TEMPUS S_JEP–12435–97
Quality Assurance System and Credit Transfer System
Course group “Foundations of Mathematics”
June 19, 1998
We agreed in the following:
1. We determined the intersection of the mathematics being taught at our universities; it is
contained in the first 7 topics from the 12 topics below.
2. In the future we will determine what is mutually acceptable in these topics?
3. During preparing slides we will elaborate whole topics .
4. Altogether we have to produce 225 slides (it means 50 slides/month; in the first year
Anna György prepares about 25 slides for graph theory).
5. We agreed in the formal requirements of slides.
6. We try to unify the notations.
Mathematical topics in question (in parenthesis we give the approximate number of slides, that
we do not stick to rigorously):
1. Mathematical logic (10)
2. Set theory (40)
3. Linear algebra (50)
4. Modern algebra (15)
5. Combinatorics (10)
6. Graph Theory (25)
7. Mathematical analysis (75)
8. Probability theory
9. Mathematical statistics
10. Information theory
11. Numerical methods
12. Operation Research
Proposal for formal requirements of slides:
Title: 24 font size, bold
Subtitle: 16 font size, bold
normal text: 14 font size
emphasised text: italic
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Each slide left-lower corner: TEMPUS S_JEP number (14 font size, bold)
Right-lower corner: Mathematics/title of topics (14-es font size, bold)
Detailed proposals for the contents of the topics:
1. Mathematical logic:
 Proposition calculus.
 Identities, logical equivalence.
 Normal forms.
 Proofs.
 Universal and existential quantifiers.
2. Set theory and number theory:
 Description of sets, operations, basic properties, algebra of sets.
 Power sets, Descartes product.
 Relations.
 Functions.
 Building of notion of numbers.
 Natural numbers, induction.
 Integers, rational and irrational numbers, real numbers.
 Cardinality.
 Complex numbers.
 Prime factorisation.
 Congruence.
 Polynomials, algebraic equations.
 Partial fraction decomposition.
3. Linear algebra:
 Finite dimensional vector spaces, basis, dimension, subspaces.
 Matrix algebra, inverse, theory of rank.
 Elementary properties of determinants, expansion theorem, multiplication theorem.
 Condition for solvability of a system of linear equations, description of the solution
set, determining of general solution, Cramer’s rule, method of elimination.
 Linear mappings of finite dimensional vector spaces.
 Properties of operations, their matrix representations.
 Eigenvalue, eigenvector, invariant subspace, Jordan normal form of matrices.
 Euclidean space.
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4. Modern algebra:
 Semigroup, group.
 Ring, field.
 Boole algebra, lattice.
5. Combinatorics:
 Pigeonhole principle.
 Permutations, variations, combinations.
 Binomial coefficients, binomial and polynomial theorem.
 Partition problems.
6. Graph theory:
 Directed and undirected graph.
 Adjacency matrix.
 Simple graph, subgraph, spanned subgraph.
 Path, cycle, Euler cycle, algorithm for Euler cycle, Hamiltonian cycle.
 Connectedness, components.
 Trees, determining spanning tree with minimum weight.
 Planar graph, Kuratowski’s theorem.
 Colouring graphs.
 Bipartite graphs.
7. Mathematical analysis:
 Convergence of sequences, limit of functions, continuity.
 Convergence of series, function series, power series.
 Elementary functions.
 Differentiability of real functions, differentiation rules.
 Mean value theorem of differential calculus, Taylor series.
 Analysis of functions, determining extrema.
 Riemann integral, integration rule of elementary functions.
 Riemann–Stieltjes integral.
 Mean value theorem.
 Primitive functions.
 Notion of measure and its applications, arc length, area, volume and surface.
 Differential equations. Elementary methods for solving first order ordinary differential equations.
 Functions of several variables and vector-valued functions.
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8. Probability theory:
 Algebra of events, probability, probability spaces.
 Conditional probability, theorem of full probability, Bayes formula, independence of
events.
 Random variables, distribution functions, important discrete distributions, expectation, variance, moments.
 Joint distribution and independence of random variables, conditional distribution
and conditional expectation, correlation coefficient.
 Convolution of random variables, distribution of functions of random variables.
 Laws of large numbers, central limit theorems.
9. Mathematical statistics (No detailed proposal)
10. Information theory (No detailed proposal)
11. Numerical methods:
 Approximation of functions, interpolation, method of least squares, uniform approximations.
 Numerical differentiation and integration.
 Solving non-linear equations and system of equations, approximation of roots of
polynomials.
 Approximate solution of system of linear equations, factorisation of matrices, calculation of inverse matrix and determinants, determining eigenvalues and eigenvectors.
 Approximate solution of differential equations.
 Operation research (No detailed proposal)
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