- Docente: Marilena Barnabei
- Credits: 8
- SSD: MAT/02
- Language: Italian
- Moduli: Marilena Barnabei (Modulo 1) Marta Morigi (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 8010)
Course contents
Set theory: inclusion, complement, union, intersection. Cartesian product.
Relations. Functions. Injective, surjective, and bijective functions. Inverse of a bijective function. Composition.
Discrete sets. The number of functions between two finite sets. The number of injective functions between two finite sets. Falling factorial and factorial. Permutations. Binomial coefficients: definition and properties. Fibonacci numbers. Inclusion-exclusion principle.
Natural numbers: Peano axioms, the induction principle. Finite and countable sets. Integers. The division lemma. The prime number decomposition of an integer. Greatest common divisor. The Euclidean algorithm.
Partitions of a set. Equivalence relations, quotient set, the equivalence relation associated to a map, canonical factorization of a map.
Order relations. Partially ordered sets and lattices (outlines).
Congruences mod n and related properties.
Permutations: cycle decomposition, sign of a permutation.
The notion of group, subgroup and group morphism. The group structure of Z_n.
The symmetric group. Cyclic groups and their subgroups. Quotient groups as a group modulo an equivalence relation which respects the product, the fundamental theorem of homomorphism for groups.
Readings/Bibliography
M. Barnabei – F. Bonetti: Matematica Discreta Elementare. Pitagora, Bologna, 1994
M.Artin: Algebra. Bollati Boringhieri 1997.
I.N. Herstein: Algebra. Editori riuniti, 2010.
Teaching methods
Lectures and exercise sessions
Assessment methods
Written and oral exam
Office hours
See the website of Marilena Barnabei
See the website of Marta Morigi