Maria João Chaves Marques da Cunha Oliveira

1. Combinatorics
2. Number theory
3. Recurrences
4. Graph theory

At the end of this course students are expected to able to:
Understand the basic notions and techniques of combinatorics;
Understand the basic notions and techniques of elementary number theory
Solve linear recursions;
Understand the basic notions and techniques of graph theory.

Introduction to combinatorics: basic countings problems; the pigeonhole principle, the inclusion-exclusion principle; arrangements, permutations and combinations; the binomial theorem and the Pascal triangle; properties of the binomial coefficients.
Elementary number theory: divisibility, divisibility algorithm, maximum common divisor, Euclides algorithm, minimum common divisor; prime numbers, the fundamental theorem of arithmetic; congruence relation for integer numbers and modular arithmetic; applications to coding and criptography.
Introduction to linear recurrences: Fibonacci numbers; linear recurrence relations.
Graphs: Basic properties of non-direct graphs; connectedness; paths and cycles; Eulerian and Hamiltonian paths; trees; coloring of vertexes of a graph; planar graphs; Euler theorem.

Course materials will be provided online

Complementary Bibliography: 
C. André; F. Ferreira, Matemática Finita, Universidade Aberta, 2000. ISBN: 972-674-305-2.
D. M. Cardoso; J. Szymanski; M. Rostami, Matemática Discreta: Combinatória, Teoria de Grafos e Algoritmos, Escolar Editora, 2008. ISBN: 978-972-592-237-8.
N. L. Biggs, Discrete Mathematics, Oxford University Press, 2nd Ed. 2007.

E-learning.

Continuous assessment is privileged: 2 digital written documents (e-folios) during the semester (40%) and a final digital test, Global e-folio (e-folio G) at the end of the semester (60%). In due time, students can alternatively choose to perform one final exam (100%).

No pre-requisites are required