Mathematical logic, set theory and arithmetics |
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Teaching Staff in Charge |
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Aims |
An introduction to the elements of mathematical logic concerning the logic of sentences and predicates as a completion of the high school knowledge. Presentation of Cantor's theory of sets. Study of binary relations and functions. Considerations on cardinal numbers, finite and countable sets and ordinal numbers. |
Content |
1. Elements of mathematical logic: propositional calculus, logical operators and formulae, identic true and identic false formulae, logical implication and logical equivalence, decision problem; predicate calculus, quantifiers, theorems.
2. Sets, relations, functions: set algebra, binary relations, equivalence relations and partitions, functions, one-to-one correspondence, the kernel of a function, factorization theorems, ordered sets, lattices, homomorphisms, direct product and exponentiation of sets and functions. 3. Cardinal numbers: definition, operations with cardinal numbers, ordering of cardinal numbers, countable and noncountable sets, infinite and finite sets, the set of natural numbers (the Frege-Russell construction and Peano's axiomatic study). |
References |
1. M. BECHEANU s.a.: Algebra pentru perfectionarea profesorilor, Ed. Didactica si Pedagogica, Bucuresti 1983.
2. I.T. ADAMSON: A Set Theory Workbook, Birkhauser, Boston, 1998. 3. S. BILANIUK: A Problem Course in Mathematical Logic, Trent University, Ontario 2003 4. G. GRATZER: General Lattice Theory, Birkhauser, Boston 1998. 5. P.R. HALMOS: Naive Set Theory, D. Van Nostrand Company Inc. Princeton 1967. 6. C. NASTASESCU: Introducere in teoria multimilor, Ed. Didactica si Pedagogica, Bucuresti 1981. 7. Y.I. MANIN: A Course in Mathematical Logic, Springer-Verlag, New York 1977 8. S.G. KRANTZ: Logic and Proof Techniques for Computer Science, Birkhauser Boston 2002. 9. I.A. LAVROV, L.L. MAKSIMOVA: Probleme de teoria multimilor si logica matematica, Ed. Tehnica, Bucuresti 1974. 10. A. MARCUS, C. SZANTO, L. TOTH: Logika es halmazelmelet, Sapientia Kiado, Kolozsvar 2004. 11. A. HAJNAL, L. CSIRMAZ: Matematikai logika, Egyetemi jegyzet, ELTE, Budapest 1994. 12. A. HAJNAL, P. HAMBURGER: Halmazelmelet, Tankonyvkiado, Budapest 1994. 13. P. KOMJATH: Halmazelmelet, Egyetemi jegyzet, ELTE Budapest 1999. 14. P. KOMJATH: Matematikai logika, Egyetemi jegyzet, ELTE Budapest 2000. |
Assessment |
Two control papers throughout the semester (each representing 20% of the final mark) and oral exam at the end of the semester (representing 60% of the final mark). |