## Babes-Bolyai University of Cluj-Napoca Faculty of Mathematics and Computer Science Study Cycle: Graduate SUBJECT

 Code Subject MML0003 Logic, Set Theory and Arithmetic
 Section Semester Hours: C+S+L Category Type Mathematics 1 2+2+0 fundamental compulsory Mathematics and Computer Science 1 2+2+0 fundamental compulsory Applied Mathematics 1 2+2+0 fundamental compulsory
 Teaching Staff in Charge
 Lect. MODOI Gheorghe Ciprian, Ph.D.,  cmodoimath.ubbcluj.roLect. SZANTO Csaba Lehel, Ph.D.,  szantomath.ubbcluj.roLect. SACAREA Cristian, Ph.D.,  csacareamath.ubbcluj.ro
 Aims Introductory concepts and results on mathematical logic, set theory and arithmetic.
 Content Chapter I. ELEMENTS OF MATHEMATICAL LOGIC. 1. Propositional calculus. 2. The decision problem. 3. Predicate calculus, quantifiers. Chapter II. SETS, RELATIONS, FUNCTIONS. 1. Operations with sets. 2. Binary relations, functions. 3. Injective, surjective, bijective functions. 4. Equivalence relations and factor sets; the kernel of a function. 5. Factorization theorems. 6. Ordered sets, lattices, homomorphisms. 7. Boole algebras. Applications to Logic and Computer Science. Chapter III. CARDINAL NUMBERS. 1. Definition. 2. Direct product and exponentiation of sets and functions. operations with cardinals. 2. Ordering of cardinal numbers. 3. Infinite and finite sets. 5. Elements of combinatorics. Chapter IV. NUMBERS. 1. Introduction to axiomatic set theory. 1. Natural numbers (the Frege-Russell construction and Peano's axioms). 2. The construction of integer, rational, real numbers. Chapter V. ARITHMETIC 1. Divisibility. Greatest comon divisor. 2. The division algorithm. The Euclidean algorithm. 3. Prime numbers. The unique factorization theorem.
 References 1. 1. I. PURDEA, I. POP, Algebra, Editura GIL, Zalau, 2003. 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. S.G. KRANTZ: Logic and Proof Techniques for Computer Science, Birkhauser Boston 2002. 8. A. MARCUS, C. SZANTO, L. TOTH: Logika es halmazelmelet, Sapientia Kiado, Kolozsvar 2004. 9. P. KOMJATH: Halmazelmelet, Egyetemi jegyzet, ELTE Budapest 1999. 10. P. KOMJATH: Matematikai logika, Egyetemi jegyzet, ELTE Budapest 2000. 11. I. PURDEA, C. PELEA, Probleme de algebra, EFES Cluj-Napoca 2005.
 Assessment A test paper throughout the semester representing 25% of the final mark, and an writing exam at the end of the semester (representing 75% of the final mark).
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject