1st module

A.A. 1998/99

Dott. Domenico Luminati

1. The integers

• induction
• division, greatest common divisor and least common multiple, unique factorization
• b-adic expression of an integer
• congruence modulo n and congruence classes
• recurrence relations

• counting the number of elements of some simple sets (cartesian product, subsets, maps, k-sub-sets,injective maps, permutations)
• the principle of inclusion and exclusion (surjective maps)

• composition of permutations
• decomposition as product of disjoint cycles
• decomposition as product of transpositions, sign of a permutation

• partially ordered sets and lattices
• semigroups, monoids, groups, rings
• quotients, morphisms, fundamental theorem of homomorphism
• actions polynimials, remainder theorem, multyplicity of a root
• more recurrence relations
• basic notion in universal algebra: a general setting for the studied structures

• trees and foresta, spanning trees
• weighted graphs
• eulerian and hamiltonian graphs
• directes graphs and networks

1. P. J. Cameron, Combinatorics: Topics, tecniques, algorithms, Cambridge University Press
2. A. Facchini, Algebra  per informatica, Decibel-Zanichelli
3. I. N. Herstein, Algebra, Editori Riuniti
4. N. Jacobson, Basic Algebra I, W. H. Freeman and Company
5. B. Scimemi, Algebretta, Decibel ed.