Advanced Mathematics for Informaticians – Applied Algebra and Mathematical Logic

System of linear of equations, Gaussian elimination, vectors, vector spaces, subspaces, basis, matrices, special matrices, LU and PLU decomposition of a matrix, determinants, linear transformations and its properties, computation of the eigenvalues and eigenvectors, euclidean space, orthogonality, diagonalization of a matrix, Jordan canonical form of a matrix, Jordan basis, orthogonal diagonalization, norm of vectors and matrices, singular value decomposition of a matrix, nonnegative matrices, matrix functions, Perron-Frobenius theory. The language of first order logic, an outlook to higher order languages. Formalization. Structure, valuation. The sets of true valuations. Logical consequence and comparing with the operation implication. Deduction theorem, and characterizations of logical consequence. Normal forms: conjuctive, prenex, Skolem. Compactness theorem and its applications. – Proof theory. Deductive and refutation calculi. Analitic tableaux and its semantical background. Completeness theorem and its importance. Examples for semantical and proof theoretical approaches of some logical properties. The model method. Theorems of Löwenheim-Skolem types. Model constructions. Standard and non-standard models, on the concepts on non-standard real numbers, integers, infinitesimals. Categoricity, and completeness. – Discrete and density orderings. On the limits of first order logic, incompleteness and undecidableness, the famous results of Gödel and Church. On the connection of propositonal logic and Boolean algebras.

Mathematics

weekly

42 h / 14 weeks

4

Written examination

5

Budapest University of Technology and Economics

Faculty of Natural Sciences

Institute of Mathematics