28357 - Algebra 1

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