martin, RNDr., Ph.D. (DMATH)
Lecturer:Havel Václav, prof. RNDr., DrSc. (DMAT)
Kovár Martin, RNDr., Ph.D. (DMATH)
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEEC BUT
Follow-ups:
 Algorithms (IAL), DIFS Computer Graphics Principles (IZG), DCGM Formal Languages and Compilers (IFJ), DIFS Mathematical Analysis (IMA), DMAT Modelling and Simulation (IMS), DITS Numerical Methods and Probability (INM), DMAT

Learning objectives:
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.
Description:
The sets, relations and mappings. The topology and the continuous mapping. The structures with one and two operations. Equivalences and partitions. Posets. Lattices and Boolean algebras.The proposional calculus. The normal forms of formulas. Deduction. The proving techniques. The elementary notions of the graph theory. Connectedness. Subgraphs and morphisms of graphs. Planarity. Trees and their properties. Simple graph algorithms.
Learning outcomes and competencies:
The student will obtain the basic orientation in mathematics as well as the foundations of the discrete mathematics, logic and related mathematical structures: elementary topology, binary relations, posets, lattices, groups, graphs etc.
Syllabus of lectures:

• A set intuitively. Basic set operations. The power set. The set of numbers. Binary relations. A mapping as a binary relation. Domain and co-domain. Functions and sequences. The composition of relations.
• Injective, surjective and bijective mappings. The inverse mapping. The image and the inverse image. Important collections of sets with applications. Topological definition of continuity.
• Operations on a set. Classification of the structures with one and two operations. The group of permutations of a finite set. Cominatorial properties of finite sets. The Principle of inclusion and exclusion.
• Reflective, symetric, antisymetric and transitive binary relations. Reflective, symetric and transitive closure. Equivalences and partitions with examples.
• The partially ordered sets. Lattices and their basic properties. Khalimsky's digital line and its order of specialization. The natural order of the real numbers. The Hasse diagrams. The lattice as a set with two binary operations. The Boolean algebra.
• The basic properties of Boolean algebras. The duality and the set representation of a finite Boolean algebra.
• Predicates, formulas, quantifiers and basic logical connectives. The proposional calculus and its syntaxis. The classification of formulas. Some subclasses of the proposional calculus.
• The nterpretation of formulas. Tautologies,non-performable formulas and the logic equivalence of formulas. The structure of the algebra of non-equivalent formulas.
• Prenex normal forms of formulas. The truthfulness and determinism.
• Deduction systems. The system of the natural deduction and its rules. The proof in the system of natural deduction. The techniques of proofs.
• The elementary notions of the graph theory. The Shortest path algorithm. The connectivity of graphs. The subgraphs.
• The isomorphism and the homeomorphism of graphs. The Planarity problém.
• The trees and the spanning trees and their properties. The searching of the binary tree. Selected searching algorithms.
Syllabus of computer exercises:

• Practising and modeling of the properties of sets, set operations, relations, mappings and compositions.
• Practising and modeling of the properties of injective, surjective and bijective mappings, inverse mappings, simple topologies and topological continuity.
• Practising and modeling of the properties of the structures with one and two operations, combinatorial properties of finite sets and the Principle of inclusion and exclusion.
• Investigating the reflectivity, the symetry, the antisymetry and the transitivity of binary relations. The construction of the corresponding closures, equivalences and partitions. Investigating the properties of some simple groups.
• Examples of various posets. Khalimsky digital line and the real line. Hasse diagrams. Boolean algebras.
• Practising and modeling of the properties of the finite Boolean algebras. Their duality and their set reprezentation.
• Practising of the basics of the proposional calculus.
• Investigating and practising of the properties of the formulas.
• Constructing of the prenex normal forms of the formulas.
• Practising of the proof techniques.
• The construction and usage of the Shortest path algorithm. The investigating of the connectedness of graphs.
• The investigating of the planarity of graphs. The investigating and modeling of trees and spanning trees of graphs.
• The minimal spanning tree algortithms. The searching of the binary tree. Sorting.
Fundamental literature:

• Hellerstein N.S., Diamond, A Paradox Logic, World Scientific, Singapore, 1996.
• Johnsonbaugh R., Discrete mathematics, Macmillan Publ. Comp., New York, 1984.
• Jablonskij S. V., Úvod do diskrétnej matematiky, Alfa,
• Kolář J., Štěpánková, O., Chytil, M., Logika, algebry a grafy, STNL, Praha 1989.
• Kolibiar M. a kol., Algebra a príbuzné disciplíny, Alfa, Bratislava, 1992.
• Kučera L., Kombinatorické algoritmy, SNTL, Praha 1983.
• Lipschutz S., Lipson M. L., 2000 Solved Problems in Discrete Mathematics, McGraw-Hill, New York, 1992.
• Manna Z., Matematická teorie programů, SNTL, Praha 1981.
• Novák V., Fuzzy množiny a jejich aplikace, SNTL, Praha 1986.
• 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.
• Štěpán J., Diskrétní matematika, UP, Olomouc, 1990 (skriptum).
• Vickers S., Topology Via Logic, Cambridge University Press, Cambridge, 1989.
Study literature:

• 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.
• Lipschutz S., Lipson M. L., 2000 Solved Problems in Discrete Mathematics, McGraw-Hill, New York, 1992.
• Manna Z., Matematická teorie programů, SNTL, Praha 1981.
• 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.
• Štěpán J., Diskrétní matematika, UP, Olomouc, 1990 (skriptum).
Controlled instruction:
Pass out the practices in the determined range.
Progress assessment:
The active attendance of the computer practices.
Exam prerequisites:
A class accreditation is not defined.